INDUCTION, BOUNDING, WEAK COMBINATORIAL PRINCIPLES,

AND THE HOMOGENEOUS MODEL THEOREM

DENIS R. HIRSCHFELDT, KAREN LANGE, AND RICHARD A. SHORE

Abstract. Goncharov and Peretyat’kin independently gave necessary and sufficient conditions for when a set of types of a complete theory T is the type spectrum of some homogeneous model of T. Their result can be stated as a principle of second order arithmetic, which we call the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. We show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense. We do the same for an analogous result of Peretyat’kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.

Along the way, we analyze a number of related principles. Some of these turn out to fall into well-known reverse mathematical classes, such as ACA_{0}, IΣ_{2}^{0}, and BΣ_{2}^{0}. Others, however, exhibit complex interactions with first order induction and bounding principles. In particular, we isolate several principles that are provable from IΣ_{2}^{0}, are (more than) arithmetically conservative over RCA_{0}, and imply IΣ_{2}^{0} over BΣ_{2}^{0}. In an attempt to capture the combinatorics of this class of principles, we introduce the principle Π_{1}^{0}GA, as well as its generalization Π_{n}^{0}GA, which is conservative over RCA_{0} and equivalent to IΣ_{n+1}^{0} over BΣ_{n+1}^{0}.

Date: July 19, 2015. 2010 Mathematics Subject Classification. Primary 03B30; Secondary 03C07,

03C15, 03C50, 03C57, 03D45, 03F30, 03F35.

Hirschfeldt was partially supported by the National Science Foundation of the United States, grants DMS-0801033 and DMS-1101458. Lange was partially supported by NSF Grants DMS-0802961 and DMS-1100604. Shore was partially supported by NSF Grants DMS-0554855, DMS-0852811, and DMS-1161175, John Templeton Foundation Grant 13408, and a short term visiting position at the University of Chicago as part of the Mathematics Department’s visitor program.

Contents

1. Introduction 2 1.1. The thematic level 3 1.2. The specifics 8 1.3. An outline of the paper 18 2. Definitions 19 2.1. Reverse mathematics 20 2.2. Model theoretic notions 22 2.3. Other notions 25 3. The Atomic Model Theorem and related principles 26 4. Defining homogeneity 32 5. Closure conditions and model existence 48 5.1. Type spectra of homogeneous models 48 5.2. Type spectra of general models 50 5.3. Comparing closure conditions 52 5.4. Spectrum enumeration existence theorems 54 5.5. Model existence theorems 58 6. Extension functions and model existence 59 6.1. Extension functions 59 6.2. The reverse mathematics of extension functions 66 7. The reverse mathematics of model existence theorems 90 7.1. Extension function approximations, AMT, and ATT 91 7.2. Comparing model existence theorems 101 7.3. Computability theoretic equivalences 102 8. Open questions 104 Appendix A. Approximating generics 105 Appendix B. Atomic trees 109 Appendix C. Saturated models 111 References 112

1. Introduction

This paper began as an investigation into the difficulty (in terms of the axioms needed) of proving certain theorems of classical model theory about the existence of models of given theories with specific properties (in particular homogeneity). We were also motivated by, and interested in illuminating, the relations of these proof theoretic analyses with ones of the computational complexity of constructing these models. In analyzing these questions we were led into several byways: proof theoretic, computational, and combinatorial. The paper has thus become a case study in reverse mathematics and its relations to effective or computable mathematics.

We begin our introduction with a thematic explanation of the format and goals of reverse mathematics as well as its connections in general to computable mathematics. We then describe the nature of our results and the places that they occupy within reverse mathematics on the same thematic level. After discussing some previous related results (both reverse mathematical and computability theoretic), we give a more specific, even formal, description of the questions considered and the results provided. Finally, we provide an outline of the full paper in Section 1.3.

1.1. The thematic level. The general goal of reverse mathematics is to calibrate the complexity of mathematical theorems by determining precisely the systems of axioms needed to prove the given theorems. More specifically, the subject deals with what might be called countable mathematics in that usually it restricts its attention to theorems about countable structures (algebraic, logical, or combinatorial) or ones in the parts of analysis and topology (such as separable metric spaces) that can be approximated by countable sets. (For some approaches to the uncountable see [25] and [45].)

One direction of this calibration is standard. We know what it means for a theorem to be provable from a set of axioms. The other direction provides both the preciseness of the answers and the name of the subject. We reverse the usual situation by proving the axioms from the theorem. Formally, this means starting with some common weak base set of axioms B and an axiom system S extending B that proves the theorem of interest, Φ. Then one establishes a reversal by showing that B ∪ {Φ} proves S. Thus, relative to the weak base theory B, the set of axioms S proving Φ and Φ itself are logically equivalent. So the proof theoretic, or better reverse mathematical, strength of Φ is determined to be precisely that of S.

Given that our subjects of interest are countable, the usual and natural setting for reverse mathematics is second order arithmetic (rather than say set theory). The language here is that of ordinary first order arithmetic (+, ×, ⩽, 0, and 1) with first order logic (including the usual variables x, y, z, … ranging over “the numbers”) supplemented by a collection of distinct variables X, Y, Z … (and perhaps constant symbols A, B, …) ranging over (or denoting specific) “sets of numbers” along with a binary relation symbol x ∈ X representing membership of numbers x in sets X. The axioms for a system of second order arithmetic begin with the standard basic ones for arithmetic with +, ×, ⩽, 0, and 1 and an induction principle saying that every set containing 0 and closed under successor contains all the numbers (or, equivalently, that every nonempty set has a least element). It may then be augmented by additional comprehension axioms asserting that sets with some properties (e.g. definable by formulas in some given class) exist. One may also add axioms asserting induction principles of the form that if 0 has some property (e.g. as specified by a formula of some class) and the numbers with this property are closed under successor then every number has the specified property (or the corresponding least number principle as above). We give more precise definitions and examples in Section 2.1.

There is a close connection between reverse and effective mathematics and between the standard axiomatic systems of reverse mathematics and the calibration schemes from computability theory for the complexity of sets and functions. To make this correspondence clear (and for many other purposes), we need to specify the semantics for axiom systems for second order arithmetic. A structure for this language is one of the form M = ⟨M, S, +, ×, ⩽, 0, 1, ∈⟩ where M is a set (the set of the “numbers” of M) over which the first order quantifiers and variables of our language range; S ⊆ 2^{M} is the collection of subsets of the “numbers” in M over which the second order quantifiers and variables of our language range; as usual, + and × are binary functions on M; and ⩽ is a binary relation on M while 0 and 1 are members of M. We always interpret ∈ as the usual membership relation between elements of M and elements of S.

The standard weak base system for reverse mathematics, RCA_{0}, contains, in addition to the usual basic axioms for arithmetic: IΣ_{1}^{0}, induction for Σ_{1}^{0} formulas (i.e., ones with one (or a block of) existential quantifier(s) followed by a quantifier free matrix); and Δ_{1}^{0}-CA_{0}, comprehension for sets determined by equivalent Σ_{1}^{0} and Π_{1}^{0} formulas (the latter being ones with one universal quantifier or block of universal quantifiers). This system corresponds to computable (or recursive) mathematics in the sense that the models of this theory whose numbers M are just the usual natural numbers ℕ are precisely the ones whose sets are closed under join (i.e., effective union) and Turing reducibility; i.e., if A, B ∈ M then any set computable from the pair A ⊕ B is also in M. (The structures M with M = ℕ are called ω-models.) Thus the theorems of RCA_{0} are essentially theorems of computable mathematics and the converse usually holds as well except at times when the classical computability theory style proofs rely on more than Σ_{1}^{0} induction. In this paper we also can often follow proofs from computable model theory to derive their analogs in RCA_{0}. More frequently, though, RCA_{0} does not suffice.

Reverse mathematics had its beginnings in the work of Harvey Friedman in the late ’60s and early ’70s ([11, 12, 13]). Its early development is well chronicled in Simpson’s classic text [46], which is also the source for many important ideas and results. It has four basic systems in addition to RCA_{0}, and the ω-models of each correspond in a similar way to closure conditions on their sets familiar from computability theory. The primary story of the first few decades of reverse mathematics was that almost every standard theorem from the literature of classical mathematics that was analyzed was equivalent over RCA_{0} (in the sense of reverse mathematics described above) to one of these five systems. They were thus dubbed the “big five”. Moreover, each of these systems corresponds to a well known philosophical system for mathematics as well as to a standard level of complexity in computability theory. (For details, here and elsewhere, as well as a thorough general introduction to reverse mathematics, we refer the reader to the second edition of the standard text by Simpson [47]. For a shorter introduction focused on combinatorial and model theoretic principles, see Hirschfeldt [20]. A very brief introduction can be found in Shore [44].)

In addition to RCA_{0}, the two next stronger of the big five are the ones most relevant to this paper: ACA_{0} and WKL_{0}. The first extends RCA_{0} by adding on Σ_{n}^{0}-CA_{0} for every n ∈ ℕ, i.e., arithmetic comprehension: all sets defined by formulas of first order arithmetic (with set parameters) exist. It corresponds to closure under the Turing jump (i.e., the halting problem for machines with oracles in the structure). (The Turing jump and its iterates provide the most common measure of complexity in computability theory.) The second (and weaker) system corresponds to structures in which the collection of sets forms what is called a Scott set; i.e., every infinite binary tree in the collection has an infinite path in it as well. Each of these three systems has many equivalents in most areas of mathematics. Following the classical story of reverse mathematics we provide more equivalences in this paper.

On the other hand, a very interesting and, in our view, important trend in reverse mathematics has emerged in the past decade or so. Many theorems of standard mathematics have been found to lie outside the scope of the big five systems. Some are stronger than all of them, some are weaker even than WKL_{0}, and some are incomparable with WKL_{0} but still below ACA_{0}. Starting with papers such as [5], [22], and [23], combinatorics and model theory have been primary sources of such examples. Earlier examples can be found in papers like [42], but these more recent papers began a trend of exploring the reverse mathematics of large classes of related combinatorial and model theoretic principles not in the scope of the big five. (Expositions and discussions of these phenomena can be found in [44] and much more extensively in [20].) In this paper we continue along this road with new examples from model theory of theorems with comprehension strength lying outside the big five.

A yet newer phenomenon in reverse mathematics is the appearance of a large number of examples of principles escaping the standard systems because of issues that lie along the induction rather than (or in addition to) the comprehension spectrum. One striking example at the high end of the standard systems is provided by work of Montalbán [32] and Neeman [38, 39]. In the last of these three papers, Neeman showed that in the proof of an equivalence in the second paper (sharpening results of the first paper), the use of IΣ_{1}^{1} (induction for formulas with leading second order existential quantifiers followed by a first order matrix) cannot be eliminated. Another example where high levels of induction are required is in Montalbán and Shore [35]. They prove a result in ATR_{0} (the system from the big five after ACA_{0}) using transfinite induction. In particular, they use induction on arbitrary well orderings (instead of just of ℕ) for classes defined by Π_{1}^{1} formulas (so second order with one leading universal quantifier). However, the theorem fails without the extra induction assumption.

In this paper, in addition to cases where extra comprehension axioms are needed, we often run into issues at the lowest levels of the induction spectrum. Indeed, this happens first at the stage of choosing the “right” definitions. In our case, there is one important variation of homogeneity that requires comprehension axioms (ACA_{0}) to prove its equivalence to the others (Proposition 4.2). This equivalence is thus better viewed as a theorem and, indeed, is usually cited as such in the literature. On the other hand, there are a number of variations on the standard definitions that are used interchangeably and not distinguished in either the classical or computability theoretic literature. These variations are ones for which the usual proofs of equivalence use different levels of induction-like axioms. Here, we show reversals, in that the equivalences imply the axioms used to prove them over RCA_{0}. (Examples include Theorems 4.3 and 4.4.) Which definition is chosen can play a crucial role in getting the “right” theorem characterizing the strength of our model theoretic principles.

In addition, even with the “right” choice of definition, more induction than is available in RCA_{0} is used in the classical or computability theoretic proofs at a number of crucial junctures (e.g. for Σ_{2}^{0} rather than Σ_{1}^{0} formulas). Here, three routes are followed for different theorems. One applies the method of Shore blocking from generalized recursion theory ([43]) to carry out, in RCA_{0}, arguments that seem to need IΣ_{2}^{0}. (Examples include Theorem 6.10 as well as Theorem 4.1 of [23].) The second provides reversals to show that the theorems of interest are reverse mathematically equivalent to some induction-like principle. (Examples are Theorem 4.3 for IΣ_{2}^{0} and Theorem 4.4 for BΣ_{2}^{0}, the bounding principle for Σ_{2}^{0} formulas, which will be defined in Section 2.1 and has been shown by Slaman [48] to be equivalent to Δ_{2}^{0}-induction.) The third, and most interesting, route is one that leads to new axiomatic systems, theorems, and combinatorial principles of strength intermediate among, or incomparable with, the standard hierarchy of induction type axioms (which includes the usual bounding axioms). The driving examples include entries labeled 3 or 4 in Figure 2 on page 67 that lead to the combinatorial principle Π_{1}^{0}GA used extensively in Section 6. This principle is generalized in Appendix A to principles Π_{n}^{0}GA that fit into all levels of the induction hierarchy in a most unusual way. Each follows from the next level of induction: RCA_{0}+IΣ_{n+1}^{0} ⊢ Π_{n}^{0}GA. However, Π_{n}^{0}GA is strictly weaker in two ways. First, even BΣ_{2}^{0} does not follow from any Π_{n}^{0}GA. Second, for every n, we have RCA_{0} ⊢ Π_{n}^{0}GA + BΣ_{n+1}^{0} ↔ IΣ_{n+1}^{0}.

A foundationally challenging phenomenon arising in this paper is provided by theorems or constructions that have essentially two provably different proofs. In particular, we have theorems of model theory that are provable from, for example, either WKL_{0} or IΣ_{2}^{0} and others from WKL_{0} ∨ BΣ_{2}^{0} (as described, e.g., following Proposition 4.5) or the disjunction of other pairs of comprehension and induction type axioms (see, e.g., the remarks preceding Question 8.5 and items labeled 3 or 4 in Figure 2). When such a principle is not provable in RCA_{0}, there may not be any canonical “best” proof or axiomatic system for it. The only earlier example of this phenomenon of which we are aware is the fact that the principle that iterations of continuous functions are continuous is equivalent to WKL_{0} ∨ IΣ_{2}^{0} (Friedman, Simpson, and Yu [14]). Many additional examples of this phenomenon and related ones can be found in recent work of Belanger [3]. (Some of his equivalents for WKL_{0} ∨ IΣ_{2}^{0} in terms of amalgamation of types are mentioned in Theorem 5.12 below.) An interesting question then is what is the impact of such results on the philosophical or foundational program of reverse mathematics? (A different foundational question is posed by other results by Belanger ([2, 3]) where model theoretic facts are shown to be equivalent to ACA_{0} ∨ ¬WKL_{0}.)

1.2. The specifics. In order to describe the actual problems analyzed in our results, we require specific formal definitions of the standard axiomatic systems being used (and some variations on them) as well as the model theoretic notions being analyzed in both the computability theoretic and reverse mathematical settings. These are provided in Section 2. Here we give a description for the reader who already has a basic familiarity with reverse mathematics and elementary model theory or who will refer to Section 2 as needed. For now we just note that all languages (and so theories) and structures are assumed to be countable. Additionally, we assume all theories to be complete and consistent. So in the setting of computable model theory, for us, the notions of computable and decidable coincide for theories. In the setting of reverse mathematics, we include the full elementary diagram in the presentation of a structure. (From the viewpoint of effective model theory then, we study only decidable structures for which the full elementary diagram is computable rather than computable structures for which only the atomic diagram is assumed to be computable.)

The initial motivation for our study was the investigation by Hirschfeldt, Shore, and Slaman of the reverse mathematical complexity of several classical model theoretic theorems in [23] and related work from the viewpoint of computability theory that both preceded and followed it, in particular the analysis of what in [23] is called AMT, the Atomic Model Theorem: every atomic theory has an atomic model.

Harrington [19] and independently Goncharov and Nurtazin [16] proved an important early result on atomic models: An atomic, (complete) decidable theory T has a decidable atomic model if and only if there exists a uniformly computable listing of all of the principal types of T, i.e., all the types realized in the atomic model. From the relativized version of the above characterization, it easily follows that every atomic, decidable theory has a 0′-decidable atomic model, i.e., one whose full elementary diagram is computable in 0′. Csima [7] greatly improved this result by showing that such a theory always has an atomic model decidable in some low degree. Csima, Hirschfeldt, Knight, and Soare [9] studied atomic bounding degrees, where a degree d is atomic bounding if every atomic, decidable theory has a ddecidable atomic model. They showed that the Δ_{2}^{0} atomic bounding degrees x (i.e., x ⩽_{T} 0′) are exactly the nonlow_{2} ones (i.e., x′′ >_{T} 0′′). (These computability theoretic papers used the word “prime” in place of “atomic”. The two notions are classically equivalent (for countable models) but differ in our context, as discussed extensively in [23] and a bit in this paper.) The question of which degrees not below 0′ are atomic bounding is more complicated (see Conidis [6]) but is connected in various ways to the reverse mathematical strength of AMT as analyzed in [23].

In [23], it is shown that AMT is not provable in RCA_{0}, is incomparable with WKL_{0}, and is quite a weak theory. Indeed, it is strictly weaker than all the combinatorial principles considered by Hirschfeldt and Shore [22] that are not Π_{1}^{1}-conservative over RCA_{0}. It is also shown in [23] that AMT is more than Π_{1}^{1}-conservative over RCA_{0} and is Π_{1}^{1}-conservative over (RCA_{0}+)BΣ_{2}^{0}. (A system of axioms S is Γconservative over another system T (for Γ a class of formulas) if for any Φ ∈ Γ, if S + T ⊢ Φ then T ⊢ Φ. So this is a way of saying that S + T is not stronger than T at least as far as theorems in Γ are concerned.)

Since classically every (countable) atomic model is homogeneous, it is natural to study how degree theoretic and reverse mathematical results on homogeneous models compare with ones on atomic models. Now there are two ways to view the possible generalization of AMT from atomic to homogeneous models. The first views atomic models as ones whose elements satisfy a particular property. (They each realize some atom, i.e., a formula generating a principal type.) The second characterizes the atomic models by their type spectra, the sets of types they realize. (They realize precisely the principal types.) Since these characterizations are classically equivalent, the notion of atomic bounding degree captures the question of how hard it is to build an atomic model of any decidable atomic theory and is equivalent to the question of how hard it is to construct a model that realizes precisely the set of principal types of such a theory. These questions then both reasonably correspond to AMT in reverse mathematics.

For homogeneous models the two views are different. The first asks the simpler question of how hard it is to build a homogeneous model of any given decidable theory. Here the degree theoretic answer given by Csima, Harizanov, Hirschfeldt, and Soare [8] is that it is exactly the PA degrees (the ones of complete extensions of Peano arithmetic or equivalently of Scott sets) that suffice for these constructions. As would then be expected, the reverse mathematical analog is that the existence of a homogeneous model for every theory is equivalent to WKL_{0} (Lange [26]; see also Belanger [3]). This is also the computational and reverse mathematical complexity of being able to produce a completion of any (computable) consistent set S of sentences, and hence a model of S (see [47, Theorem IV.3.3]).

INDUCTION, BOUNDING, WEAK COMBINATORIAL PRINCIPLES & HMT 10

In this paper, we are concerned with the more difficult question of determining when is there a homogeneous model with a given type spectrum. (It is worth mentioning that classically a homogeneous model is determined up to isomorphism by its type spectrum; see Proposition 4.7 for a reverse mathematical analysis of this fact.) The first question, even classically, is then which sets of types are type spectra of homogeneous models. Consider the situation in computable model theory. Clearly, for any decidable model, there is a computable enumeration of the types it realizes. Goncharov [15] and Peretyat’kin [41] independently showed that there are additional conditions on a computable listing of types X that are necessary and sufficient for there to be a decidable homogeneous model A such that X is an enumeration of the type spectrum of A. Classically, it is easy to see that the set of types realized in a homogeneous model must satisfy certain amalgamation properties for formulas and types. To get a decidable homogeneous model, an additional effectiveness condition is needed on the listing of types. It must have what we call a computable extension function approximation that indicates (in terms of their indices) how types and formulas can be amalgamated. (See Section 6.1 for a formal definition.) The results of Goncharov and Peretyat’kin also answer the question classically if one ignores the computability theoretic restrictions and simply requires that the set of types be closed under the appropriate amalgamation procedures (see Theorem 5.1).

In [27] and [28], Lange studied the degree theoretic analog of the second view of AMT for homogeneous models. She showed that for any computable list of types X with the appropriate amalgamation properties, there is a model A decidable in some low degree such that X is an enumeration of the type spectrum of A. She also investigated the question of which degrees d have the property that, given any X as above, there is a d-decidable homogeneous model A such that X is an enumeration of the type spectrum of A. These we might call the homogeneous bounding degrees. (Lange and Soare [29] called them the 0-bounding degrees. Note that the papers [8, 9, 29] use the term “homogeneous bounding degrees” in the other sense mentioned above, where the class of such degrees coincides with that of the PA degrees.) Lange showed that, as for atomic models, the homogeneous bounding degrees d ⩽ 0′ are precisely the d with d′′ > 0′′. Comparing these degree theoretic results (and others) with the aforementioned ones of Csima [7] and Csima, Hirschfeldt, Knight, and Soare [9] for atomic models suggested a deeper connection between model existence principles for atomic and homogeneous models. However, the proofs of these results for homogeneous models and for atomic models seemed quite different. In this kind of situation, where analogies between disparate principles suggest but do not immediately provide a precise connection, reverse mathematical and computability theoretic analysis can often clarify the situation. In this paper, we show that the model existence principle studied by Lange and AMT are indeed connected in a strong way, as seen from the viewpoints of reverse mathematics and computability theory. In particular, our results explain the similarity between Lange’s results for homogeneous models and the ones for atomic models by showing that, in a precise sense, both sets of results are dealing with the same principle, in two different guises.

First, we must provide an analog to the principle AMT. Recall that the view of AMT that we are trying to emulate states that every atomic theory T has a model whose type spectrum consists of the principal types of T. In the homogeneous case, we want a principle, the Homogeneous Model Theorem (HMT), asserting that if a list of types could be an enumeration of the type spectrum of a homogeneous model, then there in fact exists a homogeneous model whose type spectrum is enumerated by this list. The characterization of such lists of types in the computability theoretic setting in [15] and [41] mentioned above provides us with the obvious starting point. In Section 5.1, we present several variations on the conditions given in these characterizations that are classically equivalent but differ when viewed reverse mathematically. In Section 4, we give several definitions of homogeneity that are also classically equivalent, but differ in the setting of reverse mathematics. We obtain three versions of the HMT principle depending on how we describe the closure conditions on the list of types and which definition of homogeneity we use. In Section 7.2, we show that these versions of HMT imply AMT and that AMT implies one of them over RCA_{0} (and the other ones with sufficient additional induction assumptions). Thus the “right” versions supply us with the theorem on the existence of homogeneous models with specified type spectra that is reverse mathematically equivalent to the atomic model theorem.

Our proofs of these reverse mathematical results actually provide degree theoretic information that greatly strengthens the computational analogy between atomic and homogeneous models (from the second viewpoint above). Indeed, we show (Corollary 7.12) that the atomic bounding degrees are precisely what we have called the homogeneous bounding degrees.

At this point, we also note that while we were originally motivated by the idea of the analogy between existence principles for atomic and homogeneous models, and the analogy proved precise for the second view of AMT, there is also a close analogy between these principles and existence principles for arbitrary models. As we mentioned above, the pure model existence problem for sets of consistent sentences is reverse mathematically (and degree theoretically) equivalent to WKL_{0} (the PA degrees), as is the existence problem for homogeneous models of theories (or consistent sets of sentences). If we move to the second view of AMT, where we require the existence of a model with any given “possible” type spectrum, and move from arbitrary consistent sets of sentences to theories, our analysis shows that the existence of arbitrary models, AMT, and HMT are both reverse mathematically (with the “right” definitions or induction assumptions) and degree theoretically equivalent. (Notice that specifying a set of types automatically specifies a (complete) theory.)

Of course, one needs the appropriate necessary and sufficient conditions for a set of types to be the type spectrum of an arbitrary model. These are given in [41] for both the classical and computability theoretic settings. In Theorem 5.7 we present these conditions, and in the following sections we carry out the analysis for arbitrary models along with the one for homogeneous models. The outcome of this analysis is that one of the variants (WMT) of the existence theorem for arbitrary models is reverse mathematically equivalent to HMT (Theorem 7.6). Moreover, the analogously defined class of model bounding degrees is the same as the classes of atomic and homogeneous bounding degrees (Corollary 7.12). As with HMT there are variants for arbitrary models that are classically and computably equivalent but may differ depending on the types of induction-like axioms available. Before turning to these questions, we note that not only are AMT, HMT, and WMT equivalent over RCA_{0} but, if we work over IΣ_{2}^{0}, they are also equivalent to all the variants that we are about to discuss, as well as the combinatorial principles Π_{1}^{0}G and ATT that we consider below. Thus AMT is a candidate for a system that, like each of the big five, is what Montalbán ([33] and [34]) has called robust, a term that still lacks a precise definition but is informally defined in [34] as “equivalent to small perturbations of itself”. Another example of a robust system outside the big five is WWKL_{0} (see [47, Section X.1] for a definition of this system).

As we mentioned above, issues connected to induction arise already when we try to formalize the definition of homogeneous model in Section 4. The usual classical definition requires that, given two sequences of the same type and an extension of one of them, we can find a corresponding extension of the other such that the extended sequences continue to have the same type. One can vary this idea by considering extensions of length 1 or of arbitrary finite length, and also by requiring that extensions can be found for finitely many pairs of given sequences as in Definition 4.1(i)–(iii). All these variants are classically and computably equivalent since the proofs of equivalence require only simple induction arguments. These proofs are not quite simple enough, however, to work in RCA_{0}. There is one additional standard version (Definition 4.1(iv)), which we call strong 1-homogeneity, that requires the existence of an automorphism taking one of the two given sequences with the same type to the other. This is “obviously” a much stronger condition. The usual proof is not computable and seems to require the jump of the model to carry it out. Not surprisingly, we can show that this level of complexity is computationally and reverse mathematically necessary. The equivalence of this characterization of homogeneity to any of the others is itself reverse mathematically equivalent to ACA_{0} (Proposition 4.2). On the other hand, given this result, it is surprising to note that the existence theorem that every theory has a homogeneous model in this strong sense remains at the level of WKL_{0} (Lange [26] and Belanger [3]).

The analysis of the relations among the other variations on homogeneity (Section 4) are more interesting and unusual, as are the ones (Section 5) of the variations on the amalgamation closure conditions in the definition of the “possible” sets of types that could be type spectra of homogeneous models. These two sets of variations are corresponding ones, in that the closure conditions are the ones that obviously hold for models satisfying the corresponding notion of homogeneity. (Thus there is a match up for possible candidates for the “right theorem”.) The analyses of each set of variations and the relations between them provide reverse mathematical equivalents to induction type axioms: IΣ_{2}^{0} (Theorems 4.3, 5.11, and 5.14 and the entries labeled 2 in Figure 2) and BΣ_{2}^{0} (Theorems 4.4, 5.15, and 5.16).

Now there are many logical, computability theoretic, and combinatorial equivalents of the standard versions of these induction type axioms at all levels of their hierarchies (IΣ_{n}^{0} and BΣ_{n}^{0}). Some can be found, for example, in the standard (and quite comprehensive) text on first order arithmetic by Hájek and Pudlák [17, Section A.I.2]. Others in the setting of second order arithmetic are scattered throughout the reverse mathematical literature. Our results here, however, are about model theoretic properties. Their proofs are often given by delicate and at times complicated constructions of theories and specific models. The models constructed are designed to satisfy one variant of homogeneity but such that, if they satisfied another, we could prove one of the standard equivalents of IΣ_{2}^{0} or BΣ_{2}^{0}.

Much more unusual and surprising are the roles that induction plays in the analysis of the construction of some of the variations of homogeneous models. Here, we want to construct a model with a suitable given type spectrum. Of course, we are assuming that the potential type spectrum satisfies some version of the conditions required for there to be such a model in the computability theoretic setting. (As mentioned above, these conditions were given by both Goncharov [15] and Peretyat’kin [41] independently.) Examples of such constructions include Theorems 5.1 and 6.4. In particular, we assume that there is an “extension function approximation” (Definition 6.2) that approximates the action (in terms of indices) on the given list of types that the amalgamation relations require. Here we refer the reader especially to the entries marked 3 of Figure 2 on page 67.

The ones in the column labeled “pairwise full amalgamation” are all construction principles for homogeneous or arbitrary models with given type spectra of the appropriate kinds. (The pairwise full amalgamation closure conditions form one of our versions of the Goncharov-Peretyat’kin conditions mentioned above.) Our original proofs for them (and all the entries marked 3) showed that the constructions can be carried out assuming either IΣ_{2}^{0} or the comprehension principle Π_{1}^{0}G from [23] which essentially asserts the existence of generics for (i.e., meeting each one of) any given uniformly Π_{1}^{0} collection of dense sets. Moreover, each of them is equivalent to IΣ_{2}^{0} over BΣ_{2}^{0}. This presented us with a very unusual situation. Let us consider one of these examples.

P: For every type sequence X satisfying the (pairwise full) amalgamation closure conditions and having an extension function approximation, there is a (1-)homogeneous model realizing exactly the types in X.

(1) P is essentially the formulation of the computable construction principle for homogeneous models that appears in the literature (see [15] and [41]).

(2) P makes no mention of recursiveness, Turing reducibility, arithmetic, or formulas of any specified quantifier rank.

(3) The proof of P is nontrivial. (It requires a priority argument.)

(4) P is a consequence of IΣ_{2}^{0}.

(5) P is Π_{1}^{1}-conservative over IΣ_{1}^{0} and so does not imply even BΣ_{2}^{0}, let alone IΣ_{2}^{0}.

(6) P + BΣ_{2}^{0} is equivalent to IΣ_{2}^{0}.

Thus P is quite unusual reverse mathematically. It is a result in the standard literature with a proof not primarily by induction. Indeed, it makes no explicit mention of induction, recursiveness, or formulas in the arithmetic hierarchy. Nonetheless, it occupies a place that is within the usual hierarchy of induction axioms but is different from all the standard ones. We then thought that perhaps we could capture the combinatorial essence of these arguments in a way that would isolate the reverse mathematical issues involved. This attempt led us to an approximation version of Π_{1}^{0}G that we call Π_{1}^{0}GA. Basically, it asserts not that there is a generic for uniformly Π_{1}^{0} classes of dense sets but rather that there is a function approximating such a generic. (Here approximation is in the sense of the Shoenfield limit lemma which provides computable functions f(x, s) approximating any function g ⩽_{T} 0′ such that (∀n)(∃t)(∀s ⩾ t)(∀x ⩽ n)[f(x, s) = g(x)]. The formal definition is given after the proof of Theorem 3.1.)

This principle, Π_{1}^{0}GA, is indeed sufficient to prove all the entries marked 3 in Figure 2 (as well as those marked 4) and has much the same quite unusual reverse mathematical status. It is not provable in RCA_{0}. It is provable from both IΣ_{2}^{0} and Π_{1}^{0}G, so it is strictly weaker than each of them, and indeed (like Π_{1}^{0}G), is highly conservative over RCA_{0} and so, in particular, does not prove, for instance, BΣ_{2}^{0} or WKL_{0}. On the other hand, like the principle P above, it joins BΣ_{2}^{0} up to IΣ_{2}^{0} in the sense that IΣ_{2}^{0} is strictly stronger than each of these principles and is implied by their conjunction. We do not know of any principles other than the ones considered in this paper (except for trivial ones such as BΣ_{2}^{0} → IΣ_{2}^{0}) with such properties. Our investigations also led to some properties of trees and their isolated paths. One gives an equivalent of BΣ_{2}^{0} (Proposition B.1). A new one (FATT) is strictly weaker even than BΣ_{2}^{0} but nonetheless not provable in RCA_{0} (Theorem B.2).

In the other direction, we also move up the induction hierarchy in Appendix A. There we present natural generalizations Π_{n}^{0}GA of Π_{1}^{0}GA to all levels of the arithmetic hierarchy. Essentially they say that there are approximations (in the sense of the iterated limit lemma) to generics for all uniformly Π_{n}^{0} sequences of dense sets. The primary reverse mathematical properties of Π_{1}^{0}GA are then replicated all the way up the hierarchy:

(1) IΣ_{n+1}^{0} ⊢ Π_{n}^{0}GA (Theorem A.1).

(2) Π_{n}^{0}GA ⊬ BΣ_{2}^{0}. (The discussion before Theorem A.1 shows that Π_{n}^{0}GA is Π_{1}^{1} conservative (and more) over RCA_{0}.)

(3) RCA_{0} ⊢ Π_{n}^{0}G + BΣ_{n}^{0} ↔ IΣ_{n}^{0} (Theorem A.2).

Thus our reverse mathematical investigations of one model theoretic theorem have led us to a new hierarchy of combinatorial principles that are interspersed among the standard induction hierarchy in a very interesting and unusual way.

A number of questions about the model theoretic and combinatorial principles that we have investigated, and especially their reverse mathematical strength, are still open. We collect the most natural ones in Section 8. Perhaps the most interesting from a foundational rather than simply technical point of view are the ones asking for a precise determination of the reverse mathematical strength of groups of principles. In particular Question 8.6 asks whether all the entries marked 3 in Figure 2 are equivalent and, if so, whether they are all equivalent to Π_{1}^{0}GA. If, indeed, all of these principles are equivalent, this would give some evidence for the robustness of Π_{1}^{0}GA as discussed above.

Returning to the relationship between reverse and computable model theory, we would like to point to a general theme brought out by our investigations here as well as those in [23]. The motivating issue is the phenomenon in computable model theory of what one might call partial effectivization. One begins with a classical type of model such as atomic (prime) or (strongly 1-) homogeneous and the question of how hard it is to construct such a model. Typically, the effective version assumes that the underlying theory is decidable and asks about the the possible degrees of models of the desired kind and, in particular, whether (or under what conditions) there is a decidable one.

Our investigations of these questions from the viewpoint of reverse mathematics point out that the standard computability theoretic work ignores the issue of the complexity of verifying that the model constructed is of the desired kind and just accepts the classical proof. Often, there is no difference. When, however, the definition of the kind of model is not first order but calls for the existence of morphisms, for example, then additional effectiveness considerations can be raised about the complexity of the morphism required to verify the construction. A reverse mathematical analysis automatically levels the playing field for both the construction and verification. In the reverse mathematical setting, one can show that a model constructed is prime or strongly 1-homogeneous only by showing that the morphisms required can also be constructed in the same system. It makes sense to carry over these questions to computable model theory and ask how complicated (in terms of Turing degree) it is to show that the morphisms exist.

As an illustration, we take the two aforementioned topics investigated reverse mathematically in [23], [3], and this paper: the existence of prime models for every atomic theory (PMT) and the existence of strongly 1-homogeneous models for every theory.

As we have mentioned, [9] shows that given a decidable atomic theory T, every d ⩽ 0′ with d′′ > 0′′ computes a prime model M of T. However, if we look at the verification that the M computable from d is prime (rather than atomic), we are required to produce, for every N ⊨ T, an elementary embedding f : N → M. Now even if N is decidable there may be no such f computable even in d. From the viewpoint of reverse mathematics [23] shows that PMT is equivalent to ACA_{0}. Thus in terms of a closure operation one needs the jump operator and the morphisms are arithmetic in the data. However, the computable model theorist should look more carefully at the question. What is then seen is that one can always construct the required f computably in T′ ⊕M⊕N (for any prime model M of T and any model N of T). The proof of the reverse mathematical equivalence then shows that this level of complexity is necessary.

Turning to strongly 1-homogeneous models, [8] shows that given a decidable T one can construct a 1-homogeneous model M computably in any PA degree. To verify that the model M is strongly 1-homogeneous one must construct, for every pair of tuples ā and b̄ from M with the same type, an automorphism f of M taking ā to b̄. Now for an arbitrary 1-homogeneous model, it takes the jump of M to construct such an automorphism. This can be seen from the proof that reverse mathematically this general implication is equivalent to ACA_{0} (Proposition 4.2). The computable model theorist can easily verify that M′ is also sufficient. On the other hand, as we have noted, Lange [26] and Belanger [3] have shown that the existence of strongly 1-homogeneous models for every theory is equivalent to WKL_{0}. Belanger’s proof explicitly shows that if one constructs M still computably in a PA degree d but carefully, then one can also simultaneously construct all the required automorphisms computably in d.

Thus our reverse mathematical investigations suggest a general type of problem for computable model theory. When one proves that there is a decidable or d-decidable model M with some property that requires (in its definition) the existence of more than the elementary diagram of M, then one should ask how hard it is to compute the other objects (e.g. morphisms) required to verify that the definition holds of M. Other natural examples for investigation include issues dealing with categoricity for theories (as is common in model theory, rather than for structures as is common in computable model theory), universality, and saturation. Some of these questions have recently been investigated reverse mathematically by Belanger in [2] and [3].

Another general connection between reverse and computable mathematics (as well as degree classes more broadly) is suggested by the various notions of bounding degrees and their analyses as discussed above. Given any principle expressed by a sentence of the form ∀X∃Y Φ(X, Y) we can form a bounding operator B_{Φ} : D → P(D) (where D is the set of Turing degrees) defined by B_{Φ}(x) = {y | (∀Z ⩽_{T} x)(∃W ⩽_{T} y)Φ(Z, W)}. Examples considered in this paper include the ones generated by relativizing the atomic, homogeneous, and model bounding degrees to an arbitrary x. Indeed, most principles studied in reverse mathematics are of this form with Φ arithmetic, or at times Π_{1}^{1}. Note that the images of these operators are always upward closed subsets of D. More generally, we can consider almost jump operators J that take degrees x to upward closed subsets J (x) of D (typically contained in the cone above x). Examples include ones generated from all the usual (definable) jump operators J, from the Turing jump to sharps, by taking the image of x to be the cone above J(x). The natural partial order ⩽ (and so the resulting equivalence relation ≡) on both of these classes of operators is given by reverse inclusion, i.e., J ⩽ K ⇔ (∀x)[K(x) ⊆ J (x)]. Our results on the model theoretic questions studied in this paper say that the operators corresponding to AMT, HMT, and WMT are all equivalent.

The methods of comparing the strength of principles using bounding operators and computable entailment are closely connected to the computability theoretic reduction procedures between Π_{2}^{1} principles studied in papers such as [21]. Many other results in reverse and computable mathematics can be seen as saying interesting things about the natural ordering on bounding operators in general, or about specific instances of it. One that does both not only shows the existence of a minimal operator (above the one induced by the identity map on degrees) but also identifies it with a model theoretic construction principle. The obvious degree theoretic definition of this operator is given by Φ(X, Y) ≡ X <_{T} Y. That it corresponds to a model theoretic assertion (AST) about the existence of an atomic model of a theory with a “nice” enumeration of its types is the content of Theorem 6.3 of [23]. Another correspondence between reverse mathematical principles and standard degree classes studied in computability theory (other than the obvious ones given by the PA degrees and the Turing jump) from [23] is a type omitting theorem (OPT) shown there to be equivalent to the existence of hyperimmune sets (relative to any given set). There are many other examples that shed light on the structure of the ordering of bounding and almost jump operators. So, too, are there many questions that can be raised in this setting and approached from both the computability theoretic and reverse mathematical viewpoints.

1.3. An outline of the paper. In Section 2, we define several notions used throughout the paper.

In Section 3, we establish some basic facts about AMT and related principles. In particular, we introduce the principle Π_{1}^{0}GA and prove that it has the properties mentioned above. As the versions of Π_{1}^{0}GA for higher levels of the arithmetic hierarchy do not form a part of our main narrative, we leave their discussion to Appendix A. The principle FATT mentioned above, which is also related to AMT, is discussed in Appendix B.

In Section 4, we compare several classically equivalent definitions of homogeneity from the reverse mathematical point of view. We also analyze the strength of theorems relating homogeneous models to atomic, prime, and saturated models. One tangential result on saturated models is left to Appendix C.

In Section 5, we introduce the characterizations by Goncharov and Peretyat’kin of possible type spectra of (homogeneous) models. We give several classically equivalent versions of the conditions in these results, and compare them reverse mathematically. We also analyze the strength of the easier direction of these results (that the type spectrum of a (homogeneous) model must satisfy these conditions). Finally, in Section 5.5, we introduce HMT and its variants, including ones such as WMT concerning the existence of general models (rather than homogeneous ones).

In Section 6.1, we discuss the effective versions of the above characterizations (also due to Goncharov and Peretyat’kin). In particular, we introduce the notions of extension function and extension function approximation central to these effective versions, and compare them reverse mathematically. In Section 6.2, we study the reverse mathematics of several versions of the computability theoretic results of Goncharov and Peretyat’kin, summarizing our results in Figures 2 and 3 (see also Figures 3 and 4). As mentioned above, along with versions provable in RCA_{0} and ones equivalent to IΣ_{2}^{0}, we obtain several that exhibit behavior similar to that of Π_{1}^{0}GA.

In Section 7, we study principles asserting the existence of extension function approximations under various versions of the conditions of Goncharov and Peretyat’kin, and use our results to compare versions of AMT and HMT reverse mathematically and computability theoretically. We show in particular that AMT, HMT, and WMT are equivalent, both over RCA_{0} and in the sense of uniform computability theoretic reducibility.

Finally, in Section 8, we gather several open questions arising from our work.

2. Definitions

A few definitions given in this section have already been mentioned in the introduction, but we include them here as well for ease of reference.

2.1. Reverse mathematics. We assume familiarity with the basics of reverse mathematics, but briefly describe some of its commonly studied axiom systems. For a complete introduction to the field, see Simpson [47]. For a shorter introduction focused on combinatorial and model theoretic principles, see Hirschfeldt [20]. We work in the language of second order arithmetic, with lower case letters representing number variables and uppercase letters representing set variables. In particular, every set we consider is assumed to be countable. We think of first order objects, such as finite strings, as encoded by natural numbers, and of second order objects, such as trees or models, as encoded by sets of natural numbers (see [47] for more details).

Let P consist of axioms stating that the natural numbers form a discrete ordered commutative semiring, together with set induction:

(0 ∈ X ∧ (∀n)[n ∈ X → n + 1 ∈ X]) → (∀n)n ∈ X.

Let IΣ_{n}^{0} be the following induction principle, expressed as an axiom scheme.

(φ(0) ∧ (∀n)[φ(n) → φ(n + 1)]) → (∀n)φ(n)

for all Σ_{n}^{0} formulas φ.

Our base axiom system is RCA_{0}, which consists of P together with IΣ_{1}^{0} and the following set existence axiom scheme, which is just strong enough to prove the existence of computable sets.

(∀n)[φ(n) ↔ ψ(n)] → (∃X)(∀n)[n ∈ X ↔ φ(n)]

for all Σ_{1}^{0} formulas φ and Π_{1}^{0} formulas ψ in which X does not occur free.

A useful principle provable in RCA_{0} is bounded Σ_{1}^{0}-comprehension (see [47]), which for our purposes we state as follows. For any finite sequence s of natural numbers and any Σ_{1}^{0} property P, there is a subsequence of s consisting of those elements of s that satisfy P. Of course, by taking complements, bounded Π_{1}^{0}-comprehension also holds in RCA_{0}.

Another useful fact is that IΣ_{2}^{0} is equivalent over RCA_{0} to the finite Π_{1}^{0}-recursion principle (see [22]), which states that if φ is a Π_{1}^{0} formula defining a total function, then for each z and n, there is a sequence x_{0}, …, x_{n} such that x_{0} = z and φ(x_{i}, x_{i+1}) holds for all i < n.

Another collection of axioms schemes, related to IΣ_{n}^{0}, is that of restricted bounding principles. The bounding principle BΣ_{n}^{0} is given by the axiom scheme

(∀m)[(∀i < m)(∃u)φ(i, u) → (∃v)(∀i < m)(∃u < v)φ(i, u)]

for all Σ_{n}^{0} formulas φ in which v does not occur free. Over RCA_{0}, IΣ_{n}^{0} implies BΣ_{n}^{0}, and BΣ_{n}^{0} implies IΣ_{n−1}^{0}; these implications are strict (Paris and Kirby [40]; see also Hájek and Pudlák [17]).

The principle BΠ_{n}^{0} is defined in the analogous way. We use the fact that BΣ_{2}^{0} and BΠ_{1}^{0} are equivalent over RCA_{0} (see [17]). Another useful principle equivalent to BΣ_{2}^{0} over RCA_{0} is the finite axiom of choice for Π_{1}^{0} properties (see [17]): for any sequence n_{0}, …, n_{k} and any Π_{1}^{0} property P, if for each i ⩽ k there is an m with P (n_{i}, m), then there is a sequence m_{0}, …, m_{k} such that P (n_{i}, m_{i}) for each i ⩽ k.

WKL_{0} is the axiom system that includes RCA_{0} and the principle known as Weak König’s Lemma, which states that every infinite subtree of 2^{<ℕ} has an infinite path. (Here a tree is a set of strings closed under initial segments.) This system is strongly related to degrees of completions of Peano arithmetic.

The last major system we consider here is ACA_{0}, which consists of RCA_{0} together with arithmetic comprehension:

(∃X)(∀n)[n ∈ X ↔ φ(n)]

for every arithmetic formula φ in which X does not occur free, where a formula is arithmetic if it has no set quantifiers. Computability theoretically, ACA_{0} corresponds to the existence of the Turing jump.

The system ACA_{0} is strictly stronger than WKL_{0}, which is in turn strictly stronger than RCA_{0} (see Simpson [47]). We use the fact that ACA_{0} is equivalent over RCA_{0} to the statement that for every function f : ℕ → ℕ, the range of f exists, i.e., there is a set R such that n ∈ R if and only if (∃m)f(m) = n (see [47]).

For a system S, a principle R, and a collection of sentences of second-order arithmetic Γ, we say that R is conservative for Γ, or Γconservative, over S if every sentence in Γ provable in S +R is provable in S. For example, a classic result of Harrington (see [47]) is that Weak König’s Lemma is Π_{1}^{1}-conservative over RCA_{0}.

A model in the language of second order arithmetic consists of a first order part M, which is a structure in the usual language of first order arithmetic, a subset S of the power set of the domain of M, and the membership relation ∈ between elements of M and elements of S. If M is the standard natural numbers, we say that this model is an ω-model. We denote the natural numbers by ℕ. Of course, when interpreting a statement in a model (M, S) as above, ℕ is interpreted as M. An object is finite if it is coded by a natural number. For example, when we refer to a finite sequence n_{0}, …, n_{k} of natural numbers, we mean that ⟨n_{0}, …, n_{k}⟩ ∈ ℕ, where ⟨·, …, ·⟩ is any standard encoding of tuples of natural numbers as single natural numbers. A set is infinite if it is unbounded. Thus, when we say that there are infinitely many natural numbers x such that some property P (x) holds, we mean that for each n ∈ ℕ there is an x > n such that P (x) holds. Note that, for a model (M, S) in the language of second order arithmetic, there may be A ⊂ M that are neither finite nor infinite in the above sense, though if (M, S) ⊨ RCA_{0}, then such an A cannot be in S.

2.2. Model theoretic notions. In this section, we introduce some notation and discuss a few concepts and results of basic model theory. See [4] or [30] for an introduction to model theory. For an overview of computable model theory, see [1] or [18]. We assume that all theories are consistent and complete, and that all languages include equality (although our results for the most part do not depend on the latter assumption; see Remark 5.2 for the one exception, which is easily handled). As noted above, we think of second order model theoretic objects such as theories and models as encoded by sets of natural numbers. In particular, we assume that all languages and models are countable. We use calligraphic letters such as A for models, and use the corresponding roman letters such as A for their domains. Whenever we mention a tuple of variables, we assume the variables in the tuple are distinct. (When we mention a tuple of elements of a model, on the other hand, we do not make this assumption.) Definition 2.1. Let T be a theory and A a model of T.

(i) A formula θ(x̄) is consistent with T if T ∪ {(∃x̄)θ(x̄)} is consistent, or equivalently, because of the completeness of T, if (∃x̄)θ(x̄) ∈ T.

(ii) An n-type p(x̄) of T in the n-tuple of variables x̄ is a maximal

set of formulas in variables taken from x̄ consistent with T. (iii) For a k-subtuple ȳ of an n-tuple x̄, and an n-type p(x̄), let p ↾ ȳ

be the k-type consisting of all the formulas in p whose variables are all in ȳ. We call such a type a subtype of p.

(iv) Let S_{n}(T) be the set of all n-types of T (in all n-tuples of

variables), and let S(T) = ⋃_{n∈ℕ} S_{n}(T).

(v) We say p ∈ S_{n}(T) is a principal type if there exists a formula ψ ∈ p such that T ⊢ ψ → θ for all θ ∈ p. We call such a ψ a generator of p and say ψ generates or isolates p.

(vi) A formula φ(x̄) is an atom of T if for every formula ψ(x̄), either

T ⊢ φ → ψ or T ⊢ φ → ¬ψ, but not both. (vii) The theory T is atomic if for every formula ψ consistent with T,

there is an atom φ of T such that T ⊢ φ → ψ. (In other words, T is atomic if every formula consistent with T is an element of some principal type of T.)

(viii) An n-tuple ā ∈ A realizes an n-type p(x̄) ∈ S_{n}(T) if A ⊨ θ(ā) for all θ(x̄) ∈ p(x̄). In this case we also say that A realizes p. The type of ā in the variables x̄ (where |x̄| = |ā|) is the type p(x̄) that ā realizes.

(ix) The type spectrum of A is

𝕋(A) = {p ∈ S(T) : A realizes p}.

(x) The model A is atomic if every tuple of elements of A satisfies an atom of T. (In other words, A is atomic if 𝕋(A) consists of principal types.)

(xi) The model A is prime if it is elementarily embeddable in each

model of T. (xii) The model A is saturated if for all ā ⊆ A and p ∈ S_{n}(Th(A, ā)),

the type p is realized in A, where (A, ā) is the expanded structure obtained by adding to A constants for the elements of ā, and Th(A, ā) is the theory of this structure.

(xiii) A partial type of T in the variables x̄ is a (not necessarily maximal) set of formulas in variables taken from x̄ consistent with T. The model A is strongly saturated if for all ā ⊆ A, each partial type of Th(A, ā) is realized in A. (This notion is of course classically equivalent to being saturated, but this equivalence is not provable in RCA_{0}, and is in fact equivalent to WKL_{0}; see Appendix C.)

Since we are interested in statements about the existence of homogeneous models realizing particular sets of types, we need a way to encode lists (i.e., enumerations) of types.

Definition 2.2. Let T be a theory and let {θ_{i}}_{i∈ℕ} be an effective listing of the formulas in the language of T. We identify p ∈ S(T) with the function f_{p} ∈ 2^{ℕ} such that f_{p}(i) = 1 if θ_{i} ∈ p and f_{p}(i) = 0 otherwise. Let p ↾ s = p ∩ {θ_{i}}_{i<s}. We identify p ↾ s with the function f_{p} ↾ s.

Let X = {p_{i}}_{i∈ℕ} ⊂ S(T). We encode X as a function f(i, j) such that for each i, the first value f(i, 0) encodes the set of free variables of p_{i}, and f(i, k + 1) = f_{p_{i}}(k).

Note that, in both the computability theoretic and the reverse mathematical settings, we have to deal with enumerations of potential type spectra, because even if {p_{i}}_{i∈ℕ} and {q_{i}}_{i∈ℕ} are equal as sets of types, there may be no effective way to go from an index i to an index j such that q_{j} = p_{i}. We say that a list of types {p_{i}}_{i∈ℕ} is computable if the function f in Definition 2.2 is computable.

In computable model theory, a model A can be associated with its atomic diagram or with its elementary diagram. Let L_{A} be the expansion of the language of A obtained by adding a constant symbol for each element of A. The elementary diagram D^{e}(A) of A is the set of all sentences of L_{A} that are true in A (under the obvious interpretation of the new constant symbols). The atomic diagram D^{a}(A) of A is the set of all atomic sentences of L_{A} that are true in A. Alternatively, we can associate these diagrams with satisfaction predicates. A predicate S is an elementary (atomic) satisfaction predicate for A if for any (atomic) formula θ(c̄) with constants from A, the predicate S holds of θ(c̄) if and only if θ(c̄) is true in A. In computable model theory, we measure the complexity of a model by the degree theoretic complexity of these diagrams. We say that A is computable if D^{a}(A) is computable and that A is decidable if D^{e}(A) is computable, and we similarly define the notions of d-computable and d-decidable for any degree d. (Note that for a theory T, saying that T is decidable is just another way of saying that T is computable.)

When investigating the strength of theorems of model theory such as AMT and HMT, decidable models and their relativizations are the more relevant notions. Accordingly, in reverse mathematics, a model A includes a function interpreting the terms of the language in the model, as well as the full satisfaction predicate for A. For a formal definition of a model in this setting, see Simpson [47, Definition II.8.3].

In several places below, we specify a theory by giving a set A of axioms that generates it. In each case, the language of A does not include any function symbols, so all terms are either variables or constant symbols. To show that such a set A does in fact specify a theory T, we use quantifier elimination. Recall that to show that A admits effective quantifier elimination, it is enough to give a procedure to find a quantifier-free formula ψ(x̄) that is A-equivalent to a given existential formula (∃y)φ(x̄y), where φ(x̄y) is a conjunction of literals, i.e., atomic formulas and negations of atomic formulas. (We always assume that we have propositional constants T and F for true and false, respectively.) It then follows that the deductive closure T of A is Δ_{1}^{0}-definable from A and complete, assuming that the set of quantifier-free sentences consistent with A is itself Δ_{1}^{0} definable from A. (It is easy to see that this assumption holds if the set of literal sentences (i.e., literals with no free variables) consistent with A is Δ_{1}^{0}, and hence that it always holds when the language of A does not include constant symbols, as then the only literal sentences are T and F.) It also follows that every type of T is determined by the literals it contains, so we can identify the types of T with atomic types, i.e., sets of literals p(x̄) such that for each atomic formula ψ with variables among the elements of x̄, either ψ ∈ p(x̄) or ¬ψ ∈ p(x̄).

Remark 2.3. Let L be the language of T, and let L′ be any finite sublanguage of L that includes all symbols occurring in φ. Say that a formula ψ(ū) is an L′-atomic diagram if it is a conjunction of literals in L′; for every atomic θ in L′ with variables among those in ū, exactly one of θ and ¬θ occurs in ψ; and ψ is consistent with the usual rules for equality. It is easy to see that there is a procedure that, given any conjunction δ of literals in L′, finds a disjunction of L′-atomic diagrams that is equivalent to δ (or returns F if δ contains both θ and ¬θ for some θ, or is not consistent with the rules for equality). Since existential quantifiers distribute over disjunctions, in performing an effective quantifier elimination procedure as above, we may assume that φ is itself an L′-atomic diagram.

If u = y occurs in φ, where u ∈ x̄ or u is a constant symbol, then we may replace all occurrences of y in φ by u and drop the (∃y) quantifier to obtain a quantifier-free formula equivalent to (∃y)φ(x̄y). Similarly, if u = v occurs in φ, where u ∈ x̄ or u is a constant symbol, and v ∈ x̄ is distinct from u, then we may replace all occurrences of v in φ by u and move u = v outside the existential quantifier. If we now have both θ and ¬θ occurring in φ, then ⊢ (∃y)φ(x̄y) ↔ F. Thus we may assume that φ is an L′-atomic diagram that includes ¬u = v for all u such that u ∈ x̄ or u is a constant symbol, and all v ∈ x̄y distinct from u.

Suppose there is a binary relation symbol E in L′ such that A asserts that E is an equivalence relation. Then there clearly is a procedure that, given a L′-atomic diagram ψ, decides whether ψ is consistent with the assertion that E is an equivalence relation (that is, whether ψ includes E(u, v) whenever it includes u = v, includes E(u, v) whenever it includes E(v, u), and includes E(u, w) whenever it contains E(u, v) and E(v, w), where in each case, each of u, v, and w is a variable or a constant symbol). If this is not the case, then A ⊢ (∃y)ψ(x̄y) ↔ F, so we may assume that our L′-atomic diagram φ is consistent with the assertion that E is an equivalence relation for any such E.

2.3. Other notions. We define a few basic notions about trees that appear in our proofs below. We discuss the connection between trees and theories in Section 3; for more on this topic, see Lange and Soare [29].

Definition 2.4. (i) A (binary) tree T is a subset of 2^{<ℕ} closed under initial segments, i.e., if σ ∈ T and τ ≺ σ then τ ∈ T . (As this is the only kind of tree considered below, we omit the word “binary”.) The set of (infinite) paths of T is

[T ] = {f ∈ 2^{ℕ} : (∀n)f ↾ n ∈ T }.

(ii) A tree T is extendible if for every σ ∈ T , at least one of σ0 and σ1 is in T . (In other words, T is extendible if every σ ∈ T can be extended to an infinite path of T .)

(iii) Suppose T is extendible. A node τ ∈ T is an atom of T if for each n ⩾ |τ|, there is exactly one σ ∈ T of length n extending τ. In other words, τ ∈ T is an atom of T if there is only one f ∈ [T ] extending τ. We say that this f is isolated, that τ is a generator of f, and that τ isolates or generates f.

(iv) An extendible tree T is atomic if for every σ ∈ T there is an atom τ ∈ T extending σ. It is strongly atomic if for every finite collection σ_{0}, …, σ_{n} ∈ T there is a finite collection τ_{0}, …, τ_{n} of atoms of T such that each τ_{i} extends σ_{i}. (Of course, every atomic tree is strongly atomic, but this fact cannot be proved in RCA_{0}; see Appendix B.)

3. The Atomic Model Theorem and related principles

In this section, we discuss a few principles that play a significant role below. As mentioned in the introduction, Hirschfeldt, Shore, and Slaman [23] studied the reverse mathematical strength of the following basic model theoretic fact.

AMT (Atomic Model Theorem): Every atomic theory has an atomic model.

They showed that AMT is a fairly weak principle, but is nevertheless not provable in WKL_{0}. (See [23] for details.)

AMT is also closely related to the following genericity principle. (When we say that D is a Π_{1}^{0} set of strings, we mean that D is a Π_{1}^{0} predicate on 2^{<ℕ}, and write σ ∈ D to mean that D holds of σ. This notation does not imply that D exists as a set.)

Π_{1}^{0}G: For any uniformly Π_{1}^{0} collection of sets D_{i}, each of which is dense in 2^{<ℕ}, there is a G such that (∀i)(∃m)[G ↾ m ∈ D_{i}].

Conidis [6] showed that AMT and Π_{1}^{0}G are computability theoretic equivalent. In the context of reverse mathematics, his result shows that RCA_{0}+AMT and RCA_{0}+Π_{1}^{0}G have the same ω-models. Subsequently, Hirschfeldt, Shore, and Slaman [23] showed that Π_{1}^{0}G implies AMT over RCA_{0}, while AMT implies Π_{1}^{0}G over RCA_{0} + IΣ_{2}^{0}, but not over RCA_{0}. They also established the following conservativity results.

Theorem 3.1 (Hirschfeldt, Shore, and Slaman [23]).

(1) AMT and Π_{1}^{0}G are restricted Π_{2}^{1}-conservative over RCA_{0}, i.e., conservative over RCA_{0} for sentences of the form

(∀A)[Θ(A) → (∃B)Φ(A, B)]

where Θ is arithmetic and Φ is Σ_{3}^{0}. In particular, AMT and Π_{1}^{0}G are Π_{1}^{1}-conservative over RCA_{0}.

(2) AMT is Π_{1}^{1}-conservative over RCA_{0} + BΣ_{2}^{0}.

On the other hand, they showed that Π_{1}^{0}G implies IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0}. In Section 6.2, we discuss several model theoretic principles provable from RCA_{0}+Π_{1}^{0}G that likewise imply IΣ_{2}^{0} over RCA_{0}+BΣ_{2}^{0}. Unlike Π_{1}^{0}G, however, these principles are true in every ω-model of RCA_{0}, and indeed are provable in RCA_{0} + IΣ_{2}^{0}. It thus becomes interesting to find a combinatorial principle, along the lines of Π_{1}^{0}G, living in that same part of the reverse mathematical universe. We obtain such a principle by replacing the statement that Π_{1}^{0}-generics exist by the statement that they can be approximated:

Π_{1}^{0}GA: For any uniformly Π_{1}^{0} collection of sets of strings D_{0}, D_{1}, …, each of which is dense in 2^{<ℕ}, there is a sequence g_{0}, g_{1}, … ∈ 2^{ℕ} such that

(∀i)(∃m)(∃t)(∀u > t)[g_{u} ↾ m = g_{t} ↾ m ∈ D_{i}].

Note that this principle is equivalent to the version where we add the requirement that (∀n)(∃t)(∀u > t)[g_{u} ↾ n = g_{t} ↾ n], since we can add to the list D_{0}, D_{1}, … all sets of the form {σ : |σ| ⩾ n}, and if m ⩾ n, then g_{u} ↾ m = g_{t} ↾ m implies g_{u} ↾ n = g_{t} ↾ n.

Of course, Π_{1}^{0}G implies Π_{1}^{0}GA, so conservativity results for the former remain true of the latter. In particular, Π_{1}^{0}GA does not imply BΣ_{2}^{0} over RCA_{0}. The following two theorems show that Π_{1}^{0}GA is in fact equivalent to IΣ_{2}^{0} over BΣ_{2}^{0}.

Theorem 3.2. RCA_{0} + IΣ_{2}^{0} ⊢ Π_{1}^{0}GA.

Proof. We argue in RCA_{0} + IΣ_{2}^{0}. Let D_{0}, D_{1}, … be a uniformly Π_{1}^{0} collection of sets of strings, each of which is dense in 2^{<ℕ}. Let D_{i}[s] be the stage s approximation to D_{i}. For each s, define σ_{s,i} for i ⩽ s by recursion as follows. Let σ_{s,0} be the length-lexicographically least string in D_{0}[s]. Given σ_{s,i−1}, let σ_{s,i} be the length-lexicographically least extension of σ_{s,i−1} in D_{i}[s]. Each D_{i}[s] contains D_{i}, and hence is dense, so the σ_{s,i} are all well-defined. Let g_{s} = σ_{s,s}0^{ℕ}.

Consider the statement

(∃t ⩾ i)(∀u > t)[σ_{u,i} = σ_{t,i}].

This statement clearly holds of i = 0, since there are a string σ and a t such that σ ∈ D_{0} and τ ∉ D_{0}[t] for all τ that are lengthlexicographically less than σ. By a similar argument, if the statement holds of i, then it holds of i + 1. Thus, by IΣ_{2}^{0}, the statement holds of all i. Fix i and let t be as in the above statement. Then σ_{t,i} ∈ D_{i}, since σ_{t,i} = σ_{u,i} ∈ D_{i}[u] for all u ⩾ t. Furthermore, g_{u} extends σ_{t,i} for all u ⩾ t, so (∃m)(∃t)(∀u > t)[g_{u} ↾ m = g_{t} ↾ m ∈ D_{i}]. □

The following proof is similar to that of the corresponding result for Π_{1}^{0}G in Hirschfeldt, Shore, and Slaman [23].

Theorem 3.3. RCA_{0} + BΣ_{2}^{0} + Π_{1}^{0}GA ⊢ IΣ_{2}^{0}, and hence RCA_{0} + BΣ_{2}^{0} ⊢ Π_{1}^{0}GA ↔ IΣ_{2}^{0}.

Proof. Fix a model of RCA_{0} + BΣ_{2}^{0} + Π_{1}^{0}GA with first order part M and second order part S. Suppose for the sake of a contradiction that IΣ_{2}^{0} fails in this model. Then there is a Σ_{2}^{0} cut I in M; i.e., x ∈ I ⇔ (∃y)(∀z)φ(x, y, z) for some Δ_{0}^{0} formula φ and I is a proper initial segment of M closed under successor. Let D_{i} be the set of all σ ∈ 2^{<ℕ} such that, thinking of σ as a finite set, the last two elements of σ, in order, are w and ⟨i, x, y⟩, and

(∃y′ < y)(∀z)[φ(x, y′, z) ∧ (∀y′ ⩽ w)(∃z′ < y)¬φ(x, y′, z′)].

Clearly the D_{i} are uniformly Π_{1}^{0}. As I is a cut and so not a member of S, there is no bound on the least witnesses y such that (∀z)φ(x, y, z) for x ∈ I, while by IΣ_{1}^{0} every x ∈ I has such a least witness y, and by BΣ_{2}^{0} there is a bound on the z′ needed to show that y is in fact the least witness for x. Thus each D_{i} is dense, for if we consider any τ with last element w there is an x ∈ I whose least witness y is larger than w and a z > y such that counterexamples for all y′ ⩽ w can be found below z. So if we let σ extend τ by adding on as its only new element ⟨i, x, z⟩ we have our extension of τ in D_{i} as required.

Let g_{0}, g_{1}, … be as in Π_{1}^{0}GA and let a be any number above all those in I. Let E_{i} = {⟨m, t⟩ : (∀u ⩾ t)[g_{u} ↾ m = g_{t} ↾ m ∈ D_{i}[u]]}. Note that the E_{i} are uniformly Π_{1}^{0}. We have (∀i < a)(∃⟨m, t⟩)⟨m, t⟩ ∈ E_{i}, so by the finite axiom of choice for Π_{1}^{0} properties, there is a function f with domain a that is M-finite (i.e., coded in M by a number) and such that for each i < a there is an ⟨m, t⟩ ∈ E_{i} with the last element of g_{t} ↾ m of the form ⟨i, f(i), y⟩. Since ⟨m, t⟩ ∈ E_{i} ⇒ g_{t} ↾ m ∈ D_{i}, every f(i) is in I.

We claim that f is one-to-one. Since I is a proper subset of a, this fact gives us a contradiction, as we can show in RCA_{0} that no such function exists. So assume for a contradiction that f(i) = f(j) = x for i ≠ j. Then there are ⟨m, t⟩ ∈ E_{i} and ⟨m′, t′⟩ ∈ E_{j}, and there are v, w, y_{i}, y_{j}, such that the next to last and last elements of g_{t} ↾ m are v and ⟨i, x, y_{i}⟩, and the next to last and last elements of g_{t′} ↾ m′ are w and ⟨j, x, y_{j}⟩. Let u = max(t, t′). By the definitions of E_{i} and E_{j}, we have g_{u} ↾ m = g_{t} ↾ m and g_{u} ↾ m′ = g_{t′} ↾ m′. Thus we cannot have m = m′, since g_{t} ↾ m and g_{t′} ↾ m′ have different last elements. So, without loss of generality, we may assume that m < m′ and so v < ⟨i, x, y_{i}⟩ ⩽ w < ⟨j, x, y_{j}⟩. We now have a contradiction, as the definitions of D_{i} and D_{j} then say that (∃y′ < y_{i})(∀z)φ(x, y′, z) and (∀y′ ⩽ w)(∃z′ < y)¬φ(x, y′, z′), respectively, but y_{i} ⩽ w. □

Corollary 3.4. AMT does not imply Π_{1}^{0}GA over RCA_{0}.

One can think of Π_{1}^{0}GA as an attempt to capture the first order difference between AMT and Π_{1}^{0}G, but we do not know whether AMT+ Π_{1}^{0}GA implies Π_{1}^{0}G over RCA_{0}.

There are principles analogous to Π_{1}^{0}GA for higher levels of the arithmetic hierarchy, and results corresponding to the above two theorems hold for these principles. Since these results, while of independent interest, are not needed in the rest of this paper, we leave them to Appendix A.

Let T be an atomic theory. From an atomic model of T, we can easily obtain a listing of the principal types of T. By applying Shore blocking to a priority argument of Goncharov and Nurtazin [16] and Harrington [19], Hirschfeldt, Shore, and Slaman [23] showed that it is also provable in RCA_{0} that if there is a listing of the principal types of T, then T has an atomic model. Thus AMT can be restated as “If T is an atomic theory, then there is a listing of the principal types of T.”

There is a fairly straightforward translation between theories and extendible trees. In one direction, given a theory T, we can consider the tree T of types of T. More precisely, we define T as follows. For each n, let φ_{0}^{n}(x_{0}, …, x_{n−1}), φ_{1}^{n}(x_{0}, …, x_{n−1}), … list the formulas with free variables among x_{0}, …, x_{n−1} in the language of T. For σ ∈ 2^{<ℕ}, let

φ_{σ}^{n} ≡ ⋀_{i:σ(i)=1} φ_{i}^{n} ∧ ⋀_{i:σ(i)=0} ¬φ_{i}^{n}.

Let T consist of all strings of the form 0^{n} and all strings of the form 0^{n}1σ such that φ_{σ}^{n} is consistent with T. It is easy to show in RCA_{0} that T is an extendible tree, that if T is atomic then T is strongly atomic, and that if there is a listing of the isolated paths of T , then there is a listing of the principal types of T.

In the other direction, we have the following construction.

Definition 3.5. Let L be the language {P_{i}}_{i∈ℕ}, where each P_{i} is a unary relation symbol. For σ ∈ 2^{<ℕ}, let P^{σ}(y) be the formula ⋀_{n=0}^{|σ|−1} P_{n}^{σ(n)}(y), where P_{n}^{1}(y) denotes the formula P_{n}(y) and P_{n}^{0}(y) denotes the formula ¬P_{n}(y). For a k-tuple s = (σ_{0}, …, σ_{k−1}) of elements of 2^{<ℕ}, let P^{s}(y_{0}, … y_{k−1}) be the formula ⋀_{i<k} P^{σ_{i}}(y_{i}).

Given an extendible tree T ⊆ 2^{<ℕ}, let T (T ) be the deductive closure in L of the following set R(T ) of axioms.

(1) (∃^{>m}x)P^{σ}(x) for all σ ∈ T and m ∈ ℕ.

(2) ¬(∃x)P^{σ}(x) for all σ ∉ T .

We can clearly prove the existence of R(T ) in RCA_{0}. The following lemma shows that we can do more, and establishes the correspondence between isolated paths of T and principal types of T (T ).

Proposition 3.6. The following hold in RCA_{0} for an extendible tree T .

(1) T (T ) exists and is a theory, i.e., complete and consistent.

(2) If T is strongly atomic, then T (T ) is an atomic theory.

(3) From an enumeration of the principal types of T (T ), we can build an enumeration of the isolated paths of T .

Proof. We argue in RCA_{0}.

(1) We first show that R(T ) admits effective quantifier elimination, as discussed at the end of Section 2.2. Fix an existential formula (∃y)φ(x̄y), where φ(x̄y) is a conjunction of literals. Let m be such that no P_{i} with i > m occurs in φ, and let L′ be the language consisting of all P_{i} with i ⩽ m. As discussed in Remark 2.3, we may assume that φ is an L′-atomic diagram, and that it includes ¬x = y for all x ∈ x̄. Let σ be such that P^{σ}(y) equals the conjunction of literals in φ of the form P_{i}(y), and let ψ be the conjunction of literals in φ not involving y (which is just T if there are no such literals). It is easy to see that if σ ∈ T then R(T ) ⊢ (∃y)φ(x̄y) ↔ ψ, and otherwise, R(T ) ⊢ (∃y)φ(x̄y) ↔ F. Thus R(T ) admits effective quantifier elimination, and so its deductive closure exists and is complete if consistent, and hence a theory as required.

To show that T (T ) is consistent, we provide a model A of T (T ). Since T is extendible, there is a function f : 2^{<ℕ} × ℕ → 2 such that for each σ ∈ T , if we think of f(σ, ·) as an element α of 2^{ℕ}, then α is the leftmost path of T extending σ. Let the domain of A be ℕ. Since R(T ) admits effective quantifier elimination and there are only unary relation symbols, we need to specify only what unary relations hold of each n ∈ ℕ. Let {τ_{i}}_{i∈ℕ} be an enumeration of T . For each i, j, k, let P_{k}^{f(τ_{i},k)}(⟨i, j⟩) hold in A. (We assume our pairing function is bijective.) Clearly A is a model of T (T ), and hence T (T ) is consistent.

(2) Suppose that T is strongly atomic, and fix a formula θ(x_{0}, …, x_{n}) consistent with T (T ). By quantifier elimination, we may assume that θ is quantifier-free, and hence that it is a disjunction of conjunctions of literals. Since any disjunction is implied by each of its disjuncts, and θ has at least one disjunct that is consistent with T (T ), we may in fact assume that θ is a conjunction of literals. Since we are working in a language with only unary relation symbols, we can split θ into formulas θ_{0}(x_{0}), …, θ_{n}(x_{n}) consistent with T (T ), and it suffices to show that each θ_{i} is implied over T (T ) by an atom of T (T ). Since each θ_{i}(x_{i}) is a conjunction of literals consistent with T (T ), there are σ_{0}, …, σ_{n} ∈ T such that each P^{σ_{i}}(x_{i}) logically implies θ_{i}(x_{i}). Since T is strongly atomic, there are atoms τ_{0}, …, τ_{n} of T such that τ_{i} extends σ_{i}. It is easy to check that each P^{τ_{i}}(x_{i}) is an atom of T (T ) and implies θ_{i}(x_{i}) over T (T ).

(3) From an enumeration of the principal types of T (T ), we can build an enumeration p_{0}(x), p_{1}(x), … of the principal 1-types of T (T ) in the variable x. Let f_{i}(k) = 1 if P_{k}(x) ∈ p_{i}, and let f_{i}(k) = 0 otherwise. Then it is easy to see that f_{0}, f_{1}, … is an enumeration of the isolated paths of T . □

Thus we can further restate AMT as “If T is a strongly atomic extendible tree, then there is a listing of the isolated paths of T .” If we weaken the hypothesis of this statement, we get a different principle:

ATT (Atomic Tree Theorem): If T is an atomic extendible tree, then there is a listing of the isolated paths of T .

ATT implies AMT by definition. The finite axiom of choice for Π_{1}^{0} properties, and hence BΣ_{2}^{0}, implies that every atomic tree is strongly atomic. (We show in Appendix B that this statement is in fact equivalent to BΣ_{2}^{0}.) Thus ATT is provable in RCA_{0} + BΣ_{2}^{0} + AMT. It is easy to see that ATT is also provable in RCA_{0} + Π_{1}^{0}G. These two facts together imply that ATT is restricted Π_{2}^{1}-conservative over RCA_{0} and Π_{1}^{1}-conservative over RCA_{0} + BΣ_{2}^{0}. They also imply that ATT is strictly weaker than both Π_{1}^{0}G and AMT + BΣ_{2}^{0}.

We do not know whether AMT (or even AMT + Π_{1}^{0}GA) implies ATT over RCA_{0}. Assuming AMT does not imply ATT, ATT is strictly intermediate in strength between AMT and AMT+BΣ_{2}^{0}, which suggests the idea of “miniaturizing” ATT, as we did to Π_{1}^{0}G to obtain Π_{1}^{0}GA, yielding a principle strictly weaker than BΣ_{2}^{0} but not provable in RCA_{0}. We show in Appendix B that this idea can indeed be implemented.

4. Defining homogeneity

In this section, we compare several classically equivalent definitions of homogeneity.

Definition 4.1. Let M be a model. By an M-tuple we mean a tuple of elements of M. For M-tuples ā and b̄, we write ā ≡ b̄ to mean that ā and b̄ have the same type. (If the language of M does not include equality, then the definition of ≡ should also include the requirement that (a_{0}, …, a_{n}) ≡ (b_{0}, …, b_{n}) implies that a_{i} = a_{j} if and only if b_{i} = b_{j}.)

(i) A model M is 1-point homogeneous if for each pair of M-tuples ā ≡ b̄ and each c ∈ M, there is a d ∈ M such that āc ≡ b̄d.

(ii) A model M is 1-homogeneous if for each pair of M-tuples ā ≡ b̄

and each M-tuple c̄, there is an M-tuple d̄ such that āc̄ ≡ b̄d̄. (iii) A model M is homogeneous if for each finite sequence of pairs

of M-tuples ā_{0} ≡ b̄_{0}, …, ā_{n} ≡ b̄_{n} and each finite sequence of Mtuples c̄_{0}, …, c̄_{n}, there are M-tuples d̄_{0}, …, d̄_{n} such that ā_{i}c̄_{i} ≡ b̄_{i}d̄_{i} for all i ⩽ n.

(iv) A model M is strongly 1-homogeneous if for each pair of Mtuples ā ≡ b̄, there is an automorphism of M taking ā to b̄.

These definitions are easily seen to be classically equivalent by back and forth constructions. ACA_{0} is clearly powerful enough to carry out these constructions, so all of our definitions are equivalent in ACA_{0}. However, as we show in this section, in RCA_{0} the situation is different. Of course, in RCA_{0} we do have (iii) ⇒ (ii) ⇒ (i), as well as (iv) ⇒ (ii). We now consider the remaining implications. Figure 1 summarizes our results. The entry in row R and column C describes the strength of the principle stating that every model with property R has property C, in terms of a principle equivalent to it over RCA_{0}.

1-pt. hom 1-hom. hom. strongly 1-hom.
1-pt. hom  IΣ_{2}^{0} IΣ_{2}^{0}  ACA_{0}
1-hom.  RCA_{0}  BΣ_{2}^{0}  ACA_{0}
hom.  RCA_{0} RCA_{0}  ACA_{0}
strongly 1-hom. RCA_{0} RCA_{0} BΣ_{2}^{0}

Figure 1. Versions of homogeneity

Definition (i) is the standard one, but (iii) is better behaved than either (i) or (ii) from the reverse mathematical point of view. In particular, it is the one that yields a version of HMT equivalent to AMT, as shown in Theorem 7.6. By Theorem 4.3 below, the implication from (i) to (ii) is equivalent to IΣ_{2}^{0}, and by Theorem 4.4, the implications from (ii) to (iii) and from (iv) to (iii) are both equivalent to BΣ_{2}^{0}. Since (i)–(iii) assert the existence of first order objects, while (iv) asserts the existence of a second order object (namely, an automorphism of M), it is not surprising that the implications from (i)–(iii) to (iv) are at the level of ACA_{0}. We have already noted that they all follow from ACA_{0}. We now show that the weakest of them implies ACA_{0}, using an example built by Hirschfeldt, Shore, and Slaman [23, Theorem 2.3] to show that the statement that any two atomic models of the same theory are isomorphic implies ACA_{0} over RCA_{0}.

Proposition 4.2. The statement that every homogeneous model is strongly 1-homogeneous, i.e., (iii) ⇒ (iv) in Definition 4.1, implies ACA_{0} over RCA_{0}.

Proof. The example in [23] consists of two models, A and B, in a language with unary relation symbols R_{i} and R_{i,j} for i, j ∈ ℕ, defined as follows. The domains of both A and B are ℕ and in each the interpretation of R_{i} is {⟨i, n⟩ | n ∈ ℕ}. Let ∅′_{s} be the set of elements entering the halting set ∅′ by stage s of its enumeration. (The basic constructions of computability theory, including that of the sequence ∅′_{0}, ∅′_{1}, …, can be carried out in RCA_{0}; see for instance [20] or [47].) If i ∉ ∅′_{s} then both interpretations of R_{i,s} are empty. Suppose that i enters ∅′ at stage t. In A we put {i} × [0, t] into R_{i,s} for each s ⩾ t, split the rest of {i} × ℕ into two computable infinite pieces, put one into R_{i,s} for each s ⩾ t, and keep the other out of all the R_{i,s}. In B we keep {i} × [0, t] out of all the R_{i,s}, split the rest of {i} × ℕ into two computable infinite pieces, put one into R_{i,s} for each s ⩾ t, and keep the other out of all the R_{i,s}.

We modify this example by adding a binary relation symbol E to the language, and letting M be the model obtained by letting the interpretation of E be an equivalence relation splitting the domain into two computable infinite equivalence classes, one of which is a copy of A and the other a copy of B. To be precise, let M = ℕ and let E^{M}(x, y) hold if and only if x and y are both even or both odd. Let R_{i}(2n) hold in M if and only if R_{i}(n) holds in A, let R_{i}(2n + 1) hold in M if and only if R_{i}(n) holds in B, and act similarly for R_{i,s}.

A simple modification of the argument in the proof of Theorem 2.3 of [23] shows that M provably exists in RCA_{0}. (All we need to add to the proof of quantifier elimination for the theory of M is a check of the consistency restrictions imposed by the axioms for the equivalence relation E, as discussed in Remark 2.3.) It is also easy to show in RCA_{0} that there are a, b ∈ M such that a ≡ b and ¬E^{M}(a, b), so if M is strongly 1-homogeneous, then A ≅ B, which implies ACA_{0}, as shown in the proof of Theorem 2.3 of [23]. Thus it suffices to argue in RCA_{0} that M is homogeneous.

It is not difficult to see that, working in RCA_{0}, we can define a function f such that for each a, x, and i < 2, we have f(a, x, i) > x and f(a, x, i) ≡ a (i.e., the elements f(a, x, i) and a of M have the same 1-type), and E^{M}(a, f(a, x, i)) if and only if i = 1.

Let ā_{0} ≡ b̄_{0}, …, ā_{n} ≡ b̄_{n} and c̄_{0}, …, c̄_{n} be M-tuples. For each m ⩽ n, proceed as follows. Let ā_{m} = (a_{m,0}, …, a_{m,k}), let b̄_{m} = (b_{m,0}, …, b_{m,k}), and let c̄_{m} = (c_{m,0}, …, c_{m,l}). For each j ⩽ l, proceed as follows. If c_{m,j} = a_{m,j′} for some j′, then let d_{m,j} = b_{m,j′}. Otherwise, let x > b_{m,p} for all p ⩽ k. If E^{M}(a_{m,0}, b_{m,0}) then let d_{m,j} = f(c_{m,j}, x, 1); otherwise, let d_{m,j} = f(c_{m,j}, x, 0).

Let d̄_{m} = (d_{m,0}, …, d_{m,l}). It is easy to check that ā_{m}c̄_{m} ≡ b̄_{m}d̄_{m} for all m ⩽ n. Thus M is homogeneous. □

The remaining implications between our definitions of homogeneity are more subtle, involving induction and bounding principles.

Theorem 4.3. The statement that every 1-point homogeneous model is 1-homogeneous, i.e., (i) ⇒ (ii) in Definition 4.1, is equivalent to IΣ_{2}^{0} over RCA_{0}.

Proof. Let M be 1-point homogeneous. Let ā ≡ b̄ and c̄ = (c_{0}, …, c_{n}) be M-tuples. For each i ⩽ n, the existence of a sequence d_{0}, …, d_{i−1} such that (ā, c_{0}, …, c_{i−1}) ≡ (b̄, d_{0}, …, d_{i−1}) is a Σ_{2}^{0} property of i, so Σ_{2}^{0}-induction suffices to show that such sequences exist for all i, in particular i = n, which implies that M is 1-homogeneous.

We now argue in RCA_{0} to obtain the reversal. We use the equivalence between IΣ_{2}^{0} and finite Π_{1}^{0}-recursion mentioned in Section 2.1. Let φ be a Π_{1}^{0} formula defining a total function, and fix z and n. We build a 1-point homogeneous model M such that if M is 1-homogeneous then there is a sequence x_{0}, …, x_{n} with x_{0} = z and φ(x_{i}, x_{i+1}) for all i < n.

We work in the language with unary relation symbols P_{i} for i ∈ ℕ and a binary relation symbol E. Let A be the following set of axioms, which we write informally, as formalizing them in our language is straightforward.

(1) P_{i+1}(x) → P_{i}(x) for all i ∈ ℕ.

(2) E is an equivalence relation with two classes.

(3) Within each E-equivalence class, for each i ∈ ℕ there are infinitely many x such that P_{i}(x) ∧ ¬P_{i+1}(x), and there are infinitely many x such that ¬P_{0}(x).

It is easy to show that A admits effective quantifier elimination, as discussed at the end of Section 2.2: Fix an existential formula (∃y)φ(x̄y), where φ(x̄y) is a conjunction of literals. Let m be such that no P_{i} with i > m occurs in φ, and let L′ be the language consisting of E and all P_{i} with i ⩽ m. As mentioned in Remark 2.3, we may assume that φ is an L′-atomic diagram that includes ¬x = y for all x ∈ x̄ and is consistent with the assertion that E is an equivalence relation. Let ∼ be the equivalence relation on the variables in x̄y defined by letting u ∼ v hold if and only if φ contains E(u, v). If ∼ has more than two equivalence classes, or φ contains both P_{j}(y) and ¬P_{i}(y) for i < j, then A ⊢ (∃y)φ(x̄y) ↔ F. Otherwise, let ψ be the result of removing from φ all literals involving y (and hence is just T if all literals in ψ involve y). Then it is easy to see that A ⊢ (∃y)φ(x̄y) ↔ ψ.

We now specify a 1-point homogeneous model M of A. By effective quantifier elimination, it suffices to specify the atomic diagram of M. Let s_{0}, s_{1}, … list all nonempty finite sequences of natural numbers with first element z. Let θ(k) hold if and only if, writing s_{k} = (z = m_{0}, …, m_{l−1}), we have φ(m_{i}, m_{i+1}) for all i < l − 1. Note that θ is Π_{1}^{0}, so there is a sequence of sets R_{0} ⊇ R_{1} ⊇ ⋯ such that θ(k) if and only if k ∈ R_{i} for all i. We may assume that for each i, there are infinitely many k such that i is the least number for which k ∉ R_{i}. Note also that if θ(k) and θ(k′) both hold for k ≠ k′, then s_{k} and s_{k′} must have different lengths, since φ defines a function.

Let the domain M of M be all numbers of the form ⟨i, j, k⟩ with i, j ∈ {0, 1} and k ∈ ℕ. Let E^{M} hold of two such numbers if and only if they have the same first coordinate. Let P_{m}^{M}(⟨i, 0, k⟩) hold if and only if k = ⟨l, x⟩ for some l > m and x ∈ ℕ. Let P_{m}^{M}(⟨0, 1, k⟩) hold for all k ∈ ℕ. Let P_{m}^{M}(⟨1, 1, k⟩) hold if and only if k ∈ R_{m}. Clearly, M is a model of A.

To see that M is 1-point homogeneous, let ā = (a_{0}, …, a_{m−1}) ≡ b̄ = (b_{0}, …, b_{m−1}) be M-tuples and let c ∈ M. If c = a_{i} for some i < m then we can let d = b_{i} and have āc ≡ b̄d, so we may assume that is not the case. Say that ā and b̄ are on opposite sides if there is an i < m such that a_{i} and b_{i} have different first coordinates (in which case, the same must hold of all pairs a_{i}, b_{i}). Otherwise, say that ā and b̄ are on the same side. Let l be large enough so that every number of the form ⟨i, j, x⟩ for x > l is bigger than all the b_{i}. We have several cases.

(1) If ā and b̄ are on the same side and c is of the form ⟨i, 0, ⟨k, x⟩⟩, then let d = ⟨i, 0, ⟨k, x′⟩⟩ for some x′ such that ⟨k, x′⟩ > l.

(2) If ā and b̄ are on the same side and c is of the form ⟨0, 1, j⟩, then let d = ⟨0, 1, j′⟩ for some j′ > l.

(3) If ā and b̄ are on the same side, c is of the form ⟨1, 1, j⟩, and θ(j) does not hold, then let k be least such that j ∉ R_{k} and let d = ⟨1, 0, ⟨k, x⟩⟩ for some x such that ⟨k, x⟩ > l.

(4) If ā and b̄ are on opposite sides and c is of the form ⟨i, 0, ⟨k, x⟩⟩, then let d = ⟨1 − i, 0, ⟨k, x′⟩⟩ for some x′ such that ⟨k, x′⟩ > l.

(5) If ā and b̄ are on opposite sides, c is of the form ⟨1, 1, j⟩, and θ(j) holds, then let d = ⟨0, 1, j′⟩ for some j′ > l.

(6) Finally, if ā and b̄ are on opposite sides and c is of the form ⟨0, 1, j⟩, or ā and b̄ are on the same side, c is of the form ⟨1, 1, j⟩, and θ(j) holds, then proceed as follows. Let s be the longest string such that s = s_{j′} for some j′ such that θ(j′) and b_{i} = ⟨1, 1, j′⟩ for some i < m, if there is such a string, and let s be the empty string otherwise. (Note that we can pick out the set of j′ such that θ(j′) and b_{i} = ⟨1, 1, j′⟩ for some i < m by bounded Π_{1}^{0}- comprehension.) Then, by the hypothesis on φ, there is a string s_{j′′} of length |s| + 1 such that θ(j′′) holds. Let d = ⟨1, 1, j′′⟩, and note that d ≠ b_{i} for all i < m.

In any case, it is easy to check that āc ≡ b̄d.

Now suppose that M is 1-homogeneous. Since clearly ⟨0, 0, 0⟩ ≡ ⟨1, 0, 0⟩, there exist d_{0}, …, d_{n−1} such that

(⟨0, 0, 0⟩, ⟨0, 1, 0⟩, …, ⟨0, 1, n − 1⟩) ≡ (⟨1, 0, 0⟩, d_{0}, …, d_{n−1}).

We must have d_{i} = ⟨1, 1, k_{i}⟩ for each i < n. The numbers k_{i} are all distinct and all satisfy θ, so at least one s_{k_{i}} has length at least n. This string witnesses the fact that the instance of Π_{1}^{0}-recursion corresponding to φ, n, and z holds. □

Theorem 4.4. The following are equivalent over RCA_{0}.

(1) BΣ_{2}^{0}

(2) Every 1-homogeneous model is homogeneous, i.e., (ii) ⇒ (iii) in Definition 4.1.

(3) Every strongly 1-homogeneous model is homogeneous, i.e., (iv) ⇒ (iii) in Definition 4.1.

Proof. We first argue in RCA_{0} + BΣ_{2}^{0} to prove (2), and hence (3). Let M be 1-homogeneous, let ā_{0} ≡ b̄_{0}, …, ā_{n} ≡ b̄_{n} be pairs of M-tuples, and let c̄_{0}, …, c̄_{n} be M-tuples. By 1-homogeneity, for each i ⩽ n, there is an M-tuple d̄ such that ā_{i}c̄_{i} ≡ b̄_{i}d̄. The condition that two given Mtuples have the same type is Π_{1}^{0}, so by the version of the finite axiom of choice mentioned in Section 2.1, there is a sequence of M-tuples d̄_{0}, …, d̄_{n} such that ā_{i}c̄_{i} ≡ b̄_{i}d̄_{i} for each i ⩽ n.

We now argue in RCA_{0} to show that (3) implies BΣ_{2}^{0}, and hence so does (2). Assume that every strongly 1-homogeneous model is homogeneous. We show that BΠ_{1}^{0} holds. (Recall that BΠ_{1}^{0} and BΣ_{2}^{0} are equivalent over RCA_{0}.) Let Ψ be a Π_{1}^{0} formula and n a number such that (∀i < n)(∃u)Ψ(i, u). We build a strongly 1-homogeneous model M such that if M is homogeneous then there is a v for which (∀i < n)(∃u < v)Ψ(i, u). Let Φ be a quantifier-free formula such that Ψ(i, u) ≡ (∀z)Φ(i, u, z).

A first approach. Since our model is somewhat complicated, we begin by sketching a simpler version that does not prove the theorem but helps motivate our full construction. We work in the language with unary relation symbols U_{i} for i < n and R_{i} for i ∈ ℕ, and a binary relation symbol E. Let Â be the following set of axioms.

(1) The U_{i} partition the universe into pairwise disjoint sets.

(2) E is an equivalence relation. If E(x, y) holds then U_{i}(x) and U_{i}(y) both hold for some i < n, and E splits each U_{i} into two infinite classes.

(3) R_{k+1}(x) → R_{k}(x) for all i ∈ ℕ.

(4) For each i < n and each m ∈ ℕ there are infinitely many x in each E-equivalence class of U_{i} such that R_{k}(x) holds if and only if k < m.

It is easy to show that Â admits effective quantifier elimination. Let N be the model of Â defined as follows. The universe N of N consists of all numbers of the form ⟨i, j, m⟩ with i < n, j ∈ {0, 1}, and m ∈ ℕ. Let U_{i}^{N} (x) hold if and only if the first coordinate of x is i. Let E^{N} (x, y) hold if and only if x and y have the same first and second coordinates. For j ∈ {0, 1}, let R_{k}^{N} (⟨i, j, 2m⟩) hold if and only if k < m. Let R_{k}^{N} (⟨i, 0, 2m + 1⟩) hold for all k. Let R_{k}^{N} (⟨i, 1, 2⟨m, c⟩ + 1⟩) hold if and only if (∃u < m)(∀z < k)Φ(i, u, z).

Note that (∀k)(∃u < m)(∀z < k)Φ(i, u, z) ↔ (∃u < m)Ψ(i, u). (The right-to-left-direction is obvious. For the other direction, suppose that (∀u < m)¬Ψ(i, u). Then for each u < m there is a z_{u} such that ¬Φ(i, u, z_{u}). These z_{u} can be found effectively, so their maximum k exists. Then (∀u < m)(∃z < k + 1)¬Φ(i, u, z).)

Suppose N is homogeneous. Let a_{i} = ⟨i, 0, 0⟩, let b_{i} = ⟨i, 1, 0⟩, and let c_{i} = ⟨i, 0, 1⟩. Then a_{i} ≡ b_{i} for all i < n, so there are d_{0}, …, d_{n−1} such that a_{i}c_{i} ≡ b_{i}d_{i} for all i < n. It follows from the definition of N and the remark in the previous paragraph that each d_{i} is of the form ⟨i, 1, 2⟨m_{i}, c⟩ + 1⟩ for some m_{i} such that (∃u < m_{i})Ψ(i, u). Let m = max_{i<n} m_{i}. Then (∀i < n)(∃u < m)Ψ(i, u).

However, we cannot show that N is strongly 1-homogeneous, or even 1-homogeneous. Indeed, if we assume that N is 1-homogeneous, then we can argue as above to obtain m_{0}, …, m_{n−1}, because (a_{0}, …, a_{n−1}) ≡ (b_{0}, …, b_{n−1}), so there are d_{0}, …, d_{n−1} such that

(a_{0}, …, a_{n−1}, c_{0}, …, c_{n−1}) ≡ (b_{0}, …, b_{n−1}, d_{0}, …, d_{n−1}).

To get around this problem, we introduce new binary relations between elements of different U_{i}’s. The goal is to ensure that if elements (a_{0}, …, a_{n−1}) ≡ (b_{0}, …, b_{n−1}) of a model M can be used to produce m_{0}, …, m_{n−1} using 1-homogeneity as above, then these sequences already contain enough information to obtain m_{0}, …, m_{n−1} without appealing to 1-homogeneity. More generally, we want to ensure that if (a_{0}, …, a_{k−1}) ≡ (b_{0}, …, b_{k−1}) then these sequences contain enough information to obtain an automorphism of M taking each a_{i} to b_{i}.

Axioms. We work in the language with unary relation symbols U_{i} for i < n and R_{i} for i ∈ ℕ, and binary relation symbols E and Q_{i} for i ∈ ℕ. Let A be the following set of axioms, some of which we write informally, as formalizing them in our language is straightforward. The first three families of axioms are the same as above. The fourth ensures that the Q_{i} hold only between elements of different U_{j}’s. The fifth expresses the fact that any existential statement with parameters that is not forbidden by the other axioms holds. This last axiom is necessary to ensure that A admits effective quantifier elimination.

(1) The U_{i} partition the universe into pairwise disjoint sets.

(2) E is an equivalence relation. If E(x, y) holds then U_{i}(x) and U_{i}(y) both hold for some i < n, and E splits each U_{i} into two infinite classes.

(3) R_{i+1}(x) → R_{i}(x) for all i ∈ ℕ.

(4) If Q_{i}(x, y) holds then U_{j}(x) and U_{j′}(y) hold for some j ≠ j′.

(5) Let x_{0}, …, x_{k−1} be distinct elements and let i_{0}, …, i_{k−1} be such that U_{i_{j}}(x_{j}) holds for all j < k. Let

F_{0}, …, F_{k−1}, F̂_{0}, …, F̂_{k−1}, G_{0}, …, G_{k−1}, Ĝ_{0}, …, Ĝ_{k−1}

be finite sets such that F_{j} ∩F̂_{j} = G_{j} ∩Ĝ_{j} = ∅ for all j < k. Then for each i < n and each m ∈ ℕ, there are infinitely many y in each E-equivalence class of U_{i} such that (a) ¬R_{m}(y) and, if m > 0, also R_{m−1}(y);

(b) for each j < k such that i_{j} ≠ i, we have Q_{d}(x_{j}, y) for all

d ∈ F_{j} and ¬Q_{d}(x_{j}, y) for all d ∈ F̂_{j}; and

(c) for each j < k such that i_{j} ≠ i, we have Q_{d}(y, x_{j}) for all

d ∈ G_{j} and ¬Q_{d}(y, x_{j}) for all d ∈ Ĝ_{j}.

Quantifier elimination. To show that A admits effective quantifier elimination, as discussed at the end of Section 2.2, fix a formula (∃y)φ(x̄y), where φ(x̄y) is a conjunction of literals. Let m be such that no R_{i} or Q_{i} with i > m occurs in φ, and let L′ be the language consisting of E, all U_{i} with i < n, and all R_{i} and Q_{i} with i ⩽ m. As mentioned in Remark 2.3, we may assume that φ is an L′-atomic diagram that is consistent with the assertion that E is an equivalence relation, and includes ¬u = v for all distinct u, v ∈ x̄y.

If both U_{i}(u) and U_{j}(u) for j ≠ i occur in φ, or ¬U_{i}(u) occurs in φ for all i < n, then A ⊢ (∃y)φ(x̄y) ↔ F. So we may assume that we can partition x̄y into classes C_{0}, …, C_{n−1} such that if u ∈ C_{i} then U_{i}(u) occurs in φ and ¬U_{j}(u) occurs in φ for all j ≠ i with j < n. Suppose that at least one of the following conditions holds.

(1) Both R_{i}(u) and ¬R_{j}(u) for j < i occur in φ.

(2) E(u, v) occurs in φ, and u ∈ C_{i} and u ∈ C_{j} for i ≠ j.

(3) Q_{i}(u, v) occurs in φ, and u, v ∈ C_{j} for some j < n.

(4) Let ∼ be the equivalence relation on the variables in x̄y defined by letting u ∼ v hold if and only if φ contains E(u, v). Then there is an i < n such that the restriction of ∼ to C_{i} has more than two equivalence classes.

Then A ⊢ (∃y)φ(x̄y) ↔ F. Otherwise, application of the fifth axiom family of A shows that A ⊢ (∃y)φ(x̄y) ↔ θ(x̄), where θ is the result of removing from φ all literals involving y (and hence is just T if all literals in φ involve y).

Thus the deductive closure T of A exists. We build a model M of A below, thus showing T is a theory. First we describe and name some of the types of T.

Types. By effective quantifier elimination, the types of T can be identified with atomic types.

The 1-types of T in a given variable x are then of two kinds. We have the principal type r_{i,j}(x) for i < n and j ∈ ℕ, which is determined by U_{i}(x) and ¬R_{j}(x), and, if j > 0, also R_{j−1}(x). We also have the type r_{i,∞}(x), determined by U_{i}(x) and R_{j}(x) for all j ∈ ℕ. Let r_{i,∞/m}(x) be the type determined by U_{i}(x) and R_{j}(x) if and only if (∃u < m)(∀z < j)Φ(i, u, z). Note that r_{i,∞/m} = r_{i,∞} if (∃u < m)Ψ(i, u) holds, and otherwise r_{i,∞/m} = r_{i,k} for some k ∈ ℕ. We use this type in the same way as in our simplified model N above.

There are uncountably many 2-types of T, but we are interested in only countably many of them. The 2-types p(x, y) that include both U_{i}(x) and U_{i}(y) are all realized in the model M defined below, but we do not need to refer to them directly. To describe the other 2-types in given variables x and y realized in M, we first define the partial 2-type s_{i,j,S}(x, y) for i < n and j ∈ ℕ ∪ {∞}, and a finite or cofinite set S. This type states that x has type r_{i,j} and Q_{m}(x, y) holds if and only if m ∈ S. If S is finite then we say that a 2-type extending this partial type is i-finitary. If S is cofinite then we say that a 2-type extending this partial type is i-infinitary. Now let s_{i_{0},i_{1},j_{0},j_{1},S_{0},S_{1}}(x_{0}, x_{1}) for i_{0} ≠ i_{1} be the 2-type determined by s_{i_{0},j_{0},S_{0}}(x_{0}, x_{1}) ∪ s_{i_{1},j_{1},S_{1}}(x_{1}, x_{0}).

We get around the deficiency of our simplified model N above by ensuring that in M, if x_{0}, x_{1} are not large enough to encode an m such that Ψ(i, u) holds for some u < m, and we have U_{i}(x_{0}), U_{i}(x_{1}), and ¬E(x_{0}, x_{1}), then for some l ∈ {0, 1}, whenever U_{j}(y) holds for j ≠ i, the 2-type of (x_{l}, y) is i-finitary, while the 2-type of (x_{1−l}, y) is i-infinitary. To implement this strategy, we use the following definitions.

Let s_{i,j,S/fin_{d}^{c}}(x, y) be the partial 2-type stating that x has type r_{i,j} and Q_{k}(x, y) holds if and only if k ∈ S and either k < c or (∃u < d)(∀z < k)Φ(i, u, z). Then s_{i,j,S/fin_{d}^{c}} = s_{i,j,S} if Ψ(i, u) holds for some u < d, and otherwise s_{i,j,S/fin_{d}^{c}} = s_{i,j,S′} for some finite set S′. (Of course, if S is finite, we could have S′ = S.)

Let s_{i,j,S/inf_{d}^{c}}(x, y) be the partial 2-type stating that x has type r_{i,j} and Q_{k}(x, y) holds if and only if k ∈ S or both k > c and (∀u < d)(∃z < k)¬Φ(i, u, z). Then s_{i,j,S/inf_{d}^{c}} = s_{i,j,S} if Ψ(i, u) holds for some u < d, and otherwise s_{i,j,S/inf_{d}^{c}} = s_{i,j,S′} for some cofinite set S′. (Of course, if S is cofinite, we could have S′ = S.)

Using these, we define 2-types such as s_{i_{0},i_{1},j_{0},j_{1},S_{0}/fin_{d_{0}}^{c_{0}},S_{1}/inf_{d_{1}}^{c_{1}}}(x_{0}, x_{1}) in the obvious way.

For each of the 2-types we have defined, if j, j_{0}, or j_{1} is ∞, we also consider the corresponding type with ∞ replaced by ∞/m as in the definition of r_{i,∞/m}. For instance, the type s_{i_{0},i_{1},∞/m,j_{1},S_{0},S_{1}}(x_{0}, x_{1}) is defined in the same way as the type s_{i_{0},i_{1},∞,j_{1},S_{0},S_{1}}(x_{0}, x_{1}), except that x_{0} now has type r_{i_{0},∞/m} instead of r_{i_{0},∞}.

Each type of T can be specified by listing its 2-subtypes. Note in particular that, by quantifier elimination, for tuples (a_{0}, …, a_{m}) and (b_{0}, …, b_{m}) of elements of the model M built below, if a_{i}a_{j} ≡ b_{i}b_{j} for all i, j ⩽ m, then (a_{0}, …, a_{m}) ≡ (b_{0}, …, b_{m}).

The model M. We now specify a strongly 1-homogeneous model M of A. By effective quantifier elimination, it suffices to specify the atomic diagram of M.

The domain M of M consists of all numbers of the form ⟨i, j, l, m⟩ with i < n, j ∈ {0, 1}, and l, m ∈ ℕ. For a ∈ M and k ∈ {1, 2, 3, 4}, let π_{k}(a) be the kth coordinate of a.

Let U_{i}^{M}(a) hold if and only if π_{1}(a) = i. Let E^{M}(a, b) hold if and only if π_{1}(a) = π_{1}(b) and π_{2}(a) = π_{2}(b). Let R_{k}^{M}(⟨i, j, l, 2⟨m, c⟩⟩) hold if and only if k < m. Let R_{k}^{M}(⟨i, 0, l, 2m + 1⟩) hold for all k. Let R_{k}^{M}(⟨i, 1, l, 2⟨m, c⟩ + 1⟩) hold if and only if (∃u < m)(∀z < k)Φ(i, u, z). (We assume our pairing function is onto.)

In other words, we give ⟨i, j, l, 2⟨m, c⟩⟩ type r_{i,m}, give ⟨i, 0, l, 2m + 1⟩ type r_{i,∞}, and give ⟨i, 1, l, 2⟨m, c⟩ + 1⟩ type r_{i,∞/m}. This definition ensures that for each i, there is an m_{i} such that all elements of the form ⟨i, 1, l, 2⟨m_{i}, c⟩ + 1⟩ have type r_{i,∞}, but any sequence m_{0}, …, m_{n−1} of such elements can be used to satisfy our instance of BΠ_{1}^{0}.

In defining the interpretations of the Q_{k}, the idea is to use elements with third coordinate l to provide witnesses for the satisfaction of particular 1-types over elements with third coordinates less than l. In particular, we provide witnesses to the satisfaction of the fifth axiom group of A, though we have to do more than that, since we have to realize i-infinitary types while preserving strong 1-homogeneity. To implement the strategy described above for repairing the problem with our previous model N, we also wish to ensure that the following property holds. Let x, y have different first coordinates. Let i = π_{1}(x) and m = π_{4}(x). Suppose that ¬Ψ(i, u) for all u < m. If π_{2}(x) = 0 then the type of (x, y) is i-finitary, and otherwise this type is i-infinitary.

We first define the Q_{k} on pairs of elements with the same third coordinate. For each k, l ∈ ℕ, let Q_{k}^{M}(⟨i, j, l, m⟩, ⟨i′, j′, l, m′⟩) hold if and only if i′ ≠ i and j = 1. That is, we give each pair (⟨i, 0, l, m⟩, ⟨i′, j′, l, m′⟩) with i′ ≠ i a type extending a partial type of the form s_{i,d,∅} (which is i-finitary), and each pair (⟨i, 1, l, m⟩, ⟨i′, j′, l, m′⟩) with i′ ≠ i a type extending a partial type of the form s_{i,d,∞} (which is i-infinitary).

Now let l > 0. Partition the elements of M of the form ⟨i, 0, l, 2m+1⟩ into infinite sets B_{i,0,0}, B_{i,0,1}, . . .. Let m ∈ ℕ. Partition the elements of M of the form ⟨i, 0, l, 2⟨m, c⟩⟩ into infinite sets B_{i,3m+1,0}, B_{i,3m+1,1}, . . .. Partition the elements of M of the form ⟨i, 1, l, 2⟨m, c⟩⟩ into infinite sets B_{i,3m+2,0}, B_{i,3m+2,1}, . . .. Partition the elements of M of the form ⟨i, 1, l, 2⟨m, c⟩ + 1⟩ into infinite sets B_{i,3m+3,0}, B_{i,3m+3,1}, . . .. Assign to each x ∈ ℕ

(1) a tuple (a_{0}, …, a_{k−1}) of elements of M with third coordinates less than l,

(2) a tuple of numbers

(d_{0}, …, d_{k−1}, d′_{0}, …, d′_{k−1}, c_{0}, …, c_{k−1}, c′_{0}, …, c′_{k−1}),

and

(3) a tuple of finite or cofinite sets (S_{0}, …, S_{k−1}, S′_{0}, …, S′_{k−1}) (by which we mean that some of these sets might be finite while others might be cofinite).

We call this 3-tuple, consisting of a tuple of elements, a tuple of numbers, and a tuple of sets, the parameters associated with x. Perform this assignment in such a way that each set of parameters is associated with some x.

Fix x ∈ ℕ and the parameters associated with x as above. Fix i < n. Let a_{k} ∈ ⋃_{j∈ℕ} B_{i,j,x}. Let i_{j} = π_{1}(a_{j}), let e_{j} = π_{2}(a_{j}), and let m_{j} = π_{4}(a_{j}). For j < k, let t_{j} be such that a_{j} has been given type r_{i_{j},t_{j}}. Thus t_{j} could be a natural number, ∞, or ∞/q for some q ∈ ℕ. For j < k, if e_{j} = 0 then let mod_{j} denote fin_{min(d_{j},m_{j})}^{c_{j}}, and otherwise let

mod_{j} denote inf_{min(d_{j},m_{j})}^{c_{j}}. If e_{k} = 0 then let mod_{k} denote fin_{min(d′_{j},m_{k})}^{c′_{j}},

and otherwise let mod_{k} denote inf_{min(d′_{j},m_{k})}^{c′_{j}}.

For each j < k such that i_{j} ≠ i_{k}, define the Q_{c}^{M} on (a_{j}, a_{k}) and (a_{k}, a_{j}) so that (a_{j}, a_{k}) has type s_{i_{j},i_{k},t_{j},t_{k},S_{j}/mod_{j},S′_{j}/mod_{k}}. We say that a_{k} is a witness for the tuple (a_{0}, …, a_{k−1}), as well as for the whole set of parameters associated with x.

For any a ∈ M with π_{3}(a) < l that is not among a_{0}, …, a_{k−1}, let Q_{c}^{M}(a, a_{k}) hold if and only if π_{1}(a) ≠ i_{k} and π_{2}(a) = 1, and let Q_{c}^{M}(a_{k}, a) hold if and only if π_{1}(a) ≠ i_{k} and e_{k} = 1.

We have completed the definition of M. It is clear that M satisfies the first four axiom groups of A. It also satisfies the fifth axiom group because for all elements a_{0}, …, a_{k−1} of M and all finite sets F_{0}, …, F_{k−1}, F̂_{0}, …, F̂_{k−1}, G_{0}, …, G_{k−1}, Ĝ_{0}, …, Ĝ_{k−1}, we eventually assign elements a_{k} as in the previous paragraph. The mod operation in the definition of the type of (a_{j}, a_{k}) is not a problem if the c_{j} are sufficiently large (i.e., larger than all numbers mentioned in the particular instance of the fifth axiom group we are considering).

Type translation. To show that M is strongly 1-homogeneous, we first need to discuss the process of translating types between the equivalence classes of a given U_{i}^{M}, which will help us define the required automorphism of M. Consider the type s = s_{i_{0},i_{1},t_{0},t_{1},S_{0}/fin_{d_{0}}^{c_{0}},S_{1}/inf_{d_{1}}^{c_{1}}}. If

(∃u < d_{0})Ψ(i_{0}, u) then s = s_{i_{0},i_{1},t_{0},t_{1},S_{0},S_{1}/inf_{d_{1}}^{c_{1}}}. In this case, let Ŝ_{0} = S_{0}.

Otherwise, there is a finite set Ŝ_{0} such that s = s_{i_{0},i_{1},t_{0},t_{1},Ŝ_{0},S_{1}/inf_{d_{1}}^{c_{1}}}, and this set can be found by searching for a k such that (∀u < d_{0})(∃z < k)¬Φ(i_{0}, u, z). In either case, if u is least such that Ψ(i_{0}, u) holds, then s = s_{i_{0},i_{1},t_{0},t_{1},Ŝ_{0}/inf_{u+1}^{c_{0}},S_{1}/inf_{d_{1}}^{c_{1}}}. We call this latter type the i_{0}-translate

of s. We call (u + 1, d_{1}, Ŝ_{0}, S_{1}, inf, inf) the conditions of this translate. To find this translate, we need to be able to determine Ŝ_{0} and u. The former can be obtained effectively from u, though, so in fact, all we need is u.

Similarly, if we know the least v such that Ψ(i_{1}, v), then we can find the i_{1}-translate of s, that is, a type s_{i_{0},i_{1},t_{0},t_{1},S_{0}/fin_{d_{0}}^{c_{0}},Ŝ_{1}/fin_{v+1}^{c_{1}}} equal to

s. The conditions of this translate are (d_{0}, v + 1, S_{0}, Ŝ_{1}, fin, fin). If we know both u and v, then we can also find the i_{0}, i_{1}-translate of s, that is, a type s_{i_{0},i_{1},t_{0},t_{1},Ŝ_{0}/inf_{u+1}^{c_{0}},Ŝ_{1}/fin_{v+1}^{c_{1}}} equal to s. The conditions of this

translate are (u + 1, v + 1, Ŝ_{0}, Ŝ_{1}, inf, fin). The same process applies to any type s_{i_{0},i_{1},t_{0},t_{1},S_{0}/mod0_{d_{0}}^{c_{0}},S_{1}/mod1_{d_{1}}^{c_{1}}}, where mod0 and mod1 are each either fin or inf.

For I ⊆ n, the I-translate of a type s as above is s if i_{0}, i_{1} ∉ I, the i_{l}-translate of s if i_{l} ∈ I and i_{1−l} ∉ I, and the i_{0}, i_{1}-translate of s if i_{0}, i_{1} ∈ I.

Strong 1-homogeneity. We now show that M is strongly 1-homoge-neous. Let (a_{0}, …, a_{m−1}) ≡ (b_{0}, …, b_{m−1}). We need to build an automorphism f of M permuting these M-tuples.

Let I be the set of all i for which there is a j < m such that π_{1}(a_{j}) = i (and hence π_{1}(b_{j}) = i) and π_{2}(a_{j}) ≠ π_{2}(b_{j}). (Note that then every a_{j′}, b_{j′} with first coordinate in I have different second coordinates.) It obviously has to be the case that if f(a) = b and π_{1}(a) ∈ I, then π_{2}(a) ≠ π_{2}(b). We build f so that it is also the case that if f(a) = b and π_{1}(a) ∉ I, then π_{2}(a) = π_{2}(b). Say that (a_{0}, …, a_{r−1}) ≡ (b_{0}, …, b_{r−1}) is an I-pair if I is the set of all i for which there is a j < r such that π_{1}(a_{j}) = i and π_{2}(a_{j}) ≠ π_{2}(b_{j}). Thus our starting M-tuples form an I-pair.

We in fact show the following: Let (a_{0}, …, a_{r−1}) ≡ (b_{0}, …, b_{r−1}) be an I-pair, and let a_{r} ∈ M. Then we can effectively find b_{r} ∈ M such that (a_{0}, …, a_{r}) ≡ (b_{0}, …, b_{r}) is an I-pair. (In other words, we can define a function taking each I-pair (a_{0}, …, a_{r−1}) ≡ (b_{0}, …, b_{r−1}) and a_{r} ∈ M to an appropriate b_{r}.)

This fact suffices to build f by the usual back and forth argument: Let f(a_{i}) = b_{i} for i < m. Then let a_{m} be the least element of M not among the a_{i} for i < m, find b_{m} such that (a_{0}, …, a_{m}) ≡ (b_{0}, …, b_{m}) is an I-pair, and let f(a_{m}) = b_{m}. Then let b_{m+1} be the least element of M not among the b_{i} for i ⩽ m, find a_{m+1} such that (a_{0}, …, a_{m+1}) ≡ (b_{0}, …, b_{m+1}) is an I-pair (which we can do because the notion of I-pair is a symmetric relation on tuples), and let f(a_{m+1}) = b_{m+1}. Continue in this way to build an automorphism f of M.

So fix an I-pair (a_{0}, …, a_{r−1}) ≡ (b_{0}, …, b_{r−1}) and a_{r} ∈ M. We show how to obtain a b_{r} as above. As pointed out above, to show that (a_{0}, …, a_{r}) ≡ (b_{0}, …, b_{r}), it is enough to show that a_{i}a_{r} ≡ b_{i}b_{r} for all i < r. If a_{r} = a_{j} for some j < r, then we can let b_{r} = b_{j}, so we suppose this is not the case.

We begin with the following observation. Suppose a, b ∈ M have different first coordinates, and let i be the first coordinate of a. Suppose further that the fourth coordinate of a is a number m such that (∀u < m)¬Ψ(i, u). It is easy to check that the construction of M ensures that if the second coordinate of a is 0 then the 2-type of (a, b) is ifinitary, while if the second coordinate of a is 1 then the 2-type of (a, b) is i-infinitary. Now suppose that ab ≡ cd are pairs of elements of M such that a and b have different first coordinates, and a and c have different second coordinates. Let i be the first coordinate of a (and hence of c) and let m be the maximum of the fourth coordinates of a and c. Then there is a u < m such that Ψ(i, u), as otherwise one of the 2-types of (a, b) and (c, d) would be i-finitary, while the other would be i-infinitary.

We now claim there is a sequence (u_{i})_{i∈I} such that u_{i} is the least number for which Ψ(i, u_{i}) holds. If all the a_{j} with j < r have the same first coordinate, then |I| ⩽ 1, so the claim is true by hypothesis (together with bounded Π_{1}^{0}-comprehension). Otherwise, fix i ∈ I, let j be such that π_{1}(a_{j}) = i and π_{2}(a_{j}) ≠ π_{2}(b_{j}), and let j′ be such that π_{1}(a_{j′}) ≠ i. Since a_{j}a_{j′} ≡ b_{j}b_{j′}, the observation of the previous paragraph implies that, letting m_{i} be the maximum of the fourth coordinates of a_{j} and b_{j}, there is a u < m_{i} such that Ψ(i, u). By bounded Π_{1}^{0}-comprehension (with the bound being max_{i∈I} m_{i}), we can find (u_{i})_{i∈I}. As discussed above, this fact means that we can determine the I-translate of any type of the form s_{i_{0},i_{1},t_{0},t_{1},S_{0}/mod0_{d_{0}}^{c_{0}},S_{1}/mod1_{d_{1}}^{c_{1}}}, where mod0 and mod1 are each either fin or inf.

We begin the process of determining b_{r} by ensuring that it has the same 1-type as a_{r}. To do so, we define a set D from which b_{r} is drawn. We ensure that every element of D has the same 1-type as a_{r}, and has the same second coordinate as a_{r} if and only if π_{1}(a_{r}) ∉ I. We also ensure that D contains witnesses for every set of parameters in the definition of M. (Recall the definition of witnessing a set of parameters from the section of this proof in which we defined the model M.)

Let i = π_{1}(a_{r}), let j = π_{2}(a_{r}), and let k = π_{4}(a_{r}). Let D be the set of all elements of M of the form ⟨i, j′, l, k′⟩ where l ∈ ℕ and j′, k′ are as follows. If i ∉ I then j′ = j; otherwise j′ = 1 − j. If k is of the form 2⟨m, c⟩, then k′ is of the form 2⟨m, d⟩. If i ∉ I, j = 0, and k is odd, then k′ is odd. If i ∉ I, j = 1, and k is of the form 2⟨m, c⟩ + 1, then k′ is of the form 2⟨m, d⟩ + 1. If i ∈ I, j = 0, and k is odd, then k′ is of the form 2⟨u_{i} + 1, d⟩ + 1. If i ∈ I, j = 1, and k is of the form 2⟨m, c⟩ + 1 with m ⩽ u_{i}, then k′ is of the form 2⟨q, d⟩ where q is least such that (∀u < m)(∃z < q)¬Φ(i, u, z) (which must exist by the definition of u_{i}). If i ∈ I, j = 1, and k is of the form 2⟨m, c⟩ + 1 with m > u_{i}, then k′ is odd.

It is easy to check that D has the properties mentioned above. The fact that we choose b_{r} from D is enough to ensure that a_{j}a_{r} ≡ b_{j}b_{r} if π_{1}(a_{j}) = π_{1}(a_{r}), and that if (a_{0}, …, a_{r}) ≡ (b_{0}, …, b_{r}) then these tuples form an I-pair.

Let V be the set of j < r such that π_{1}(a_{j}) ≠ π_{1}(a_{r}), and let W be set of j ∈ V such that either a_{r} is a witness for a tuple that includes a_{j} or a_{j} is a witness for a tuple that includes a_{r}. Let P be the set of all expressions that are either in ℕ, equal to ∞, or equal to ∞/m for some m ∈ ℕ (i.e., all expressions t such that r_{i,t} is one of our named 1-types).

If j ∈ W then from the construction of M we can determine t, t′ ∈ P, finite or cofinite sets S and S′, numbers d, d′, c, c′, and expressions mod0 and mod1, each equal to either fin or inf, such that (a_{j}, a_{r}) has type s_{π_{1}(a_{j}),π_{1}(a_{r}),t,t′,S/mod0_{d}^{c},S′/mod1_{d′}^{c′}}. Let (d_{j}, d′_{j}, Ŝ_{j}, Ŝ′_{j}, mod0j, mod1j) be the conditions of the I-translate of this type. Let c_{j} = c and c′_{j} = c′.

If j ∈ V \ W then from the construction of M we can determine t, t′ ∈ P and sets S and S′, each equal to either ∅ or ℕ, such that (a_{j}, a_{r}) has type s = s_{π_{1}(a_{j}),π_{1}(a_{r}),t,t′,S,S′}. If S = ∅ then let mod0 be fin, and otherwise let mod0 be inf. If S′ = ∅ then let mod1 be fin, and otherwise let mod1 be inf. Then s = s_{π_{1}(a_{j}),π_{1}(a_{r}),t,t′,S/mod0_{0}^{0},S′/mod1_{0}^{0}}.

Let (d_{j}, d′_{j}, Ŝ_{j}, Ŝ′_{j}, mod0j, mod1j) be the conditions of the I-translate of this type. Let c_{j} = c′_{j} = 0.

Now consider the tuples (b_{j})_{j∈V}, (d_{j})_{j∈V} (d′_{j})_{j∈V} (c_{j})_{j∈V} (c′_{j})_{j∈V}, and (S_{j})_{j∈V} (S′_{j})_{j∈V}. At some point in the construction of M, witnesses for this set of parameters are provided. It is easy to check that there is such a witness b_{r} that is taken from D. By the definition of the I-translate, we have a_{j}a_{r} ≡ b_{j}b_{r} for all j ∈ V. As mentioned above, we also have a_{j}a_{r} ≡ b_{j}b_{r} for all j < r that are not in V. Thus (a_{0}, …, a_{r}) ≡ (b_{0}, …, b_{r}), which, as mentioned above, implies that these tuples form an I-pair.

Using homogeneity. Finally, suppose that M is homogeneous. Let a_{i} = ⟨i, 0, 0, 0⟩, let b_{i} = ⟨i, 1, 0, 0⟩, and let c_{i} = ⟨i, 0, 0, 1⟩. Then a_{i} ≡ b_{i} for all i < n, so there are d_{0}, …, d_{n−1} such that a_{i}c_{i} ≡ b_{i}d_{i} for all i < n. Since the type of c_{i} is r_{i,∞}, each d_{i} is of the form ⟨i, 1, l_{i}, 2⟨m_{i}, k_{i}⟩ + 1⟩ for some m_{i} such that (∃u < m_{i})Ψ(i, u). Then max_{i<n} m_{i} witnesses the satisfaction of our instance of BΠ_{1}^{0}. □

One of the important facts about homogeneity is that atomic, prime, and saturated models are homogeneous. The standard proof of this fact carries through in ACA_{0} for any of our definitions of homogeneity. In RCA_{0}, we have the following version. (Recall the definition of strong saturation from Definition 2.1.)

Proposition 4.5. The following are provable in RCA_{0}.

(1) Every atomic model is homogeneous.

(2) Every prime model is homogeneous.

(3) Every saturated model is 1-homogeneous.

(4) Every strongly saturated model is homogeneous.

Proof. (1) Let M be atomic, and let ā_{0} ≡ b̄_{0}, …, ā_{n} ≡ b̄_{n} and c̄_{0}, …, c̄_{n} be M-tuples. There is a formula ψ(x̄_{0}ȳ_{0} … x̄_{n}ȳ_{n}) isolating the type of ā_{0}c̄_{0} … ā_{n}c̄_{n}. Then for each i ⩽ n, the formula

φ_{i}(x̄_{i}ȳ_{i}) ≡ (∃x̄_{0}ȳ_{0} … x̄_{i−1}ȳ_{i−1}x̄_{i+1}ȳ_{i+1} … x̄_{n}ȳ_{n})ψ(x̄_{0}ȳ_{0} … x̄_{n}ȳ_{n})

isolates the type of ā_{i}c̄_{i}. For each i ⩽ n, we have M ⊨ (∃ȳ)φ_{i}(ā_{i}ȳ), so M ⊨ (∃ȳ)φ_{i}(b̄_{i}ȳ), and thus there are M-tuples d̄_{0}, …, d̄_{n} such that M ⊨ φ_{i}(b̄_{i}, d̄_{i}) for each i ⩽ n. Then ā_{i}c̄_{i} ≡ b̄_{i}d̄_{i} for each i ⩽ n.

(2) Hirschfeldt, Shore, and Slaman [23] noted that the proof that every prime model is atomic carries through in RCA_{0}.

(3) Let M be saturated, and let ā ≡ b̄ and c̄ be M-tuples. Let Γ = {ψ(b̄ȳ) : M ⊨ ψ(āc̄)}. Then Γ is a type of the theory of (M, b̄), so it is realized by some M-tuple d̄. Then āc̄ ≡ b̄d̄.

(4) Let M be strongly saturated, and let ā_{0} ≡ b̄_{0}, …, ā_{n} ≡ b̄_{n} and c̄_{0}, …, c̄_{n} be M-tuples. Let Γ = {ψ(b̄_{i}ȳ_{i}) : i ⩽ n ∧ M ⊨ ψ(ā_{i}c̄_{i})}. Then Γ(ȳ_{0} … ȳ_{n}) is a partial type of the theory of (M, b̄), so it is realized by some M-tuple d̄_{0} … d̄_{n}. Then ā_{i}c̄_{i} ≡ b̄_{i}d̄_{i} for each i ⩽ n. □

Proposition 4.5, combined with Theorem 4.4, implies that BΣ_{2}^{0} suffices to show that every saturated model is homogeneous. Theorem C.1 below shows that WKL_{0} also suffices to prove this statement, which implies that the statement is Π_{1}^{1}-conservative over RCA_{0}, and hence in particular does not imply BΣ_{2}^{0}. We do not, however, know the exact strength of this statement.

It is straightforward to argue as in [23] to show that the model M defined in the proof of Proposition 4.2 is atomic. Thus we have the following fact.

Proposition 4.6. The statement that every atomic model is strongly 1-homogeneous is equivalent to ACA_{0} over RCA_{0}.

We do not know the exact strengths of the statements that every prime model is strongly 1-homogeneous, and that every (strongly) saturated model is strongly 1-homogeneous

Another important property of homogeneous models is that they are completely determined by their type spectra. That is, two homogeneous models with the same type spectra are isomorphic. The following result shows that, for any of our definitions of homogeneity, this principle is equivalent to ACA_{0}.

Proposition 4.7. The following are equivalent over RCA_{0}. (1) ACA_{0}

(2) If M and N are 1-point homogeneous and realize the same types, then M ≅ N.

(3) If M and N are homogeneous and realize the same types, then M ≅ N.

(4) If M and N are strongly 1-homogeneous and realize the same types, then M ≅ N.

Proof. ACA_{0} is powerful enough to carry out the classical back and forth construction needed to prove the statements above, and (2) clearly implies (3) and (4), so we are left with showing that (3) and (4) imply ACA_{0}.

Let A and B be the models from [23] described in the proof of Proposition 4.2. As shown in [23], A and B are both atomic models of the same theory, and hence realize the same types. By Proposition 4.5, A and B are homogeneous. Let (a_{0}, …, a_{n}) ≡ (b_{0}, …, b_{n}) ∈ A. Let f be the function defined by f(a_{i}) = b_{i}, f(b_{i}) = a_{i}, and f(x) = x for all other x. Then f is an automorphism of A, since the language of A is unary. Thus A is also strongly 1-homogeneous. Similarly, B is also strongly 1-homogeneous. As shown in [23], the existence of an isomorphism between A and B implies ACA_{0}. □

5. Closure conditions and model existence

In this section, we consider various closure conditions on sets of types. Recall that, in the reverse mathematical setting, an infinite set of types is coded as a list by a single second order object (see Definition 2.2). Suppose we have defined such a listing of types X = {p_{i}}_{i∈ℕ}. When we refer to a finite sequence q_{0}, …, q_{n} of types in X, we always think of this set as encoded by a finite sequence i_{0}, …, i_{n} of natural numbers such that q_{k} = p_{i_{k}} for all k ⩽ n, and similarly for finite sets of types in X.

5.1. Type spectra of homogeneous models. As mentioned above, Goncharov [15] and Peretyat’kin [41] characterized the sets of types that are type spectra of homogeneous models. We say that types p_{0}, …, p_{n} are basically consistent if whenever p_{i} and p_{j} share a tuple of variables x̄, we have p_{i} ↾ x̄ = p_{j} ↾ x̄. A formula φ(x̄ȳ) is consistent with a type p(x̄) if (∃ȳ)φ ∈ p.

Theorem 5.1 (Goncharov [15] and Peretyat’kin [41]). Let T be a theory, and let X ⊂ S(T) be countable. Then X is the type spectrum of a homogeneous model if and only if it satisfies the following conditions.

(1) T ∈ X.

(2) X is closed under variable substitutions, i.e., if p(x̄) ∈ X and |ȳ| = |x̄| then p(ȳ) ∈ X.

(3) X is closed under taking subtypes.

(4) Extension: If p(x̄) ∈ X and φ(x̄ȳ) is consistent with p, then there is a q(x̄ȳ) ∈ X such that p ∪ {φ} ⊆ q.

(5) Pairwise Full Type Amalgamation: For each basically consistent p_{1}, p_{2} ∈ X, there is a q ∈ X such that p_{1} ∪ p_{2} ⊆ q.

Remark 5.2. Suppose that X satisfies conditions (1)–(4) above, and let p(x_{0}, …, x_{m−1}) ∈ X. Let f : n → m be a function, and let q(x_{0}, …, x_{m−1}, y_{0}, …, y_{n−1}) be the unique type of T containing p and y_{i} = x_{f(i)} for all i < n. Then a simple application of extension shows that q ∈ X. However, this application requires our assumption that the language of T includes equality. If we did not make this assumption, we would want to extend the definition of closure under variable substitutions slightly to say that if p ∈ X and q are as above, then q ∈ X. The only place below where this issue matters is in the proof of Lemma 6.9, which is used in the proof of Theorem 6.10.

We name the above set of closure conditions.

Definition 5.3. An X ⊂ S(T) satisfies the pairwise full amalgamation closure conditions if X satisfies all the conditions in Theorem 5.1.

We study effective versions of the extension condition (4) below, but first we define various other versions of type amalgamation and study their relationships. We begin with the notion of finite full type amalgamation, which is classically equivalent to pairwise full type amalgamation. Basic consistency, in general, is not enough to guarantee that finite sets of types can be amalgamated in general. We can have types p(x, y), q(x, z), r(y, z) such that p ↾ x = q ↾ x, p ↾ y = r ↾ y, and q ↾ z = r ↾ z, yet p, q, r cannot be amalgamated at all (i.e., there is no type of the given theory that amalgamates them). For example, consider a graph with four elements a, b, c, d, with edges from a to c and b to d, and no other edges. Let p(x, y) be the type of a, b; let q(x, z) be the type of a, c, which is the same as the type of b, d; and let r(y, z) be the same as q(x, z) with x substituted by y.

We say that types q_{0}(ȳ_{0}), …, q_{n}(ȳ_{n}) are consistent if they are basically consistent and there is an i ⩽ n such that all the shared variables are in ȳ_{i}. (That is, we cannot have z ∈ ȳ_{j} ∩ ȳ_{k} for j ≠ k unless z ∈ ȳ_{i}.) This condition ensures that this finite set of types can be amalgamated in the classical setting. We modify condition 5, pairwise full type amalgamation, as follows.

Finite Full Type Amalgamation: For each consistent set of types q_{0}, …, q_{n} ∈ X, there is a q ∈ X such that q_{0} ∪ ⋯ ∪ q_{n} ⊆ q.

Note that, as mentioned at the beginning of this section, we think of finite subsets of X as coded by indices, so the above is in fact a first order statement, as are other similar statements considered below. (That is, the universal quantification over sets of types q_{0}, …, q_{n} ∈ X is really over finite sets of indices of elements of X = {p_{i}}_{i∈ℕ}, and the existential quantification over q ∈ X is really over indices of elements of X.)

As with 1-homogeneity and homogeneity, the reason that pairwise amalgamation does not immediately yield finite amalgamation when working in RCA_{0} is that the property of being an amalgamator of a set of types is Π_{1}^{0}.

Suppose that X = {p_{i}}_{i∈ℕ} satisfies finite full type amalgamation. Let i_{0}, …, i_{n} ∈ ℕ. For each k ⩽ n, let q_{k} be a (not necessarily proper) subtype of a type obtained from p_{i_{k}} by variable substitution. Suppose that q_{0}, …, q_{n} form a consistent set of types. It is in the spirit of the notion of finite type amalgamation that there should be a type in X amalgamating q_{0}, …, q_{n}. However, even in the presence of the first four conditions in Theorem 5.1, it might not be provable in RCA_{0} that this is indeed the case, because there might not be a sequence j_{0}, …, j_{n} ∈ ℕ such that p_{j_{k}} = q_{k} for all k ⩽ n. Again, the issue here is that having p_{j_{k}} = q_{k} is a Π_{1}^{0} property. To avoid this problem, we introduce the following condition: X = {p_{i}}_{i∈ℕ} is closed under sequences of variable substitutions if for every sequence i_{0}, …, i_{n} ∈ ℕ and every sequence of types q_{0}, …, q_{n} such that q_{k} is a (not necessarily proper) subtype of a type obtained from p_{i_{k}} by variable substitution, there is a sequence j_{0}, …, j_{n} ∈ ℕ such that p_{j_{k}} = q_{k} for all k ⩽ n.

Remark 5.4. If X satisfies the first four conditions in Theorem 5.1, then it is provable in RCA_{0} + BΣ_{2}^{0} that X is closed under sequences of variable substitutions. (The proof is a straightforward application of the finite axiom of choice for Π_{1}^{0} properties.)

Definition 5.5. An X ⊂ S(T) satisfies the finite full amalgamation closure conditions if X satisfies the conditions in Theorem 5.1 with pairwise full type amalgamation replaced by finite full type amalgamation, and is also closed under sequences of variable substitutions.

Pairwise full type amalgamation is to 1-homogeneity as finite full type amalgamation is to homogeneity, an analogy that is made more precise below. We also consider a weaker version of pairwise full type amalgamation, which corresponds to 1-point homogeneity.

1-Point Full Type Amalgamation: For each basically consistent p_{1}(x̄y), p_{2}(x̄z) ∈ X, there is a q ∈ X such that p_{1} ∪ p_{2} ⊆ q.

Definition 5.6. An X ⊂ S(T) satisfies the 1-point full amalgamation closure conditions if X satisfies the conditions in Theorem 5.1 with pairwise full type amalgamation replaced by 1-point full type amalgamation.

5.2. Type spectra of general models. Theorem 5.1 characterizes the sets of types that are type spectra of homogeneous models. As noted by Peretyat’kin [41], it can easily be modified to characterize the sets of types that are type spectra of models in general.

Theorem 5.7 (Peretyat’kin [41]). Let T be a theory, and let X ⊂ S(T) be countable. Then X is the type spectrum of a model if and only if it satisfies the following conditions (the first four of which are the same as in Theorem 5.1).

(1) T ∈ X.

(2) X is closed under variable substitutions.

(3) X is closed under taking subtypes.

(4) Extension: If p(x̄) ∈ X and φ(x̄ȳ) is consistent with p, then there is a q(x̄ȳ) ∈ X such that p ∪ {φ} ⊆ q.

(5) Pairwise Free Type Amalgamation: For each p_{1}, p_{2} ∈ X whose variables do not overlap, there is a q ∈ X such that p_{1} ∪ p_{2} ⊆ q.

As in the full amalgamation case, we have the following notion, classically equivalent to pairwise free type amalgamation.

Finite Free Type Amalgamation: For each finite set of types p_{0}, …, p_{n} ∈ X whose variables do not overlap, there is a q ∈ X such that p_{0} ∪ ⋯ ∪ p_{n} ⊆ q.

Definition 5.8. An X ⊂ S(T) satisfies the pairwise free amalgamation closure conditions if it satisfies all the conditions in Theorem 5.7; and it satisfies the finite free amalgamation closure conditions if it satisfies these conditions with pairwise free type amalgamation replaced by finite free type amalgamation, and is also closed under sequences of variable substitutions.

Remark 5.9. Suppose that X satisfies the finite free amalgamation closure conditions. Then X satisfies a “finite sequence” version of the extension condition (condition (4) in Theorems 5.1 and 5.7), in the following sense. We first describe a notation that is also useful below. When we write φ(x̄) for a formula φ, to express the fact that free variables of φ are among the ones in the tuple x̄, we subsequently write φ[ȳ], where ȳ = (y_{0}, …, y_{n}) is a tuple of variables of the same length as x̄, for the formula obtained by replacing each free occurrence of x_{i} by y_{i}. We also adopt the analogous notation for types. Let p_{0}(x̄_{0}), …, p_{n}(x̄_{n}) ∈ X and let φ_{0}(x̄_{0}ȳ_{0}), …, φ_{n}(x̄_{n}ȳ_{n}) be formulas such that each φ_{k} is consistent with p_{k}. Let z̄_{0}, …, z̄_{n} and w̄_{0}, …, w̄_{n} be pairwise disjoint tuples of variables such that each z̄_{i} has the same arity as x̄_{i}, and each w̄_{i} has the same arity as ȳ_{i}. Then we can argue in RCA_{0} as follows. The sequence p_{0}[z̄_{0}], …, p_{n}[z̄_{n}], which exists as a sequence of elements of X by closure under sequences of variable substitutions, is amalgamated by some q ∈ X. The formula φ = ⋀_{k⩽n} φ_{k}[z̄_{k}w̄_{k}] is consistent with q, so there is an r ⊇ q ∪ {φ} in X. Thus there are r_{0}, …, r_{n} ∈ X such that p_{k}[z̄_{k}]∪{φ_{k}[z̄_{k}w̄_{k}]} ⊆ r_{k} for each k ⩽ n, which implies that there are s_{0}, …, s_{n} ∈ X such that p_{k}(x̄_{k})∪{φ_{k}(x̄_{k}ȳ_{k})} ⊆ s_{k} for each k ⩽ n. (The existence of the sequence r_{0}, …, r_{n} and that of the sequence s_{0}, …, s_{n} both follow by closure under sequences of variable substitutions.)

5.3. Comparing closure conditions. We study the reverse mathematics of Theorems 5.1 and 5.7 below, but first we discuss the relationships between the closure conditions introduced in this section. Note that if X satisfies the pairwise full amalgamation closure conditions, then X satisfies the pairwise free amalgamation conditions, and similarly for the finite versions. We also highlight the following obvious fact for future reference. (Note that, in the reverse mathematical setting, a set of types is encoded as a list of types, as in Definition 2.2. Two lists of types {p_{i}}_{i∈ℕ} and {q_{i}}_{i∈ℕ} can be equal as sets of types, meaning that for each i there is a j such that q_{j} = p_{i} and for each i there is a j such that p_{j} = q_{i}, but have different closure properties. It is worth keeping in mind, for instance, that when working in a model of second order arithmetic with nonstandard first order part, the fact that {p_{i}}_{i∈ℕ} and {q_{i}}_{i∈ℕ} are equal as sets of types does not necessarily imply that for every sequence i_{0}, …, i_{n} there is a sequence j_{0}, …, j_{n} such that q_{j_{k}} = p_{i_{k}} for all k ⩽ n.)

Proposition 5.10. The following is provable in RCA_{0}. If the lists of types X and Y are equal as sets of types and X satisfies the pairwise (free or full) type amalgamation condition, then so does Y, and similarly for 1-point full amalgamation, and each of the first four conditions in Theorem 5.1.

We show in Theorem 5.16 that this proposition does not hold for finite amalgamation.

The difference between the 1-point, pairwise, and finite versions of our closure conditions turns out to be IΣ_{2}^{0}. We record the full result here, although its proof depends on two results proved in the next section.

Theorem 5.11. The following are equivalent over RCA_{0}.

(1) IΣ_{2}^{0}

(2) If a list of types satisfies the 1-point full amalgamation closure conditions then it satisfies the pairwise full amalgamation closure conditions.

(3) If a list of types satisfies the pairwise full amalgamation closure conditions then it satisfies the finite full amalgamation closure conditions.

(4) If a list of types satisfies the pairwise free amalgamation closure conditions then it satisfies the finite free amalgamation closure conditions.

(5) If a list of types satisfies the pairwise full amalgamation closure conditions then it satisfies the finite free amalgamation closure conditions.

Proof. To show that (1) implies (2), let X be a list of types satisfying the 1-point full amalgamation closure conditions. Let p(x̄, y_{0}, …, y_{m}) and q(x̄, z_{0}, …, z_{n}) be basically consistent elements of X. By assumption, there is a type r_{0}(x̄, y_{0}, z_{0}) amalgamating p ↾ x̄y_{0} and q ↾ x̄z_{0}. Then there are types r′_{0}(x̄, y_{0}, z_{0}, y_{1}) amalgamating p ↾ x̄y_{0}y_{1} and r_{0}, and r′′_{0}(x̄, y_{0}, z_{0}, z_{1}) amalgamating q ↾ x̄z_{0}z_{1} and r_{0}. Hence there is a type r_{1}(x̄, y_{0}, z_{0}, y_{1}, z_{1}) amalgamating r′_{0} and r′′_{0}. By finite Π_{1}^{0}-recursion (which, as mentioned in Section 2.1, is provable from IΣ_{2}^{0}), we can continue in this fashion to eventually obtain a type amalgamating p and q.

To show that (1) implies (3), and hence that (1) implies (5), let X be a list of types satisfying the pairwise full amalgamation closure conditions, and let q_{0}, …, q_{n} ∈ X be a consistent sequence of types. By the definition of consistency, we may assume that all free variables shared by two or more of the q_{i} are among the free variables of q_{0}. Let P be the property that holds of i ⩽ n if and only if there is a q ∈ X such that q_{0} ∪ ⋯ ∪ q_{i} ⊆ q. It is easy to check that P is Σ_{2}^{0}. Furthermore, P (0) holds, and if P (i) holds for i < n, then, letting q be the witness to this fact, q, q_{i+1} are consistent, so there is an r ∈ X such that q_{0} ∪ ⋯ ∪ q_{i} ⊆ q ∪ q_{i+1} ⊆ r; that is, P (i + 1) holds. Thus, by IΣ_{2}^{0}, we have P (n). (Note that closure under sequences of variable substitutions also holds, by Remark 5.4.) The proof that (1) implies (4) is similar.

That (2), (3), (4), and (5) all imply (1) follows from Theorems 6.11 and 6.12 below (using condition (a) in item (3) of Theorem 6.11), together with the equivalence between IΣ_{2}^{0} and the finite Π_{1}^{0}-recursion principle. □

Belanger [3] has shown that, for listings of all types of a theory with countably many types, the situation is different

Theorem 5.12 (Belanger [3]). The following statements are equivalent over RCA_{0}.

(1) The disjunction WKL_{0} ∨ IΣ_{2}^{0}.

(2) If a list of all the types of a theory satisfies the pairwise full amalgamation closure conditions then it satisfies the finite full amalgamation closure conditions.

(3) If a list of all the types of a theory satisfies the pairwise free amalgamation closure conditions then it satisfies the finite free amalgamation closure conditions.

(4) If a list of all the types of a theory satisfies the pairwise full amalgamation closure conditions then it satisfies the finite free amalgamation closure conditions.

5.4. Spectrum enumeration existence theorems. Theorems 5.1 and 5.7 are “if and only if” statements. The “if” directions are the main focus of this paper. In this subsection, we analyze the reverse mathematics of the “only if” directions. These are easily seen to hold in RCA_{0}, though in the case of Theorem 5.1, we have to choose the appropriate notion of homogeneity.

Proposition 5.13. The following is provable in RCA_{0}. Let A be a model. Then 𝕋(A) has an enumeration X satisfying the finite free amalgamation closure conditions. If A is 1-point homogeneous then X can be chosen to satisfy the 1-point full amalgamation closure conditions. If A is 1-homogeneous then X can be chosen to satisfy the pairwise full amalgamation closure conditions. If A is homogeneous then X can be chosen to satisfy the finite full amalgamation closure conditions.

Proof. Let A be a model, let c̄_{0}, c̄_{1}, … be an enumeration of the Atuples, and for each i, let x̄_{i,0}, x̄_{i,1}, … be an enumeration of the tuples of variables of the same length as c̄_{i} (chosen so that if |c̄_{i}| = |c̄_{i′}| then x̄_{i,j} = x̄_{i′,j} for all j). Let X = {q_{i}(x̄_{i,j})}_{i,j∈ℕ}, where q_{i} is the type of c̄_{i} in the variables x̄_{i,j}. Then X is an enumeration of 𝕋(A). It is easy to see that X satisfies conditions (1)–(4) in Theorem 5.1 and is closed under sequences of variable substitutions.

Let q_{i_{0}}(x̄_{i_{0},j_{0}}), …, q_{i_{n}}(x̄_{i_{n},j_{n}}) be elements of X whose free variables do not overlap. Let i be such that c̄_{i} is the tuple c̄_{i_{0}} … c̄_{i_{n}}, and let j be such that x̄_{i,j} is the tuple x̄_{i_{0},j_{0}} … x̄_{i_{n},j_{n}}. Then q_{i_{0}}(x̄_{i_{0},j_{0}}), …, q_{i_{n}}(x̄_{i_{n},j_{n}}) ⊆ q_{i}(x̄_{i,j}). Thus X satisfies the finite free type amalgamation condition.

Now suppose that A is homogeneous and let q_{i_{0}}(x̄_{i_{0},j_{0}}), …, q_{i_{n}}(x̄_{i_{n},j_{n}}) be a consistent sequence of elements of X. By the definition of consistency, we may assume that all the free variables shared by two or more of these types are in x̄_{i_{0},j_{0}}, so we can write each q_{i_{k}}(x̄_{i_{k},j_{k}}) as q_{i_{k}}(ȳ_{i_{k}} z̄_{i_{k}}), where ȳ_{i_{k}} ⊆ x̄_{i_{0},j_{0}} and the z̄_{i_{k}} are disjoint from x̄_{i_{0},j_{0}} and pairwise disjoint. The types q_{i_{k}}(ȳ_{i_{k}} z̄_{i_{k}}) ↾ ȳ_{i_{k}} are all compatible, so homogeneity implies that for each k ⩽ n, there is an i′_{k} such that q_{i′_{k}}(x̄_{i′_{k},j_{k}}) = q_{i_{k}}(x̄_{i_{k},j_{k}}) and if the pth element of x̄_{i_{0},j_{0}} equals the qth element of x̄_{i′_{k},j_{k}}, then the pth element of c̄_{i_{0}} equals the qth element of c̄_{i′_{k}}. For each k ⩽ n, let d̄_{k} be the elements of c̄_{i_{k}} corresponding to the variables in z̄_{i_{k}}. Let i be such that c̄_{i} is the tuple c̄_{i_{0}} d̄_{i_{1}} … d̄_{i_{n}}. Let j be such that x̄_{i,j} is the tuple x̄_{i_{0},j_{0}} z̄_{i_{1},j_{1}} … z̄_{i_{n},j_{n}}. Then q_{i_{0}}(x̄_{i_{0},j_{0}}), …, q_{i_{n}}(x̄_{i_{n},j_{n}}) ⊆ q_{i}(x̄_{i,j}). Thus X satisfies the finite full type amalgamation condition.

A similar argument shows that if A is 1-point homogeneous then X satisfies the 1-point full type amalgamation condition, while if A is 1-homogeneous then X satisfies the pairwise full type amalgamation condition. □

The proof of Theorem 4.3 can easily be modified to obtain the following result.

Theorem 5.14. The following are equivalent over RCA_{0}.

(1) IΣ_{2}^{0}

(2) For every 1-point homogeneous model A, there is an enumeration of 𝕋(A) satisfying the pairwise full amalgamation closure conditions.

(3) For every 1-point homogeneous model A, there is an enumeration of 𝕋(A) satisfying the finite full amalgamation closure conditions.

Proof. That (1) implies (2) and (3) follows from Theorem 5.11 and Proposition 5.13. Clearly (3) implies (2). To see that (2) implies (1), let φ, n, z, θ, and M be as in the proof of Theorem 4.3 and suppose that 𝕋(M) has an enumeration X satisfying the pairwise full amalgamation closure conditions. Let p(x, y, z_{0}, …, z_{n−1}) be the type of

(⟨0, 0, 0⟩, ⟨1, 0, 0⟩, ⟨0, 1, 0⟩, …, ⟨0, 1, n − 1⟩)

in M, and let q(y, w_{0}, …, w_{n−1}) be the type of

(⟨0, 0, 0⟩, ⟨0, 1, 0⟩, …, ⟨0, 1, n − 1⟩)

in M. Then p ↾ y = q ↾ y, so there is a type

r(x, y, z_{0}, …, z_{n−1}, w_{0}, …, w_{n−1})

in X amalgamating these two types. Let (a, b, c_{0}, …, c_{n−1}, d_{0}, …, d_{n−1}) realize this type in M. Then either each c_{i} is of the form ⟨1, 1, k⟩ where θ(k) holds, or each d_{i} is of this form. In either case, as in the proof of Theorem 4.3, the instance of Π_{1}^{0}-recursion corresponding to φ, n, and z holds. □

Similarly, the proof of Theorem 4.4 yields the following result.

Theorem 5.15. The following are equivalent over RCA_{0}.

(1) BΣ_{2}^{0} (2) For every 1-homogeneous model A, there is an enumeration of

𝕋(A) satisfying the finite full amalgamation closure conditions.

(3) For every strongly 1-homogeneous model A, there is an enumeration of 𝕋(A) satisfying the finite full amalgamation closure conditions.

Proof. That (1) implies (2), and hence (3), follows from Theorem 4.4 and Proposition 5.13. To see that (3) implies (1), and hence so does (2), let Ψ, n, and M be as in the proof of Theorem 4.4, and suppose that 𝕋(M) has an enumeration X satisfying the finite full amalgamation closure conditions. Let p(x_{0}, …, x_{n−1}, y_{0}, …, y_{n−1}) be the type of

(⟨0, 0, 0, 0⟩, …, ⟨n − 1, 0, 0, 0⟩, ⟨0, 1, 0, 0⟩, …, ⟨n − 1, 1, 0, 0⟩)

in M. For i < n, let q_{i}(x_{i}, z_{i}) be the type of (⟨i, 0, 0, 0⟩, ⟨i, 0, 0, 1⟩) in M. The sequence q_{0}, …, q_{n−1} exists as a sequence of elements of X by closure under sequences of variable substitutions, since X contains the type of

(⟨0, 0, 0, 0⟩, ⟨0, 0, 0, 1⟩, …, ⟨n − 1, 0, 0, 0⟩, ⟨n − 1, 0, 0, 1⟩).

For the same reason, X contains the sequence of types q_{i}[y_{i}, w_{i}] (using the notation introduced in Remark 5.9), where the w_{i} are pairwise distinct from the z_{i}. Then there is a type

r(x_{0}, …, x_{n−1}, y_{0}, …, y_{n−1}, z_{0}, …, z_{n−1}, w_{0}, …, w_{n−1})

in X that amalgamates p(x_{0}, …, x_{n−1}, y_{0}, …, y_{n−1}) with q_{i}(x_{i}, z_{i}) and q_{i}[y_{i}, w_{i}] for all i < n. This type must be realized in M by some tuple

(a_{0}, …, a_{n−1}, b_{0}, …, b_{n−1}, c_{0}, …, c_{n−1}, d_{0}, …, d_{n−1}).

For each i < n, exactly one of c_{i} or d_{i} is of the form ⟨i, 1, l, 2⟨m_{i}, k_{i}⟩+1⟩, in which case we have (∃u < m_{i})Ψ(i, u). Thus we obtain a sequence m_{0}, …, m_{n−1} whose maximum witnesses the satisfaction of the instance of BΠ_{1}^{0} in the proof of Theorem 4.4. □

We finish this subsection with the following result, mentioned above following Proposition 5.10.

Theorem 5.16. The following are equivalent over RCA_{0}.

(1) BΣ_{2}^{0}

(2) If the lists of types X and Y are equal as sets of types and X satisfies the finite (free or full) type amalgamation condition, then so does Y.

(3) Every enumeration of the type spectrum of a model satisfies the finite free type amalgamation condition.

(4) Every enumeration of the type spectrum of a homogeneous model satisfies the finite full type amalgamation condition.

Proof. By Proposition 5.13, (3) and (4) follow from (2) over RCA_{0}. We show that (2) holds in RCA_{0} + BΣ_{2}^{0}. We do the argument for full amalgamation, as the argument for free amalgamation is essentially the same. Let X and Y be lists of types that are equal as sets, and suppose that X satisfies the finite full type amalgamation condition. Let q_{0}, …, q_{k} ∈ Y be a consistent sequence of types. The property of an element of X being equal to an element of Y is Π_{1}^{0}, and for each i ⩽ k there is an element of X equal to q_{i}, so by the finite axiom of choice for Π_{1}^{0} properties, there is a sequence p_{0}, …, p_{k} ∈ X such that p_{i} = q_{i} for all i ⩽ k. Then there is a p ∈ X amalgamating p_{0}, …, p_{k}. There is a q ∈ Y such that q = p, and this q amalgamates q_{0}, …, q_{k}.

For the other direction, we assume that at least one of (2), (3), or (4) holds and argue in RCA_{0}. Let Q be a Π_{1}^{0} property and m be such that (∀i < m)(∃u)Q(i, u). Let R be a Δ_{1}^{0} property such that Q(x, y) if and only if (∀z)R(x, y, z). Let T = 2^{<ℕ} and let T = T (T ) be as in Definition 3.5. We define a structure M in the language of T with domain ℕ. For n ∈ ℕ and α ∈ 2^{ℕ}, we write M ⊨ P^{α}(n) to mean that M ⊨ P_{k}(n) if and only if α(k) = 1. Let A_{σ} for σ ∈ 2^{<ℕ} and B_{i,j} for i < m and j ∈ ℕ be pairwise disjoint infinite sets whose union is ℕ. For each σ ∈ 2^{<ℕ}, let M ⊨ P^{σ0^{ℕ}}(n) for all n ∈ A_{σ}. For i < m and j ∈ ℕ, let α_{i,j}(k) be 1 if (∀z < k)R(i, j, z) and 0 otherwise. Let M ⊨ P^{0^{i}1α_{i,j}}(n) for all n ∈ B_{i,j}.

It is easy to check that M is a homogeneous model of T, so by Proposition 5.13, 𝕋(M) has an enumeration {p_{i}}_{i∈ℕ} satisfying the finite full amalgamation closure conditions. By effective quantifier elimination, to define a 1-type p(x) of T, it is enough to specify which P_{k}(x) are in p. For i < m, let q_{i}(x_{i}) be the 1-type containing P_{k}(x_{i}) iff k ⩾ i (i.e., the type corresponding to P^{0^{i}1^{ℕ}}(x_{i})), where the x_{i} are pairwise distinct variables. For k ∈ ℕ, let q_{m+k} = p_{k}. It is easy to check that Y = {q_{i}}_{i∈ℕ} is also an enumeration of 𝕋(M). Since we are assuming that at least one of (2), (3), or (4) holds, Y satisfies the finite free amalgamation closure conditions. Then there is a type in Y amalgamating q_{0}, …, q_{m−1}. This type is realized in M, so there is a sequence n_{0}, …, n_{m−1} ∈ ℕ such that M ⊨ P^{0^{i}1^{ℕ}}(n_{i}) for all i < m. Then n_{i} ∈ B_{i,j_{i}} for some j_{i} such that (∀z)R(i, j_{i}, z), and we can form the sequence j_{0}, …, j_{m−1}. Letting b be greater than all the j_{i}, we have (∀i < m)(∃u < b)Q(i, u). □

Note that BΣ_{2}^{0} is also enough to show that closure under sequences of variable substitutions is enumeration-independent (by an argument similar to that in the first part of the above proof), so the above theorem remains true if we replace “satisfies the finite (free or full) type amalgamation condition” by “satisfies the finite (free or full) amalgamation closure conditions”.

5.5. Model existence theorems. As discussed in the introduction, the degree-theoretic results on atomic and homogeneous models obtained by Csima [7] and Csima, Hirschfeldt, Knight, and Soare [8], on the one hand, and Lange [27, 28], on the other, are almost identical. Thus it is natural to compare AMT with analogous statements for homogeneous models. The following are three reasonable candidates, which match our sets of full amalgamation closure conditions with the corresponding notions of homogeneity. Of course, when formalizing the statements below, a set of types X is given as a list of types {p_{i}}_{i∈ℕ}, and 𝕋(A) = X means that X is an enumeration of the type spectrum of A (i.e., a type is realized in A iff it is equal to p_{i} for some i).

HMT (Homogeneous Model Theorem): Let T be a theory and let X ⊆ S(T) satisfy the finite full amalgamation closure conditions. Then there exists a homogeneous model A of T with 𝕋(A) = X.

1-HMT (1-Homogeneous Model Theorem): Let T be a theory and let X ⊆ S(T) satisfy the pairwise full amalgamation closure conditions. Then there exists a 1-homogeneous model A of T with 𝕋(A) = X.

1-PHMT (1-Point Homogeneous Model Theorem): Let T be a theory and let X ⊆ S(T) satisfy the 1-point full amalgamation closure conditions. Then there exists a 1-point homogeneous model A of T with 𝕋(A) = X.

We also name the following model existence theorems.

SMT (Strong Model Existence Theorem): Let T be a theory and let X ⊆ S(T) satisfy the pairwise free amalgamation closure conditions. Then there exists a model A of T with 𝕋(A) = X.

WMT (Weak Model Existence Theorem): Let T be a theory and let X ⊆ S(T) satisfy the finite free amalgamation closure conditions. Then there exists a model A of T with 𝕋(A) = X.

To discuss the reverse mathematical strength of these theorems, we first need to examine the effective versions of Theorems 5.1 and 5.7, which requires introducing some strengthenings of the extension condition in these theorems.

6. Extension functions and model existence

6.1. Extension functions. Theorem 5.1 gives closure conditions that a set of types must satisfy to be the type spectrum of a homogeneous model. However, these conditions do not tell us how to build such a model effectively from a listing of such a set. The following result answered a question of Morley [36].

Theorem 6.1 (Goncharov [15], Peretyat’kin [41], and Millar [31]). There is a computable list of types X satisfying all the conditions in Theorem 5.1 such that X is not an enumeration of the type spectrum of any decidable homogeneous model.

Goncharov [15] and Peretyat’kin [41] realized that in order to build a decidable homogeneous model M such that X is an enumeration of 𝕋(M), we need to effectivize one or more of the conditions in Theorem 5.1, to obtain some computable information about how the types in the computable list X extend one another. Of particular interest are the extension condition and the (pairwise full) type amalgamation condition. Since we naturally seek the weakest possible hypothesis guaranteeing the existence of a decidable homogeneous model M such that X is an enumeration of the type spectrum of M, a reasonable way to proceed is to begin by considering a decidable homogeneous model M and the natural listing X = {q_{i}(x̄_{i,j})}_{i,j∈ℕ} of its type spectrum given in the proof of Theorem 5.13, and determining which of these conditions hold effectively of X.

Let us recall the definition of X for convenience: Let c̄_{0}, c̄_{1}, … be an enumeration of the M-tuples, and for each i, let x̄_{i,0}, x̄_{i,1}, … be an enumeration of the tuples of variables of the same length as c̄_{i}. Let X = {q_{i}(x̄_{i,j})}_{i,j∈ℕ}, where q_{i} is the type of c̄_{i} in the variables x̄_{i,j}.

Let p_{⟨i,j⟩} = q_{i}(x̄_{i,j}). The effective version of pairwise full type amalgamation says that there is a computable function f such that, if p_{m} and p_{n} are basically consistent, then p_{m} ∪ p_{n} ⊆ p_{f(m,n)}. We claim this effective closure condition does not necessarily hold. To see that this is the case, let M be as in the proof of Proposition 4.2. We adopt the notation of that proof.

Let a, b ∈ M be such that a ≡ b and ¬E^{M}(a, b). Now assume for a contradiction that there is a computable function f as above. We show how to compute ∅′ using f. Given i, let c, d ∈ M be such that M ⊨ R_{i}(c) ∧ R_{i}(d) ∧ E(a, c) ∧ E(b, d). Let y_{0}, y_{1}, y_{2} be variables, let m be such that p_{m} is the type of ac in y_{0}, y_{1}, and let n be such that p_{n} is the type of bd in y_{0}, y_{2}. Then p_{m} and p_{n} are basically consistent, so p_{m} ∪ p_{n} ⊆ p_{f(m,n)}. Let i and j be such that f(m, n) = ⟨i, j⟩, and let e_{0}, e_{1}, e_{2} ∈ M be the elements of c̄_{i} corresponding to the positions of y_{0}, y_{1}, y_{2} in x̄_{i,j}, respectively. Then e_{1} ≡ c and e_{2} ≡ d. Furthermore, E^{M} holds of (e_{0}, e_{1}) and of (e_{0}, e_{2}), and hence of (e_{1}, e_{2}), so either ¬E^{M}(e_{1}, c) or ¬E^{M}(e_{2}, d). We assume that ¬E^{M}(e_{1}, c), as the other case is essentially the same. We also assume that e_{1} is even, and hence on the A side of M, and c is odd, and hence on the B side of M. Again, the other case is essentially the same.

If i ∈ ∅′_{max(e_{1},c)} then of course i ∈ ∅′. Otherwise, we claim that i ∉ ∅′. Suppose this is not the case. Then i enters ∅′ at a stage t > max(e_{1}, c). By our construction of M, we have M ⊨ R_{i,t}(e_{1}) and M ⊨ ¬R_{i,t}(c), which contradicts the fact that e_{1} ≡ c.

Thus M provides us an example of a decidable homogeneous model such that the natural listing of 𝕋(M) does not satisfy the effective version of the pairwise full type amalgamation condition. For the extension condition, however, the situation is different. Let us return to general case of a decidable homogeneous model M and the listing X = {q_{i}(x̄_{i,j})}_{i,j∈ℕ} of 𝕋(M). Again let p_{⟨i,j⟩} = q_{i}(x̄_{i,j}), and let θ_{0}, θ_{1}, … list the formulas in our language. The effective version of extension says that there is a computable function f such that if θ_{n}(ȳz̄) is consistent with p_{m}(ȳ), then p_{f(m,n)} extends both p_{m} and θ_{n}. This condition does in fact hold of X: Given such m and n, we simply let i and j be such that m = ⟨i, j⟩ and search for c̄_{k} extending c̄_{i} such that M ⊨ θ_{n}(c̄_{k}). We then let l be such that x̄_{k,l} = ȳz̄, and let f(m, n) = ⟨k, l⟩. Note that this argument does not use the assumption that M is homogeneous.

Goncharov [15] and Peretyat’kin [41] showed that the existence of such a computable extension function is in fact a sufficient condition on a computable list of types X satisfying the pairwise full amalgamation closure conditions to ensure that there is a decidable homogeneous model M such that X is an enumeration of the type spectrum of M. Indeed, they went even further, by showing that we do not need to be able to compute an extension function, but merely computably approximate one. As we will see in Theorem 6.4, this weaker condition is not only sufficient, but also necessary. The following definitions make the above notions precise. For a detailed look at the following definitions and their uses in the context of atomic and homogeneous models, see Lange and Soare [29].

Definition 6.2. Let X = {p_{i}}_{i∈ℕ} be a list of types of a theory T and let θ_{0}, θ_{1}, … be an effective enumeration of the formulas in the language of T.

(i) A function f(i, j) is an extension function for X if for every n-type p_{i}(x̄) and (n + k)-ary θ_{j}(x̄ȳ) consistent with p_{i}(x̄), the (n + k)-type p_{f(i,j)} extends both p_{i}(x̄) and θ_{j}(x̄ȳ), i.e.,

p_{i}(x̄) ∪ {θ_{j}(x̄ȳ)} ⊆ p_{f(i,j)}(x̄ȳ).

(ii) A function g(i, j, s) is an extension function approximation if for every n-type p_{i}(x̄) and (n + k)-ary θ_{j}(x̄ȳ) consistent with p_{i}(x̄),

a. lim_{s} g(i, j, s) exists, b. p_{i}(x̄) ∪ {θ_{j}(x̄ȳ)} ⊆ p_{lim_{s} g(i,j,s)}(x̄ȳ), and c. p_{g(i,j,s)} ↾ s = p_{g(i,j,s+1)} ↾ s for all s.

Thus, an extension function is one that, given an index for an ntype and an (n + k)-ary formula consistent with that type, returns an index for a type amalgamating the type and the formula. An extension function approximation, given the same data, approximates the index of such an amalgamating type, in such a way that the stage s guess at a true amalgamator agrees with the true amalgamator on the first s many formulas. (The last condition has led to this notion being called “monotone extension function approximation”, but since this is the only kind of extension function approximation we consider, we omit the word “monotone”.)

Theorem 6.3 (Goncharov [15] (for homogeneous models) and Peretyat’kin [41] (for both cases)). Let X be a computable list of types of a decidable theory T such that X satisfies the pairwise free amalgamation closure conditions and has a computable extension function approximation. Then there is a decidable model A such that X is an enumeration of 𝕋(A). If X also satisfies the pairwise full amalgamation closure conditions, then A can be chosen to be homogeneous.

The basic idea of the proof of this result is to perform a Henkin construction of a model A of T while ensuring that two families of requirements are met: ones stating that each type in X is realized by some A-tuple, and ones stating that each A-tuple realizes a type in X. To satisfy the first family of requirements, we assign types in X to A-tuples. These assignments might change in the course of the construction, but to satisfy the second family of requirements, we ensure that for each A-tuple ā, there is a type p ∈ X such that we eventually permanently assign p to ā. Since we do not have effective type amalgamation, our type assignments may turn out to be in conflict. To deal with this issue, we use a priority construction: We might assign types p, q ∈ X to the tuples āb̄ and āc̄, respectively, and find out at some later point in the construction that p ↾ ā ≠ q ↾ ā. In this case, we maintain whichever assignment has been made by the stronger priority requirement, say the assignment of p to āb̄, and reassign q to some new tuple of elements. We now have the problem of assigning some new type to āc̄. We do so by using our computable extension function approximation to guess at a type r in X extending both p and the formula representing the information we have already added to the elementary diagram of A about the tuple āb̄c̄, and assigning r ↾ āc̄ to āc̄. Of course, our extension function approximation can change its mind on what r is, but only finitely often. Thus we have a finite injury construction.

The existence of a computable extension function is enumerationdependent, in the sense that the type spectrum of a model A can have computable enumerations X and Y such that X has a computable extension function but Y does not. Extension function approximations are more robust: if some computable enumeration of 𝕋(A) has a computable extension function approximation, then every computable enumeration of 𝕋(A) has one. (See Lange and Soare [29, Section 4].) We give the reverse mathematical version of the latter fact in part (2) of Proposition 6.6.

The following result shows how extension functions can be used to characterize the computable lists of types that are enumerations of type spectra of decidable (homogeneous) models. It also shows that Theorem 6.3 is tight, in the sense that having a computable extension function approximation is in fact a necessary condition for a list of types be an enumeration of the type spectrum of some decidable model.

Theorem 6.4 (Goncharov [15] (for homogeneous models) and Peretyat’kin [41] (for both cases)). Let T be a decidable theory and A a (homogeneous) model of T such that 𝕋(A) has a computable enumeration. Then the following are equivalent.

(1) A has a decidable copy.

(2) Every computable enumeration of 𝕋(A) has a computable extension function approximation.

(3) Some computable enumeration of 𝕋(A) has a computable extension function approximation.

(4) Some computable enumeration of 𝕋(A) has a computable extension function.

In computable model theory, it is common to weaken the hypotheses of this definition by considering only the case k = 1, i.e., only formulas of the form θ_{j}(x̄y). Let us call the concepts defined in this way weak extension function and weak extension function approximation. Computability theoretically, there is no real difference: it is not difficult to show that if X has a computable weak extension function then it has a computable extension function, and if it has a computable weak extension function approximation, then it has a computable extension function approximation. In RCA_{0}, it is still easy show that the existence of a weak extension function implies that of an extension function, but the situation is less clear in the case of extension function approximations. The issue is the usual one that we cannot always show in RCA_{0} that the composition of finitely many functions with limits has a limit.

The only place below where the difference between weak extension function approximations and extension function approximations matters is in the proof of Theorem 6.10. In all other cases, when we use the existence of an extension function approximation, we actually use only the existence of a weak extension function approximation.

The principle that if X has a weak extension function approximation then it has an extension function approximation is easily provable from RCA_{0} +IΣ_{2}^{0}, and part (2) of Theorem 7.1 below implies that it also follows from ATT (assuming X in fact satisfies the extension condition), but we do not what else can be said about its strength.

Classically, the definition of extension function approximation looks a bit awkward, and it would seem better to define it as an approximation to an extension function with the additional property (c). However, in the reverse mathematical setting, the existence of an extension function approximation g does not guarantee the existence of the extension function f(i, j) = lim_{s} g(i, j, s). Indeed, while every extension function can easily be transformed into an extension function approximation of itself, there are examples of computable lists of types that have computable extension function approximations but no computable extension functions. In fact, we have the following equivalence.

Proposition 6.5. The statement that every list of types of a theory with an extension function approximation has an extension function is equivalent to ACA_{0} over RCA_{0}. This fact remains true if we restrict attention to lists of types satisfying any one of the sets of closure conditions discussed above.

Proof. ACA_{0} is clearly enough to show that the limit of an extension function approximation exists and is an extension function. For the reversal, fix a function f. We argue in RCA_{0} and build a list of types X of a theory T such that X satisfies the finite full amalgamation closure conditions and has an extension function approximation, so that if X has an extension function then the range of f exists as a set.

We work in the language with unary relation symbols P_{i} for i ∈ ℕ. Let A be the following set of axioms.

(1) P_{i+1}(x) → P_{i}(x) for all i ∈ ℕ.

(2) (∃^{>k}x)[P_{i}(x) ∧ ¬P_{i+1}(x)] for all i, k ∈ ℕ.

(3) (∃^{>k}x)¬P_{0}(x) for all k ∈ ℕ.

These axioms are the same as in the proof of Theorem 4.3, except that here we do not need the symbol E, so we can show that A admits effective quantifier elimination and that its deductive closure T is a theory using a simpler version of the argument in the proof of that theorem.

By effective quantifier elimination, the types of T can be identified with atomic types. Thus the 1-types of T in a given variable x are the principal types r_{i}(x) generated by P_{i−1}(x)∧¬P_{i}(x) (where the first conjunct is omitted if i = 0), and the nonprincipal type r_{∞}(x), which has P_{i}(x) for all i ∈ ℕ.

We now give names for all the types of T. Let E be an equivalence relation on some n ∈ ℕ, let σ = (k_{0}, …, k_{n−1}) be such that k_{i} ∈ ℕ ∪ {∞} for i < n and k_{i} = k_{j} whenever E(i, j) holds (note that E determines n uniquely), and let (y_{0}, …, y_{n−1}) be a tuple of variables. Let s_{E,σ}(y_{0}, …, y_{n−1}) be the n-type that includes y_{i} = y_{j} if E(i, j) holds and y_{i} ≠ y_{j} if ¬E(i, j) holds, and states that each y_{i} has type r_{k_{i}}. Note that these conditions completely specify an n-type, since we have only unary relation symbols. Furthermore, every type of T is equal to s_{E,σ}(ȳ) for some equivalence relation E, some compatible σ, and some tuple of variables ȳ.

Let x_{0}, x_{1}, … be the variables in our language. Clearly, we can enumerate all the types s_{E,σ}(x_{i_{0}}, …, x_{i_{n−1}}), where E and σ are as above and i_{0}, …, i_{n−1} are distinct numbers, as a list q_{0}, q_{1}, … with an extension function h (relative to a fixed listing θ_{0}, θ_{1}, … of the formulas in our language) such that if q_{i} is a type in the variables x̄ and the free variables of θ_{j} are ȳ then q_{h(i,j)} is a type in the variables x̄ ∪ ȳ. We may further assume that h is defined so that, for i and j as above, if x_{k} ∈ ȳ\ x̄ then q_{h(i,j)} ↾ x_{k} = r_{l}(x_{k}) for some l ∈ ℕ. We also assume our types are listed so that for any type p and any m ∈ ℕ, the restriction p ↾ m includes no formulas involving P_{m}.

We now define X = {p_{i}}_{i∈ℕ}, which in fact includes all types of T. Fix k, c ∈ ℕ. In defining p_{⟨k,c⟩}, there are two cases. If q_{k} = s_{E,(∞,l_{1},...,l_{n+1})}(x_{0}, …, x_{n+1}), where ¬E(0, 1) and l_{1} ∈ ℕ is such that (∀m < c)¬f(m) = l_{1}, then let p_{⟨k,c⟩}(x_{0}, …, x_{n+1}) be the type determined by stating that x_{0} ≠ x_{1}, that x_{j} has type r_{l_{j}} for j ∈ [1, n + 1], and that P_{i}(x_{0}) holds if and only if (∀m < i)¬f(m) = l_{1}. Otherwise, let p_{⟨k,c⟩} = q_{k}.

In the first case, p_{⟨k,c⟩} = q_{k} if and only if l_{1} is not in the range of f. If l_{1} is in the range of f, however, then the first case applies for only finitely many c, so if c is sufficiently large then p_{⟨k,c⟩} = q_{k}. Thus every type of T appears in X. Furthermore, if we know whether l_{1} is in the range of f, then we can determine which type p_{⟨k,c⟩} is, for any given c ∈ ℕ. By bounded Σ_{1}^{0}-comprehension, it follows that for any sequence i_{0}, …, i_{m}, there is a sequence j_{0}, …, j_{m} such that p_{i_{d}} = q_{j_{d}} for all d ⩽ m. It is now easy to see that X satisfies the finite full amalgamation closure conditions.

We define an extension function approximation g for X as follows. Fix p_{i}(x̄) and θ_{j}(x̄ȳ) consistent with p_{i}(x̄). By effective quantifier elimination and the fact that a disjunction is implied by each of its disjuncts, we may assume that θ_{j} is a conjunction of literals. Let k, c be such that i = ⟨k, c⟩. There are several cases.

(1) p_{i} is the 0-type. Then we can find an m such that q_{m} contains θ_{j} and is of the form s_{E,(l_{0},...,l_{n−1})}, where n = |ȳ| and each l_{j} is in ℕ, and let g(i, j, s) = ⟨m, 0⟩ for all s ∈ ℕ.

(2) p_{i} is an m-type for m ⩾ 2 that is defined to be q_{k}. Then we also have p_{⟨h(k,j),c⟩} = q_{h(k,j)}, so let g(i, j, s) = ⟨h(k, j), c⟩ for all s ∈ ℕ.

(3) p_{i} is an m-type for m ⩾ 2, but case 2 does not hold. Let j′ be such that θ_{j′} implies θ_{j} and the free variables of θ_{j′} are x_{0}, …, x_{m} for some m (for instance, θ_{j′} can be obtained by adding conjuncts of the form x_{l} = x_{l} to θ_{j}). Then it follows from the definition of X that p_{⟨h(i,j′),c⟩} is obtained from q_{h(i,j′)} by the same process as p_{i} is obtained from q_{k}, which implies that p_{i} ∪ {θ_{j}} ⊂ p_{⟨h(k,j′),c⟩}, so let g(i, j, s) = ⟨h(k, j′), c⟩ for all s ∈ ℕ.

(4) p_{i} is a 1-type and p_{⟨h(k,j),c⟩} is defined to be q_{h(k,j)}. Then p_{i} = q_{k}, so let g(i, j, s) = ⟨h(k, j), c⟩ for all s ∈ ℕ.

(5) None of the above cases hold. Then p_{i} = q_{k} is a 1-type and q_{h(k,j)} is of the form s_{E,(∞,l_{1},...,l_{n+1})}(x_{0}, …, x_{n+1}), where ¬E(0, 1). By our assumptions on h, it follows that p_{i} is r_{∞}(x_{0}), and that θ_{j}(x_{0}, …, x_{n+1}) does not include x_{0} = x_{1}. Let l be a number larger than all d such that θ_{j} includes P_{d}(x_{1}) and less than or equal to all d such that θ_{j} includes ¬P_{d}(x_{1}). Such an l exists because θ_{j} is consistent with T. Let j′ be such that θ_{j′} is θ_{j} ∧ ¬P_{l}(x_{1}) ∧ P_{l−1}(x_{1}), where the last conjunct is omitted if l = 0. If (∀m < s)¬f(m) = l, then let g(i, j, s) = ⟨h(k, j′), 0⟩. Otherwise, let g(i, j, s) = ⟨h(k, j′), s⟩.

In any case, it follows from the definition of X that lim_{s} g(i, j, s) exists, that p_{g(i,j,s)} ↾ s = p_{g(i,j,s+1)} ↾ s for all s ∈ ℕ, and that p_{i} ∪ {θ_{j}} ⊂ p_{lim_{s} g(i,j,s)}.

Now suppose that X has an extension function G. For each n, let i, j be such that p_{i} = r_{∞}(x_{0}) and θ_{j} = P_{n−1}(x_{1})∧¬P_{n}(x_{1}) (or just ¬P_{0}(x_{1}) if n = 0). Then G(i, j) must have the form ⟨k, c⟩ for some c such that either (∃m < c)f(m) = n, or n is not in the range of f. Thus from G we obtain the range of f as a set. □

6.2. The reverse mathematics of extension functions. From the reverse mathematical point of view, there are several versions of Theorem 6.4 that can be considered. We begin with the following simple facts.

Proposition 6.6. The following are provable in RCA_{0}.

(1) For every model A, there is an enumeration of 𝕋(A) with an extension function and the same properties as in Proposition 5.13.

(2) If two lists of types are equal as sets of types and one has an extension function approximation, then so does the other.

(3) If two lists of types are equal as sets of types and one has an extension function, then the other has an extension function approximation.

(4) For every model A, every enumeration of 𝕋(A) has an extension function approximation.

Proof. (1) The enumeration described in the proof of Proposition 5.13 clearly has an extension function.

(2) Suppose that Y = {q_{i}}_{i∈ℕ} has an extension function approximation g and that X = {p_{i}}_{i∈ℕ} is equal to Y as a set of types. Given i and an index j for an (n + c)-ary formula θ consistent with the n-type p_{i}, proceed as follows. Let f(i, j, 0) = 0, let k_{0} = 0, and let φ_{0} = θ. Given f(i, j, s), let k_{s+1} be the least k such that q_{k} is an n-type with q_{k} ↾ s = p_{i} ↾ s. If k_{s+1} = k_{s} then let φ_{s+1} = φ_{s}. Otherwise, let φ_{s+1} be the conjunction of the formulas in p_{f(i,j,s)} ↾ s and θ. Let m be the index of φ_{s+1}, and let f(i, j, s + 1) be the least l such that p_{l} ↾ s + 1 = q_{g(k,m,s+1)} ↾ s + 1. It is straightforward to check that f is an extension function approximation for X, using the fact that for each k there is a least l such that p_{k} = q_{l} (and vice-versa). This fact follows from IΠ_{1}^{0}, which is provable in RCA_{0}.

(3) follows from (2), while (4) follows from (1) and (3). □

We summarize what we know about other versions of Theorem 6.4 in Figure 2. The column heading describes which closure conditions we assume the list of types X = {p_{i}}_{i∈ℕ} satisfies. When a row heading asserts the existence of a model A, we mean that X is an enumeration of 𝕋(A). When a row heading asserts the existence of a listing, we mean that there is a list of types Y = {q_{i}}_{i∈ℕ} with the stated property such that X and Y are equal as sets of types. For a given column C and row R, let C ⇒ R be the principle stating that if X has the closure conditions given by the heading of C and has an extension function approximation, then the statement in the heading of R holds. (When the heading of R mentions amalgamation, we take it in the same sense (free or full) as in the heading of C. Note that, by Proposition 5.10, we do not need a separate “there exists a listing with pairwise amalgamation” row, as such a row would be the same as the first row in our chart.)

p’wise finite 1-pt. p’wise finite
free free full full full
amalg. amalg. amalg. amalg. amalg.
X has pairwise amalg.  √  √  2  √  √
X has finite amalg.  2  √  2  2  √
∃ listing w/ finite amalg.  3  √  2  2  √
∃ model  3  1  2  3  1
∃ 1-pt. homogeneous model  X  X  2  3  1
∃ 1-homogeneous model  X  X  2  3  1
∃ homogeneous model  X  X  2  2  1
∃ listing w/ EF  4  1  5  4  1
∃ listing w/ EF + p’wise am. 4  1  2  4  1
∃ listing w/ EF + fin. am.  3  1  2  2  1

Figure 2. Versions of Theorem 6.4

The entry in row R and column C in Figure 2 describes the strength of the principle C ⇒ R. A √ means that the principle is true by definition, and an X means that it is false. The meanings of the number codes are as follows.

• A 1 means that the principle is true in RCA_{0}.

• A 2 means that the principle is equivalent to IΣ_{2}^{0} over RCA_{0}.

• A 3 means that the principle is provable from Π_{1}^{0}GA over RCA_{0} and is equivalent to IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0} (and hence not provable in RCA_{0}, or even RCA_{0} + BΣ_{2}^{0}).

• A 4 means that the principle is provable from Π_{1}^{0}GA over RCA_{0}, but its strength is otherwise unknown.

• A 5 means that the principle is provable from IΣ_{2}^{0} over RCA_{0}, but its strength is otherwise unknown.

The proof that entries marked 1 are true in RCA_{0} is done simultaneously with the proof that entries marked 3 or 4 are provable from Π_{1}^{0}GA, and employs Shore blocking. We use versions of a single example to obtain IΣ_{2}^{0} from the entries marked 2 and from the entries marked 3 together with BΣ_{2}^{0}. We give two charts (Figures 3 and 4) summarizing the justifications of the correctness of Figure 2 in the proof of Theorem 6.13 below.

We can also think of this chart with C ⇒ R reinterpreted to mean that if X has the closure conditions given by the heading of C and has an extension function (rather than an extension function approximation), then the statement in the heading of R holds. In this case, the chart remains exactly the same, except for the third to last row, which is now all √’s; the next to last row, where the 1’s and 4’s are now √’s; and the last row, where the 1’s are now √’s. We show these changed lines in Figure 3. The justifications of correctness in this case are also given by the charts in the proof of Theorem 6.13.

p’wise finite 1-pt. p’wise finite
free free full full full
amalg. amalg. amalg. amalg. amalg.
∃ listing w/ EF  √  √  √  √  √
∃ listing w/ EF + p’wise am. √  √  2  √  √
∃ listing w/ EF + fin. am.  3  √  2  2  √

Figure 3. Changes to Figure 2 in the extension function case

As noted in Section 3, the definition of Π_{1}^{0}GA is intended to capture the reverse mathematical behavior of the principles marked with a 3 in a more combinatorial way. However, we do not know whether these principles are in fact equivalent to Π_{1}^{0}GA.

In the remainder of this section, we prove theorems that, together with results in preceding sections, are used in Theorem 6.13 below to verify the correctness of both versions of our chart. We begin by defining an auxiliary notion of type amalgamation that is implied by full amalgamation and, in the presence of Π_{1}^{0}GA, also by pairwise amalgamation. We later give a proof establishing the provability in RCA_{0} of the entries labeled 1 in our chart, the provability from IΣ_{2}^{0} of the entries labeled 2 or 5, and the provability from Π_{1}^{0}GA of the entries labeled 3 or 4. In this proof, instead of working with finite or pairwise amalgamation directly, we work with this auxiliary notion, which is designed to fit a finite injury priority construction (with Shore blocking). Here and below, we use the bracket notation introduced in Remark 5.9.

Definition 6.7. Let X = {p_{i}(z̄_{i})}_{i∈ℕ} be a list of types of a theory T. Let a_{i} = |z̄_{i}|. Let R be the set of all (i, n) with n ⩽ a_{i}. Let x_{i} and y_{i,n,j} for i, n, j ∈ ℕ be pairwise distinct variables. For (i, n) ∈ R, let q_{i,n} = p_{i}[x_{0}, …, x_{n−1}, y_{i,n,0}, …, y_{i,n,a_{i}−n−1}]. (So if (i, n) and (i′, n′) are distinct elements of R, then the first min(n, n′) many variables in q_{i,n} and q_{i′,n′} are the same, but the other variables in q_{i,n} are pairwise distinct from the other variables in q_{i′,n′}.) For F ⊆ R, let q_{F} = {q_{i,n} : (i, n) ∈ F }. Recall that our definition of consistency of a finite set of types requires that there be a single type in the set containing all the variables shared by any two types in the set. If F ⊂ R is finite, then this is the case for q_{F}, so it makes sense to speak of the consistency of q_{F}.

(1) X satisfies the eventual full type amalgamation condition if there

are finite F_{0}, F_{1}, … ⊂ R such that the following hold.

(a) For each (i, n) ∈ R, we have (i, n) ∈ F_{s} for all sufficiently large s.

(b) There are infinitely many s such that, for each S ⊆ F_{s} for which q_{S} is consistent, there is a p_{j} amalgamating all the types in q_{S}.

(2) X satisfies the eventual free type amalgamation condition if the above holds for {(i, 0) : i ∈ ℕ} in place of R.

(3) X satisfies the eventual full amalgamation closure conditions (resp. eventual free amalgamation closure conditions) if it satisfies the conditions in Theorem 5.1 with pairwise full type amalgamation replaced by eventual full type amalgamation (resp. eventual free type amalgamation).

Clearly, if a list of types of a theory satisfies the finite full (resp. free) amalgamation closure conditions then it satisfies the eventual full (resp. free) amalgamation closure conditions. For pairwise amalgamation, we have the following fact.

Lemma 6.8. The following is provable in RCA_{0} + Π_{1}^{0}GA. If a list of types of a theory satisfies the pairwise full (resp. free) amalgamation closure conditions then it satisfies the eventual full (resp. free) amalgamation closure conditions.

Proof. We prove the lemma for full amalgamation, as the free case is similar but simpler. We argue in RCA_{0}+Π_{1}^{0}GA. We adopt the notation of Definition 6.7. Let X = {p_{i}(z̄_{i})}_{i∈ℕ} be a list of types of a theory T satisfying the pairwise full amalgamation closure conditions. Let w_{i,j} for i, j ∈ ℕ be pairwise distinct variables that are also distinct from each x_{i} and each y_{i,n,j}. For (i, n) ∈ R, let r_{i,n} = q_{i,n} ↾ x_{0}, …, x_{n−1}, and let l_{s}(i, n) be the least l such that r_{l,n} ↾ s = r_{i,n} ↾ s. Let E ⊂ R be finite. We say that a pair (d, s) provides amalgamators for E if for each (i, n) ∈ E, the type p_{d} contains

p_{i}[w_{l_{s}(i,1),0}, …, w_{l_{s}(i,n),n−1}, y_{i,n,0}, …, y_{i,n,a_{i}−n−1}].

This property is Π_{1}^{0}, so there is a Δ_{0}^{0} predicate P such that (d, s) provides amalgamators for E if and only if (∀x)P (d, s, E, x).

Suppose (d, s) provides amalgamators for E, and let S ⊆ E be such that q_{S} is consistent. If (i, n), (i′, n′) ∈ S, then for all k ⩽ min(n, n′), we have r_{i,k} = r_{i′,k}, and hence l_{s}(i, k) = l_{s}(i′, k). So if we choose (i, n) ∈ S with the largest n and let p_{j} be the type obtained from p_{d} by first passing to a subtype that does not include the variables x_{0}, …, x_{n−1} and then replacing the variables w_{l_{s}(i,1),0}, …, w_{l_{s}(i,n),n−1} by x_{0}, …, x_{n−1}, then p_{j} amalgamates the types in q_{S}. So the idea is to use Π_{1}^{0}GA to find such (d, s) for larger and larger sets E.

Let (E_{0}, d_{0}), (E_{1}, d_{1}), … list all pairs consisting of a finite subset of R and a natural number, such that each such pair is listed infinitely often. For σ ∈ 2^{<ℕ}, proceed as follows to define a tuple (E_{σ}, d_{σ}, s_{σ}, t_{σ}). Let s_{σ} be the largest s such that σ(3s) = 1, or 0 if there is no such s. Let t_{σ} be the largest t such that σ(3t + 1) = 1, or 0 if there is no such t. Let k_{0} < ⋯ < k_{m−1} be the numbers such that σ(3k_{i} + 2) = 1. Let (E^{0}, d^{0}) = (∅, 0). Given (E^{i}, d^{i}), proceed as follows. If E_{k_{i}} ⊇ E^{i} and (∀x < t_{σ})P (d_{k_{i}}, s_{σ}, E_{k_{i}}, x), then let (E^{i+1}, d^{i+1}) = (E_{k_{i}}, d_{k_{i}}), and otherwise let (E^{i+1}, d^{i+1}) = (E^{i}, d^{i}). Finally, let (E_{σ}, d_{σ}) = (E^{m}, d^{m}). Note that (∀x < t_{σ})P (d_{σ}, s_{σ}, E_{σ}, x).

We now define some uniformly Π_{1}^{0} dense sets. (As in the definition of Π_{1}^{0}G and Π_{1}^{0}GA in Section 3, we refer to these Π_{1}^{0} properties as sets and use set notation for convenience, but we are not asserting their existence as sets in RCA_{0}.) We begin with the sets D_{c} of all τ such that |τ| > c and (d_{τ}, s_{τ}) provides amalgamators for E_{τ}. These sets are uniformly Π_{1}^{0}; we claim each D_{c} is also dense. To see that this is the case, fix c and σ, and let k_{0} < ⋯ < k_{m−1} be as above. It is provable in RCA_{0} that there is a t ⩾ |σ| such that, for each i < m, if (∃x)¬P (d_{k_{i}}, s_{σ}, E_{k_{i}}, x) then (∃x < t)¬P (d_{k_{i}}, s_{σ}, E_{k_{i}}, x). Let τ ≻ σ be such that |τ| > c and τ(j) = 1 if and only if σ(j) = 1 or j = 3t + 1. Then t_{τ} = t and s_{τ} = s_{σ}, while (E_{τ}, d_{τ}) = (E_{k_{i}}, d_{k_{i}}) for some i < m (or (E_{τ}, d_{τ}) = (∅, 0)). Thus the fact that (∀x < t_{τ})P (d_{τ}, s_{τ}, E_{τ}, x) implies that (∀x)P (d_{τ}, s_{τ}, E_{τ}, x), and hence (d_{τ}, s_{τ}) provides amalgamators for E_{τ}.

Now let C_{(i,n)} for (i, n) ∈ R consist of all τ such that (i, n) ∈ E_{ρ} for all ρ ≽ τ. The C_{(i,n)} are uniformly Π_{1}^{0}; we claim that each C_{(i,n)} is also dense. To see that this is the case, fix (i, n) ∈ R and σ, and let k_{0} < ⋯ < k_{m−1} be as above. Let E = ⋃_{j<m} E_{k_{j}} ∪ {(i, n)}. For each (j, k) ∈ E, the value of l_{s}(j, k) changes at most j many times, so it is provable in RCA_{0} that there is an s ⩾ |σ| such that l_{u}(j, k) = l_{s}(j, k) for all u ⩾ s and all (j, k) ∈ E. Write l(j, k) for l_{s}(j, k). As above, there is a t ⩾ |σ| such that, for each i < m, if (∃x)¬P (d_{k_{i}}, s, E_{k_{i}}, x) then (∃x < t)¬P (d_{k_{i}}, s, E_{k_{i}}, x). Let ν ≻ σ be such that ν(j) = 1 if and only if σ(j) = 1 or j = 3s or j = 3t + 1. Then for each (j, k) ∈ E_{ν}, the type p_{d_{ν}} contains

p_{j}[w_{l(j,1),0}, …, w_{l(j,k),k−1}, y_{j,k,0}, …, y_{j,k,a_{j}−k−1}].

By passing to a subtype if necessary, we may assume that p_{d_{ν}} does not contain the variable w_{l(i,c+1),c} for c < n unless there is a (j, k) ∈ E_{ν} such that c < k and l(j, c + 1) = l(i, c + 1), in which case r_{j,c+1} = r_{i,c+1}. Thus p_{d_{ν}} is consistent with

p_{i}[w_{l(i,1),0}, …, w_{l(i,n),n−1}, y_{i,n,0}, …, y_{i,n,a_{i}−n−1}],

and hence there is a type p_{d} amalgamating these two types. Let k > |ν| be such that (E_{k}, d_{k}) = (E_{ν} ∪ {(i, n)}, d), and let τ ≻ ν be such that τ(j) = 1 if and only if ν(j) = 1 or j = 3k + 2. Then it is easy to check that E_{τ} = E_{ν} ∪ {(i, n)}, and that if ρ ≽ τ, then the definition of E_{ρ} proceeds in the same way as the definition of E_{τ} for its first m + 1 many steps, whence (i, n) ∈ E_{τ} ⊆ E_{ρ}.

Now let g_{0}, g_{1}, … be as in Π_{1}^{0}GA for the uniformly Π_{1}^{0} dense sets D_{m} for m ∈ ℕ and C_{(i,n)} for (i, n) ∈ R. Let σ_{0}, σ_{1}, … be a listing without repetitions of all σ such that there is a t with σ ≺ g_{t}. Let F_{u} = E_{σ_{u}}. Let (i, n) ∈ R. Then there are k and s such that for all t ⩾ s, we have g_{t} ↾ k = g_{s} ↾ k ∈ C_{(i,n)}, and hence (i, n) ∈ F_{u} for almost all u. For each m, there are k and s such that g_{s} ↾ k ∈ D_{m}, so there are infinitely many F_{u} such that some (d, v) provides amalgamators for F_{u}. As discussed above, for any such u and any S ⊆ F_{u} such that q_{S} is consistent, there is a p_{j} amalgamating all the types in q_{S}. Thus X satisfies the eventual full type amalgamation condition. □

The next lemma gives us a convenient way to build a model with a specified type spectrum by computably approximating the types of initial segments of the domain of the model.

Lemma 6.9. The following is provable in RCA_{0}. Let X = {p_{i}}_{i∈ℕ} be a list of types of a theory T. Let q_{i,n} be as in Definition 6.7. Suppose that X satisfies the first four conditions in Theorem 5.1, and there is a binary partial function f with the following properties for all n, s, and k.

(i) p_{f(n,s)} is an n-type in the variables x_{0}, …, x_{n−1} whenever defined.

(ii) p_{f(n,s)} ↾ s = p_{f(n,s+k)} ↾ s whenever these are defined.

(iii) p_{f(n,s)} ↾ s ⊆ p_{f(n+1,s)} ↾ s whenever these are defined.

(iv) f(n, s) is defined for all sufficiently large s ∈ ℕ and lim_{s} f(n, s) exists. We denote this limit by f(n). (Note that this notation is purely an abbreviation for the limit notation; we cannot in general prove that the function taking n to f(n) exists.)

(v) For each (i, n) ∈ R, if p_{f(n)} ⊆ q_{i,n} then there exist m ∈ ℕ and distinct j_{0}, …, j_{a_{i}−n−1} ∈ [n, m − 1] such that

p_{i}[x_{0}, …, x_{n−1}, x_{j_{0}}, …, x_{j_{a_{i}−n−1}}] ⊆ p_{f(m)}.

Then the following hold.

(1) If X satisfies the finite full type amalgamation condition, then there is a homogeneous model M such that X is an enumeration of the type spectrum of M.

(2) If X satisfies the pairwise full type amalgamation condition, then there is a 1-homogeneous model M such that X is an enumeration of the type spectrum of M.

(3) Suppose that we weaken condition (v) by requiring it to hold only for n = 0. Then there is a model M such that X is an enumeration of the type spectrum of M.

Proof. (1) We obtain a homogeneous model M such that X is an enumeration of 𝕋(M) as follows. First, define an equivalence relation E on ℕ. For each i, j ∈ ℕ, let n = max(i, j) + 1. Then there is an s ∈ ℕ such that either x_{i} = x_{j} ∈ p_{f(n,s)} or x_{i} ≠ x_{j} ∈ p_{f(n,s)}. Let E(i, j) hold if and only if the first case obtains. Let the domain M of M consist of all i ∈ ℕ such that (∀j < i)¬E(i, j). To specify the satisfaction predicate of M, it is enough to say whether or not φ^{M}(m_{0}, …, m_{k−1}) holds for each formula φ(y_{0}, …, y_{k−1}) in the language of T and each tuple (m_{0}, …, m_{k−1}) of distinct elements of M, and then verify that our definition has the properties of a satisfaction predicate. Given such a φ and m_{0}, …, m_{k−1}, let n = max_{i<k} m_{i} + 1. Then there is an s ∈ ℕ such that either φ(x_{m_{0}}, …, x_{m_{k−1}}) ∈ p_{f(n,s)} or ¬φ(x_{m_{0}}, …, x_{m_{k−1}}) ∈ p_{f(n,s)}. Let φ^{M}(m_{0}, …, m_{k−1}) hold if and only if the first case obtains.

Then φ^{M} holds for all φ ∈ T, since every type in X contains T. It is also easy to check that the predicate defined in this way is deductively closed and consistent, and hence is a satisfaction predicate, so M is a model of T. By the properties of f, the fact that X is closed under subtypes, and Remark 5.2, every M-tuple realizes a type in X. For each p_{i} ∈ X, we have p_{f(0)} ⊆ p_{i}, so by the fifth property of f, the type p_{i} is realized in M.

It is easy to see that in the definition of homogeneity of a model (Definition 4.1(iii)), restricting attention to M-tuples of distinct elements yields an equivalent definition of homogeneity. Thus, to show that M is homogeneous, let ā_{0} ≡ b̄_{0}, …, ā_{n} ≡ b̄_{n} and c̄_{0}, …, c̄_{n} be M-tuples of distinct elements (by which we mean that each ā_{i}, b̄_{i}, and c̄_{i} is a tuple of distinct elements, though some of these tuples could have elements in common). Let m be larger than all of the elements of these tuples. Let k_{i} = |b̄_{i}| and l_{i} = |c̄_{i}|. Write b̄_{i} = (b_{i,0} …, b_{i,k_{i}−1}). Let z_{i,j} for i ⩽ n and j < l_{i} be pairwise distinct variables that are also pairwise distinct from the x_{i}. Let r_{i}(x_{b_{i,0}}, …, x_{b_{i,k_{i}−1}}, z_{i,0}, …, z_{i,l_{i}−1}) be the type of ā_{i}c̄_{i}. The sequence of types consisting of p_{f(m)} and r_{i} for i ⩽ n is basically consistent, so there is a type

p_{j}(x_{0}, …, x_{m−1}, z_{0,0}, …, z_{0,l_{0}−1}, …, z_{n,0}, …, z_{n,l_{n}−1})

amalgamating these types.

Let b̄ be the result of removing from b̄_{0} … b̄_{n} any b_{i,j} for which there are an i′ < i and a j′ such that b_{i′,j′} = b_{i,j}. Let x̄ be the result of removing from the list x_{b_{0,0}} … x_{b_{0,k_{0}−1}} … x_{b_{n,0}} … x_{b_{n,kn−1}} any x_{i,j} for which there are an i′ < i and a j′ such that x_{i′,j′} is the same variable as x_{i,j}. It now follows from the fifth property of f that there are d̄_{0}, …, d̄_{n} ∈ M such that b̄d̄_{0} … d̄_{n} realizes

p_{j} ↾ x̄z_{0,0} … z_{0,l_{0}−1} … z_{n,0} … z_{0,l_{n}−1}.

Then ā_{i}c̄_{i} ≡ b̄_{i}d̄_{i} for all i ⩽ n.

(2) The above argument works in this case as well, except that in the last two paragraphs, we take n = 0, which makes the amalgamation performed in that paragraph an instance of pairwise full type amalgamation.

(3) The same argument again works in this case, ignoring the last two paragraphs. □

We are now ready to establish a theorem that gives us the provability in RCA_{0} of the entries labeled 1 in our chart, the provability from IΣ_{2}^{0} of the entries labeled 2 or 5, and the provability from Π_{1}^{0}GA of the entries labeled 3 or 4.

Theorem 6.10. The following are provable in RCA_{0}. Let X = {p_{i}}_{i∈ℕ} be a list of types of a theory T, such that X has an extension function approximation.

(1) If X satisfies the finite full amalgamation closure conditions then there is a homogeneous model M such that X is an enumeration of the type spectrum of M.

(2) If X satisfies the finite free amalgamation closure conditions then there is a model M such that X is an enumeration of the type spectrum of M.

(3) If Π_{1}^{0}GA holds and X satisfies the pairwise full amalgamation closure conditions then there is a 1-homogeneous model M such that X is an enumeration of the type spectrum of M.

(4) If Π_{1}^{0}GA holds and X satisfies the pairwise free amalgamation closure conditions then there is a model M such that X is an enumeration of the type spectrum of M.

Proof. We do the full amalgamation case, as the free amalgamation case is similar. By Lemma 6.8 and the comment preceding that lemma, we may assume that X satisfies the eventual full amalgamation closure conditions. From now on, this is the assumption we use, so we prove (1) and (3) simultaneously. It is enough to work in RCA_{0} and build a function f as in Lemma 6.9. Let R and F_{0}, F_{1}, … be as in Definition 6.7.

Suppose that we build a list of types Y as follows. For each p_{i} in turn, add all of its subtypes (including p_{i} itself), and all types obtained from these subtypes by changes of variables, to Y, in an effective way. Let I be the set of all j such that the jth type added to Y is one of the original p_{i}’s. Then X and Y contain the same types, and, by Proposition 6.6, Y also has an extension function approximation. Thus, by replacing X with Y, we may assume that X is closed under sequences of variable substitutions, and indeed that there is an effective procedure that, for each p_{i} and each sequence s of types obtained from subtypes of p_{i} by a given change of variables, determines a sequence of elements of X equal to s. We may also redefine R to contain only those (i, n) such that i ∈ I. Having property (v) of f hold for such (i, n) suffices for the proof of Lemma 6.9 to go through, and we still have sets F_{0}, F_{1}, … as in Definition 6.7.

Let g be an extension function approximation for X, relative to a listing φ_{0}, φ_{1}, … of the formulas in the language of T. We may assume that g is defined so that p_{g(i,j,s)} always contains φ_{j}. Furthermore, if for each i and j such that φ_{j}(w̄z̄) is consistent with p_{i}(w̄), we replace each g(i, j, s) by the least k such that p_{k} ↾ s = (p_{g(i,j,s)} ↾ w̄z̄) ↾ s, then it is easy to check that we still have an extension function approximation. Thus we may assume that for such j and i, each p_{g(i,j,s)} is a |w̄z̄|-type in the variables w̄z̄.

We ensure properties (i)–(iii) of f as we go along the stage-by-stage construction. We think of the remaining two properties as split into requirements. For (iv), there is a requirement Q_{n} for each n ∈ ℕ, and for (v) a requirement R_{i,n} for each i, n ∈ R. We satisfy these requirements by a finite injury priority construction. Not all finite injury constructions carry through in RCA_{0}, however. The typical issue is the following. There is no problem in verifying that a given requirement eventually stops acting, given that there is a stage s at which all stronger priority requirements have stopped acting. However, concluding that this fact implies that, for each requirement, such a stage s exists might require more induction than is available in RCA_{0}. (The issue of how much induction is needed for various computability theoretic constructions has been extensively analyzed by the research program known as reverse recursion theory; see e.g. Mytilinaios [37].) To avoid this issue, we use Shore blocking. The use of blocking in the reverse mathematics of model theory is quite natural, since a finite list of requirements, each involving a particular type, can often be combined into a single requirement involving an amalgamation of the original types. For other examples of this technique, see [23].

In our case, the Q-requirements block naturally, but the R-require-ments need some care. Satisfying a single R-requirement in the absence of injuries from stronger priority requirements is no problem, if we have at least pairwise full type amalgamation. However, satisfying an arbitrary block of finitely many R-requirements at once requires finite full type amalgamation. So instead of satisfying our R-requirements directly, we make use of the sets F_{0}, F_{1}, … from Definition 6.7 and satisfy requirements S_{s} saying that there is a t ⩾ s such that R_{i,n} is satisfied for every (i, n) ∈ F_{t} with n ⩽ s. Satisfying the S_{s} is clearly enough to satisfy all R_{i,n}, but the S-requirements have the advantage that satisfying S_{s} is enough to satisfy S_{u} for all u ⩽ s.

For each c, s ∈ ℕ, we define a number b_{c,s}. At stage s, the Srequirements we attempt to satisfy are S_{b_{0,s}}, S_{b_{1,s}}, . . .. For each S_{b_{c,s}}, we need numbers from which to draw the witnesses required to satisfy the appropriate R-requirements (i.e., the numbers j_{i} mentioned in property (v) of f). So we define b_{c+1,s} so that b_{c+1,s} − b_{c,s} is at least as large as the total number of witnesses we might need, and draw our witnesses for S_{b_{c,s}} from [b_{c,s}, b_{c+1,s}). (We define b_{0,s} = 0 for all s.)

At stage s of our construction, the cth block of requirements consists of S_{b_{c,s}} and the Q_{n} such that n ∈ (b_{c,s}, b_{c+1,s}]. We try to satisfy all requirements in this block simultaneously, by defining f(b_{c+1,s}, s) to attempt to satisfy S_{b_{c,s}}, defining f(n, s) for n ∈ (b_{c,s}, b_{c+1,s}) so that p_{f(n,s)} is the appropriate subtype of p_{f(b_{c+1,s},s)} (which we can do because of the effective closure under sequences of variable substitutions discussed in the second paragraph of this proof), and then attempting to preserve these definitions. Of course, we also attempt to ensure that p_{f(b_{c,s},s)} is a subtype of p_{f(b_{c+1,s},s)}.

When we say that we define f(b_{c+1,s}, s) to attempt to satisfy S_{b_{c,s}}, we mean the following. Say that R_{i,n} is relevant at stage s if n ⩽ b_{c,s} and the hypothesis of R_{i,n} appears to hold at stage s. Item (1)(b) in Definition 6.7 implies that there is some F_{t} with t ⩾ b_{c,s} such that we can simultaneously satisfy all permanently relevant R_{i,n} with (i, n) ∈ F_{t}. We guess at such an F_{t}, and choose potential witnesses from [b_{c,s}, b_{c+1,s}) for all relevant R_{i,n} such that (i, n) ∈ F_{t}. We then define f(b_{c+1,s}, s) to attempt to ensure that these witnesses have the correct type to satisfy these requirements. Even absent interference from other blocks, such an attempt might fail, as defining f(b_{c+1,s}, s) correctly requires first having an appropriate t, then having a type amalgamating p_{f(b_{c+1,s},s)} with the various types involved in all the relevant R_{i,n}, both of which can be found only by trial and error. (Though once the block stops growing, these will eventually be found.) When such an attempt fails, we need to redefine b_{c+1,s} to create enough room for the next attempt (i.e., we choose it so that there are enough elements below b_{c+1,s} on which f has not yet been defined to provide witnesses for all the Rrequirements that might be considered in the next attempt at satisfying S_{b_{c,s}}). Another issue we need to confront in defining f(b_{c+1,s}, s) is that f(k, s−1) might have been defined for some k ∈ (b_{c,s}, b_{c+1,s}]. If so, then we need to use our extension function approximation g to determine f(b_{c+1,s}, s), as p_{f(b_{c+1,s},s)} has to contain p_{f(k,s−1)} ↾ s − 1 for each such k.

In addition, definitions made for two blocks, say the cth and dth for c < d, might turn out to be in conflict (i.e., p_{f(b_{c+1,s},s)} might turn out not to be a subtype of p_{f(b_{d+1,s},s)}). Then we injure the dth block, changing the values of f it has defined. When we do so, we increase b_{e,s} for all e > d. As we show below, this action ensures that the correctness of our construction can be verified in RCA_{0}.

To each block c, we attach parameters i_{c} and j_{c}, which might change or be undefined during the construction. These are chosen so that p_{i_{c}} is a type in variables among x_{0}, …, x_{b_{c+1,s}−1} and φ_{j_{c}}(x_{0}, …, x_{b_{c+1,s}−1}) is consistent with p_{i_{c}}. We ensure that whenever i_{c} and j_{c} are defined, then so are i_{d} and j_{d} for d < c. At the end of each stage s > 0, we define f(b_{c+1,s}, s) = g(i_{c}, j_{c}, s) if these parameters are defined. Thus i_{c} represents a type through which we are attempting to satisfy S_{b_{c,s}}, and j_{c} represents a formula that we need to amalgamate with this type to preserve properties (i)–(iii) of f. We also assign tuples of witnesses to R-requirements corresponding to S_{b_{c,s}}, which again might change during the construction. We ensure that if S_{b_{c,s}} has witnesses assigned to it, then i_{c} and j_{c} are defined.

Construction of f. At stage 0, act as follows. Let b_{j,0} = j for all j ∈ ℕ. Let e be such that p_{e} is the 0-type and let f(0, s) = e for all s ∈ ℕ. For notational convenience, we also let g(i_{−1}, j_{−1}, s) denote e for all s ∈ ℕ.

At stage s > 0, act as follows. Recall that a_{i} is the arity of p_{i}. For each c such that i_{c} and j_{c} are defined, say that (i, n) ∈ R is cactive if q_{i,n} ↾ s is consistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)} (which we can check effectively because of the monotonicity of g). Say that c is satisfied if i_{c} and j_{c} are defined, s ⩾ b_{c,s−1}, and there is a t ∈ [b_{c,s−1}, s] such that each c-active (i, n) ∈ F_{t} with n < b_{c,s−1} has witnesses j_{0}, …, j_{a_{i}−n−1} assigned to it for which

p_{i}[x_{0}, …, x_{n−1}, x_{j_{0}}, …, x_{j_{a_{i}−n−1}}] ↾ s ⊂ p_{g(i_{c},j_{c},s)}.

Say that c requires attention if i_{c} and j_{c} are defined, s ⩾ b_{c,s−1}, and c is not satisfied. If i_{c} and j_{c} are defined, then say they are untenable if p_{g(i_{c},j_{c},s)} ↾ s is inconsistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)}.

Let c be least such that i_{c} and j_{c} are undefined or untenable, or c requires attention. (Such a c must exist because we define i_{d} and j_{d} for only finitely many d at each stage.) We say that c is active at stage s. Let b_{d,s} = b_{d,s−1} for all d ⩽ c. Let k_{s} be the number of stages u < s at which c has been active and b_{c,u} = b_{c,s}, and let t = b_{c,s} + n, where n is such that k_{s} = ⟨n, l⟩ for some l. Let b_{c+1,s} be large enough so that [b_{c,s}, b_{c+1,s}) contains at least ∑_{(i,n)∈F_{t}} a_{i} many fresh elements (i.e., ones on which f has never been defined). Let b_{c+1+m,s} = b_{c+1,s} + m for all m > 0. For d > c, undefine i_{d}, j_{d}, and all witnesses associated with the dth block.

Say that R_{i,n} is relevant if it is c-active, n < b_{c,s}, and (i, n) ∈ F_{t}. Let R_{l_{0},n_{0}}, …, R_{l_{r−1},n_{r−1}} be the relevant requirements. For each i < r, let m_{i} = a_{l_{i}} − n_{i} and choose fresh distinct elements w_{0}, …, w_{m_{i}−1} ∈ [b_{c,s}, b_{c+1,s}), which we call witnesses for R_{l_{i},n_{i}}, so that the tuples of elements chosen for different i are pairwise disjoint. Let ψ be the conjunction of all formulas in p_{l_{i}}[x_{0}, …, x_{n−1}, x_{w_{0}}, …, x_{w_{m_{i}−1}}] ↾ s for i < r and in p_{g(i_{c−1},j_{c−1},s)} ↾ s. (If r = 0 then ψ is just the propositional constant T.) Let θ be the conjunction of all formulas appearing in p_{f(n,u)} ↾ u for n and u < s such that f(n, u) is defined.

All free variables shared by ψ and θ are among x_{0}, …, x_{b_{c,s}−1}, and p_{lim_{u} g(i_{c−1},j_{c−1},u)} is a type in the variables x_{0}, …, x_{b_{c,s}−1}, so to show that ψ∧θ is consistent, it is enough to show that each of ψ and θ is consistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)}. That ψ is consistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)} follows from the definition of a requirement being c-active. We show in the verification section below that θ is also consistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)}, so we assume that this is the case for now. Let i_{c} be least such that p_{i_{c}} is a type in the variables appearing in ψ, contains ψ, and is consistent with θ. Let j_{c} be such that φ_{j_{c}} is the conjunction of θ with x_{n} = x_{n} for all n < b_{c+1,s}. (These tautologies are there to ensure that φ_{j_{c}} has the correct arity.)

For each c ∈ ℕ such that i_{c} and j_{c} are defined, let f(b_{c+1,s}, s) = g(i_{c}, j_{c}, s). For n ∈ (b_{c,s}, b_{c+1,s}), if f(b_{c+1,s}, s) ≠ f(b_{c+1,s}, s − 1), then let f(n, s) be such that p_{f(n,s)} = p_{f(b_{c+1,s},s)} ↾ x_{0}, …, x_{n−1}. For all n < b_{c+1,s} such that f(n, s) has not yet been defined, let f(n, s) = f(n, s − 1). For all n > b_{c+1,s}, let f(n, s) be undefined.

Verification. Having completed the construction of f, we verify its correctness. We first need to show that the formula θ above is in fact consistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)}. By (Δ_{0}^{0}-)induction, we may assume that this property holds for all stages prior to s. Let v < s be the stage at which i_{c−1} and j_{c−1} were last defined prior to stage s. Let θ′ be the conjunction of all formulas appearing in p_{f(n,w)} ↾ w for n and w ⩽ v such that f(n, w) is defined. Then φ_{j_{c−1}} is equivalent to θ′, so θ′ ∈ p_{lim_{u} g(i_{c−1},j_{c−1},u)}. At each stage r ∈ (v, s), we maintain the consistency of each p_{f(n,r)} ↾ r such that f(n, r) is defined with p_{lim_{u} g(i_{c−1},j_{c−1},u)}, by the definition of untenability and the definition of f. Thus θ is consistent with p_{lim_{u} g(i_{c−1},j_{c−1},u)}.

We now verify that the five properties of f hold. Property (i) is clearly satisfied by the construction. Suppose f(n, s) and f(n, s+k) are both defined and n ∈ [b_{c,s+k}, b_{c+1,s+k}). Then either f(n, s+k) = f(n, s+ k−1) or i_{c} and j_{c} get defined at stage s+k, in which case p_{f(b_{c+1,s+k},s+k)} is defined so as to contain p_{f(n,s)} ↾ s, and p_{f(n,s+k)} = p_{f(b_{c+1,s+k},s+k)} ↾ x_{0}, …, x_{n−1}. Thus property (ii) holds by (Δ_{0}^{0}-)induction on k.

Now suppose f(n, s) and f(n + 1, s) are both defined. If there is a c such that n, n + 1 ∈ (b_{c,s}, b_{c+1,s}] then p_{f(n,s)} = p_{f(n+1,s)} ↾ x_{0}, …, x_{n−1}. Otherwise, there is a c such that n = b_{c,s}. If i_{c} and j_{c} do not get redefined at stage s then c is not untenable at stage s, so p_{f(b_{c+1,s},s)} ↾ s = p_{g(i_{c},j_{c},s)} ↾ s is consistent with lim_{u} g(i_{c−1}, j_{c−1}, u). Since p_{f(n,s)} ↾ s = p_{lim_{u} g(i_{c−1},j_{c−1},u)} ↾ s and p_{f(n+1,s)} = p_{f(b_{c+1,s},s)} ↾ x_{0}, …, x_{n}, we have p_{f(n,s)} ↾ s ⊆ p_{f(n+1,s)} ↾ s. If i_{c} and j_{c} do get redefined at stage s, then p_{f(b_{c+1,s},s)} contains p_{f(n,s)} ↾ s = p_{g(i_{c−1},j_{c−1},s)} ↾ s, so again p_{f(n,s)} ↾ s ⊆ p_{f(n+1,s)} ↾ s. In any case, property (iii) holds.

Thus we are left with verifying that all our requirements are satisfied, so that properties (iv) and (v) hold. Let b_{c} = lim_{s} b_{c,s}, which might be infinite. Suppose that b_{c} is finite. Every time i_{d} and j_{d} are redefined for some d < c, the value of b_{c,s} increases, so there is a stage s_{0} by which all these i_{d} and j_{d} have permanent values. In particular, i_{c−1} and j_{c−1} have permanent values (if c > 0). Then there is a stage s_{1} ⩾ s_{0} such that g(i_{c−1}, j_{c−1}, s) = g(i_{c−1}, j_{c−1}, s_{1}) for all s > s_{1}. Let h = g(i_{c−1}, j_{c−1}, s_{1}).

Let t ⩾ b_{c} be such that for each S ⊆ F_{t} for which q_{S} is consistent, there is a p_{j} amalgamating all the types in q_{S}. Let B be the set of requirements R_{i,n} such that (i, n) ∈ F_{t} and n < b_{c}. Any requirement in B that ceases to be c-active after stage s_{1} is never again c-active, so Σ_{1}^{0}-induction suffices to show that there is an s_{2} ⩾ s_{1} such that the collection R_{l_{0},n_{0}}, …, R_{l_{m−1},n_{m−1}} of c-active requirements in B has stabilized by stage s_{2}. By the definition of c-active requirement, the types p_{l_{i}}[x_{0}, …, x_{n−1}, y_{i,0}, …, y_{i,a_{l_{i}}−n_{i}−1}] for i < m form a consistent set of types, so by the choice of t, there is a p_{j} amalgamating them. Again by the definition of c-active requirement, p_{j} and p_{h} are consistent, so by pairwise full type amalgamation (which clearly follows from eventual full type amalgamation), there is a type in X amalgamating them. By Π_{1}^{0}-induction, there is a least n such that p_{n} is a type amalgamating p_{h} and all of the types p_{l_{i}}[x_{0}, …, x_{n−1}, y_{i,0}, …, y_{i,a_{l_{i}}−n_{i}−1}] for i < m. Then it is easy to check from the definition of i_{c} that there is a stage s_{3} ⩾ s_{2} such that if i_{c} is defined at a stage s ⩾ s_{3} at which k_{s} is of the form ⟨t − b_{c}, l⟩, then i_{c} = n.

Say that a stage s is c-final if c never requires attention after stage s, and i_{c} and j_{c} are never undefined or untenable after stage s. It is easy to check that if i_{c} and j_{c} are defined at a stage s ⩾ s_{3} at which k_{s} is of the form ⟨t − b_{c}, l⟩, then s is c-final. It is also easy to check that such a definition occurs unless k_{s} never reaches such a value for s ⩾ s_{3}. In this case, there must be a c-final stage, as k_{s} increases each time c is active. In either case, there is a c-final stage s. Then b_{c+1} = b_{c+1,s} is finite, and S_{b_{c}} is permanently satisfied by stage s. Furthermore, i_{c} and j_{c} have final values at stage s, and there is a stage u > s such that g(i_{c}, j_{c}, v) = g(i_{c}, j_{c}, u) for all v > u. Then f(n, v) = f(n, u) for all n ∈ (b_{c}, b_{c+1}), so every Q_{n} in the cth block is satisfied.

Now, given n ∈ ℕ, let d be least such that b_{d+1,s} ⩾ n for some s ∈ ℕ, which exists by Σ_{1}^{0}-induction. Then b_{d} is finite, so there is a final dth block of requirements, and Q_{n} is in that block, and hence is satisfied. Furthermore, b_{d+1} is also finite, so S_{b_{d+1}} is satisfied, and hence so is S_{n}, since n ⩽ b_{d+1}. □

We now establish two theorems that give us the reversals to IΣ_{2}^{0} of the entries labeled 2 or 3 on our chart.

Theorem 6.11. The following is provable in RCA_{0}. Let φ be a Π_{1}^{0} formula defining a total function. For each n and z, there is a theory T and a list X of types of T such that the following hold.

(1) X satisfies the pairwise full amalgamation closure conditions.

(2) X has an extension function.

(3) Suppose one of the following conditions holds.

(a) X satisfies the finite free amalgamation closure conditions.

(b) There is a listing of the types of X satisfying the finite full amalgamation closure conditions.

(c) There is a a listing of the types of X satisfying the finite free amalgamation closure conditions, and BΣ_{2}^{0} holds. Then there is a sequence (m_{0}, m_{1}, …, m_{n−1}) such that m_{0} = z

and φ(m_{i}, m_{i+1}) holds for all i < n − 1.

Proof. We argue in RCA_{0}. Fix n and z. We begin by defining a theory T and a list of types X to handle 3(a) and 3(c); we then slightly modify them to handle 3(b).

We work in the language with unary relation symbols P_{i} for i ∈ ℕ, a binary relation symbol E, and constant symbols c_{0}, …, c_{2^{n}−1}. We begin by defining a set A of axioms. We write some of these informally; formalizing them in our language is straightforward.

A1. P_{i+1}(x) → P_{i}(x) for all i ∈ ℕ.

A2. ¬P_{0}(c_{i}) for all i < 2^{n}.

A3. E is an equivalence relation with exactly 2^{n} many classes, each containing exactly one c_{i}.

A4. Within each equivalence class, for each i ∈ ℕ there are infinitely many x such that P_{i}(x)∧¬P_{i+1}(x), and there are infinitely many x such that ¬P_{0}(x).

It is not difficult to show that A is consistent and admits effective quantifier elimination, so that its deductive closure T exists and is a theory. (To conclude that T exists, we also need to know that the set of literal sentences consistent with A exists, as mentioned in the discussion of quantifier elimination at the end of Section 2.2. This is clearly the case, as this set consists exactly of the sentences ¬P_{k}(c_{i}) for i < 2^{n} and k ∈ ℕ, the sentences E(c_{i}, c_{i}) for i < 2^{n}, and the sentences ¬E(c_{i}, c_{j}) for i, j < 2^{n} such that i ≠ j.)

We begin by showing that A admits effective quantifier elimination, as discussed at the end of Section 2.2. Fix an existential formula (∃y)φ(x̄y), where φ(x̄y) is a conjunction of literals. Let m be such that no P_{i} with i > m occurs in φ, and let L′ be the language consisting of E, c_{0}, …, c_{2^{n}−1}, and all P_{i} with i ⩽ m. As mentioned in Remark 2.3, we may assume that φ is an L′-atomic diagram that is consistent with the assertion that E is an equivalence relation, and includes ¬u = v for all u such that u ∈ x̄ or u is a constant symbol, and all v ∈ x̄y distinct from u.

Suppose that at least one of the following conditions holds.

(1) Both P_{i}(u) and ¬P_{j}(u) for j < i occur in φ, where u is a variable or constant symbol.

(2) P_{0}(c_{i}) occurs in φ for some i < 2^{n}.

(3) E(c_{i}, c_{j}) for i ≠ j occurs in φ.

(4) There is a u ∈ x̄y such that ¬E(u, c_{i}) occurs in φ for all i < 2^{n}.

Then A ⊢ (∃y)ψ(x̄y) ↔ F. Otherwise, the fourth axiom family of A implies that A ⊢ (∃y)φ(x̄y) ↔ θ(x̄), where θ is the result of removing from φ all literals involving y (and hence is just T if all literals in φ involve y).

To show that T is consistent, we provide a model M of A. By effective quantifier elimination, it suffices to specify the atomic diagram of M. Let the domain of M consist of all numbers of the form ⟨i, j, k⟩, where i < 2^{n}. Let c_{i}^{M} = ⟨i, 0, 0⟩. Let E^{M}(x, y) hold if and only if x and y have the same first coordinate, and let P_{m}^{M}(x) hold if and only if x = ⟨i, j, k⟩ and m < j. It is easy to check that M is a model of A.

We now define our list X of types of T, which satisfies the pairwise full amalgamation closure conditions and has an extension function. By effective quantifier elimination, we can identify types of T with atomic types. Notice that the 0-type T corresponds to the set of literal sentences consistent with A discussed above, which we refer to as the 0-type. From now on, we refer to an atomic type as simply a type.

We begin by naming the types of T. Each 1-type in a given variable x includes the 0-type, as well as x = x and E(x, x). Thus, the 1-types of T in the variable x can be listed as follows.

(1) s_{i}(x) for i < 2^{n}, which includes x = c_{i}, c_{i} = x, E(x, c_{i}), and E(c_{i}, x), as well as ¬x = c_{j}, ¬c_{j} = x, ¬E(x, c_{j}), and ¬E(c_{j}, x) for j ≠ i, and ¬P_{m}(x) for all m ∈ ℕ.

(2) q_{i,k}(x) for i < 2^{n} and k ∈ ℕ, which includes ¬x = c_{i}, ¬c_{i} = x, E(x, c_{i}), and E(c_{i}, x), as well as ¬x = c_{j}, ¬c_{j} = x, ¬E(x, c_{j}), and ¬E(c_{j}, x) for j ≠ i, and also P_{m}(x) for m < k and ¬P_{m}(x) for m ⩾ k.

(3) q_{i,∞}(x) for i < 2^{n}, which includes ¬x = c_{i}, ¬c_{i} = x, E(x, c_{i}), and E(c_{i}, x), as well as ¬x = c_{j}, ¬c_{j} = x, ¬E(x, c_{j}), and ¬E(c_{j}, x) for j ≠ i, and also P_{m}(x) for all m ∈ ℕ.

Now each k-type of T in a tuple of variables (x_{0}, …, x_{k−1}) can be specified by two items: a list of 1-types t = (t_{0}, …, t_{k−1}) (i.e., each t_{j} is of the form s_{i}, q_{i,l}, or q_{i,∞}) and an equivalence relation D on [0, k) such that if D(i, j) then t_{i} = t_{j}, and if t_{i} = t_{j} = s_{l} for some l, then D(i, j). (We say that D is compatible with t.) This type, which we denote by r_{t,D}(x_{0}, …, x_{k−1}), is specified as follows.

(1) r_{t,D} includes the 0-type, and for each i < k, it includes t_{i}(x_{i}).

(2) If t_{i} and t_{j} are both among s_{m}, q_{m,l}, and q_{m,∞} for the same m, then r_{t,D} includes E(x_{i}, x_{j}). Otherwise it includes ¬E(x_{i}, x_{j}).

(3) If D(i, j) then r_{t,D} includes x_{i} = x_{j}. Otherwise it includes ¬x_{i} = x_{j}.

Let s = (m_{0}, …, m_{l−1}) be a sequence of natural numbers. If we have φ(m_{i}, m_{i+1}) for all i < l − 1 (where φ is as in the statement of the theorem), then let N_{s} = ∞. Otherwise, let N_{s} be the least stage greater than l at which we find that this condition does not hold. Let r_{t,D}^{s} be type obtained from r_{t,D} by replacing each t_{i} of the form q_{j,l′} such that l′ > N_{s} or l′ = ∞ by q_{j,N_{s}}. Note that r_{t,D}^{s} is obtained effectively from r_{t,D} and s. (Note also that we are not claiming that the function s ↦ N_{s} exists.)

For a type r_{t,D} of T with t = (t_{0}, …, t_{k−1}), let b_{t,D} be the number of i < k such that t_{i} is of the form q_{l,∞} for some l and ¬D(i, j) for all j < i. Let X be an effective listing of all types r_{t,D}^{s}(x̄), where

(1) t ranges over all lists of 1-types,

(2) D ranges over all equivalence relations compatible with t,

(3) s ranges over all sequences with first element z such that |s| ⩾ log_{2} b_{t,D}, and

(4) x̄ ranges over all tuples of variables of length |t|.

(The use of the logarithm here might seem mysterious, but is important in showing that X is closed under pairwise full type amalgamation.) We verify that X satisfies the pairwise full amalgamation closure conditions and has an extension function.

The 0-type is in X, and X is clearly closed under variable substitutions. Closure under taking subtypes is straightforward to verify. We now verify that X satisfies the extension condition, and in fact has an extension function.

Fix r_{t,D}^{s}(x_{0}, …, x_{|t|−1}) ∈ X and a ψ(x_{0}, …, x_{|t|−1}, x_{|t|}, …, x_{k−1}) consistent with this type. By effective quantifier elimination (and the fact that if a type includes a formula, then it includes any disjunction involving that formula), we may assume that ψ is a conjunction of literals. We may also assume that if ψ contains u = v, then it contains v = u, and similarly with other literals involving equality and E.

We first extend D to i with |t| ⩽ i < k as follows. If ψ includes x_{i} = x_{j}, or both x_{i} = c_{m} and x_{j} = c_{m}, then let D(i, j) hold. If ψ includes x_{i} = c_{m} and t_{j} = s_{m} for some j < |t|, then let D(i, j) hold. Then close D to make it an equivalence relation.

Now let d be least such that no P_{i} appears in ψ for i > d. We can check effectively whether N_{s} > d. If so, then proceed as follows. First define a sequence t′ = (t′_{0}, …, t′_{k−1}). For i < |t|, let t′_{i} = t_{i}. Now proceed by recursion starting with i = |t|. If ψ includes x_{i} = x_{j} for j < i then let t′_{i} = t′_{j}, and if ψ includes x_{i} = c_{m} then let t′_{i} = s_{m}. Otherwise, let m be such that ψ includes E(x_{i}, c_{m}), if there is such an m, and otherwise let m be least such that ψ does not include ¬E(x_{i}, c_{m}). Let l be the largest number such that ψ includes P_{l}(x_{i}), or −1 if there is no such number. Let t′_{i} = q_{m,l+1}. Now it is easy to check that r_{t′,D}^{s} amalgamates r_{t,D}^{s} and ψ.

Now suppose that N_{s} ⩽ d. Then we can effectively find a u such that r_{u,D} = r_{t,D}^{s}. Define t′ as in the previous paragraph, but using u in place of t. Let s′ be a sequence of natural numbers starting with z with length greater than d. Then it is easy to check that r_{t′,D}^{s′} amalgamates r_{t,D}^{s} and ψ.

Thus X satisfies the extension condition. Since the above procedure for obtaining extensions is effective, X in fact has an extension function.

Finally, we verify that X satisfies the pairwise full type amalgamation condition. Let r_{t,D}^{s}(x̄ȳ), r_{t′,D′}^{s′}(x̄z̄) ∈ X be basically consistent, where ȳ and z̄ are disjoint. If N_{s} ∈ ℕ then r_{t,D}^{s} is a principal type, so the existence of a type amalgamating r_{t,D}^{s} and r_{t′,D′}^{s′} follows from the fact that X satisfies the extension condition. The same is true if N_{s′} ∈ ℕ, so we may assume that N_{s} = N_{s′} = ∞. Without loss of generality, assume that |s| ⩾ |s′|. By the hypothesis on φ, we can extend s by one element to obtain a sequence ŝ of length |s| + 1 such that Nŝ = ∞. Since D and D′ must be equal on the variables in x̄, there is a minimal equivalence relation D̂ on the variables x̄ȳz̄ extending D ∪ D′. Since t and t′ agree on the variables in x̄, there is a t̂ that agrees with t on the variables in x̄ȳ and with t′ on the variables in x̄z̄. Then b_{t̂,D̂} ⩽ b_{t,D} + b_{t′,D′}, and hence

|ŝ| = |s|+1 ⩾ max(log_{2} b_{t,d}, log_{2} b_{t′,d′})+1 ⩾ log_{2}(b_{t,D}+b_{t′,D′}) ⩾ log_{2} b_{t̂,D̂},

so rŝ ; t̂,D̂(x̄ȳz̄) is in X and amalgamates r_{t,D}^{s}(x̄ȳ) and r_{t′,D′}^{s′}(x̄z̄).

We are now ready to prove the theorem in cases 3(a) and 3(c).

For 3(a), suppose that X satisfies the finite free amalgamation closure conditions, and let (x_{0}, …, x_{2^{n}−1}) be a tuple of variables. The sequence of types q_{0,∞}(x_{0}), …, q_{2^{n}−1,∞}(x_{2^{n}−1}) is in X, so there is a type r_{t,D}^{s}(x_{0}, …, x_{2^{n}−1}) in X amalgamating these types. Since log_{2} b_{t,D} = n, the sequence s = (z = m_{0}, …, m_{l−1}) has length at least n, and φ(m_{i}, m_{i+1}) for all i < l − 1.

For 3(c), suppose there is a listing Y = {h_{i}}_{i∈ℕ} of the types of X satisfying the finite free amalgamation closure conditions, and BΣ_{2}^{0} holds. Let (x_{0}, …, x_{2^{n}−1}) be a tuple of variables. Each q_{i,∞}(x_{i}) is in Y, and the property P (i, j) that holds if and only if h_{j} = q_{i,∞}(x_{i}) is Π_{1}^{0}, so by the finite axiom of choice for Π_{1}^{0} properties, the whole sequence of types q_{0,∞}(x_{0}), …, q_{2^{n}−1,∞}(x_{2^{n}−1}) is in Y. So Y has a type amalgamating these types, and hence so does X. Now the argument is as in the previous paragraph.

For 3(b), we modify our example by removing the constants. That is, our language now has only the P_{i} and E, and our axioms consist of A1, the first part of A3, and A4. Effective quantifier elimination and consistency follow by arguments much like the ones above. Our theory T now has only two kinds of 1-types, q_{k}(x), which includes P_{m}(x) for m < k and ¬P_{m}(x) for m ⩾ k; and q_{∞}(x), which includes P_{m}(x) for all m ∈ ℕ.

Our k-types are now of the form r_{t,D,F}, where t and D are as before and F is an equivalence relation on [0, |t|) such that if D(i, j) then F (i, j). The type r_{t,D,F} (x_{0}, …, x_{k−1}) is specified as follows.

(1) r_{t,D,F} includes the 0-type, and for each i < |t|, it includes t_{i}(x_{i}).

(2) If F (i, j) then r_{t,D,F} includes E(x_{i}, x_{j}). Otherwise it includes ¬E(x_{i}, x_{j}).

(3) If D(i, j) then r_{t,D,F} includes x_{i} = x_{j}. Otherwise it includes ¬x_{i} = x_{j}.

The types r_{t,D,F}^{s} and the list of types X are defined much as before. The proof that X is closed under the pairwise full amalgamation closure conditions and has an extension function is also a straightforward modification of the previous case.

Now suppose there is a listing of the types in X satisfying the finite full amalgamation closure conditions. By Proposition 6.6, this listing has an extension function approximation. By Theorem 6.10, there is a homogeneous model A such that X is an enumeration of 𝕋(A). Let c be an element of A satisfying q_{∞}, which must exists since q_{∞} ∈ X, and let a be an element of the same E^{A} equivalence class as c such that ¬P_{0}^{A}(a). Let b_{0}, …, b_{2^{n}−1} be a sequence consisting of representatives of each of the E^{A} equivalence classes, such that ¬P_{0}^{A}(b_{i}) for all i < 2^{n}. Then a ≡ b_{i} for each i < 2^{n}, so, by homogeneity, there are d_{0}, …, d_{2^{n}−1} such that, for each i < 2^{n}, we have ac ≡ b_{i}d_{i}, which implies that d_{i} is in the same E^{A} equivalence class as b_{i} and has type q_{∞}. The type of (d_{0}, …, d_{2^{n}−1}) is in X, and is of the form r_{t,D,F}^{s} (x̄). Then s = (z = m_{0}, …, m_{l−1}) has length at least n, and φ(m_{i}, m_{i+1}) holds for all i < l − 1. □

Theorem 6.12. The following is provable in RCA_{0}. Let φ be a Π_{1}^{0} formula defining a total function. For each n and z, there is a theory T and a list X of types of T such that the following hold.

(1) X satisfies the 1-point full amalgamation closure conditions.

(2) X has an extension function.

(3) If X satisfies the pairwise free amalgamation closure conditions, then there is a sequence (m_{0}, m_{1}, …, m_{n−1}) such that m_{0} = z and φ(m_{i}, m_{i+1}) holds for all i < n − 1.

Proof. We argue in RCA_{0}. Fix n and z. Assume without loss of generality that n is even. We use the same theory T as in the proof of Theorem 6.11 (in its original version, with constants), except that we now have n many equivalence classes, rather than 2^{n} many. We also adopt the notation of that proof for naming our types, except that r_{t,D}^{s} is now defined as the type obtained from r_{t,D} by replacing each t_{i} of the form q_{j,l} such that l > N_{s} or l = ∞ by q_{j,N_{s}}, as long as there is no j < \frac{n}{2} for which D(i, j) holds. Thus, the difference between this definition and the old one is that we do not alter the 1-types of the first \frac{n}{2} many variables in r_{t,D} (or any other variable that r_{t,D} declares to be equal to one of the first \frac{n}{2} many variables). The effect of this change is that our list X now includes all \frac{n}{2}-types of T but not all n-types unless a sequence m_{0}, …, m_{n−1} as above exists.

Let X be an effective listing of all types r_{t,D}^{s}(x̄), where

(1) t ranges over all lists of 1-types,

(2) D ranges over all equivalence relations compatible with t,

(3) s ranges over all sequences with first element z such that |s| ⩾ b_{t,D} (where b_{t,D} is as in the proof of Theorem 6.11), and

(4) x̄ ranges over all tuples of variables of length |t|. As before, the 0-type is in X, and X is closed under variable substitutions and taking subtypes. The proof that X satisfies the extension condition, and in fact has an extension function, is also as before.

To verify that X satisfies the 1-point full type amalgamation condition, let r_{t,D}^{s}(x̄y), r_{t′,D′}^{s′}(x̄z) ∈ X be basically consistent, where y and z are different variables. As before, we may assume that N_{s} = N_{s′} = ∞, as otherwise the existence of an amalgamator for these types follows from the fact that X satisfies the extension condition. Without loss of generality, assume that |s| ⩾ |s′|. By the hypothesis on φ, we can extend s by one element to obtain a sequence ŝ of length |s| + 1 such that Nŝ = ∞. Since D and D′ must be equal on the variables in x̄, there is a minimal equivalence relation D̂ on the variables x̄yz extending D ∪ D′. Since t and t′ agree on the variables in x̄, there is a t̂ that agrees with t on the variables in x̄y and with t′ on the variables in x̄z. Then b_{t̂,D̂} ⩽ b_{t,D} + 1, so rŝ ; t̂,D̂(x̄yz) is in X and amalgamates r_{t,D}^{s}(x̄y) and r_{t′,D′}^{s′}(x̄z).

Now suppose that X satisfies the pairwise free type amalgamation condition, and let (x_{0}, …, x_{n−1}) be a tuple of variables. Let t = (q_{0,∞}, …, q_{\frac{n}{2} −1,∞}), let D = \{(i, i) : i < \frac{n}{2}\}, let t′ = (q_{\frac{n}{2} ,∞}, …, q_{n−1,∞}), and let D′ = \{(i, i) : \frac{n}{2} ⩽ i < n\}. Then r_{t,D}^{s} = r_{t,D} and r_{t′,D′}^{s} = r_{t′,D′} for all s, so r_{t,D}(x_{0}, …, x_{\frac{n}{2} −1}), r_{t′,D′}(x\frac{n}{2} , …, x_{n−1}) ∈ X. Thus, there is a type r_{t̂,D̂}^{s}(x_{0}, …, x_{n−1}) ∈ X amalgamating them. Then s = (z = m_{0}, …, m_{l−1}) has length at least n, and φ(m_{i}, m_{i+1}) for all i < l−1. □

We can now give our justifications of the correctness of Figure 2.

Theorem 6.13. The implications described in Figure 2 hold.

Proof. We begin by recalling Figure 1 and its key for convenience.

p’wise finite 1-pt. p’wise finite
free free full full full
amalg. amalg. amalg. amalg. amalg.
X has pairwise amalg.  √  √  2  √  √
X has finite amalg.  2  √  2  2  √
∃ listing w/ finite amalg.  3  √  2  2  √
∃ model  3  1  2  3  1
∃ 1-pt. homogeneous model  X  X  2  3  1
∃ 1-homogeneous model  X  X  2  3  1
∃ homogeneous model  X  X  2  2  1
∃ listing w/ EF  4  1  5  4  1
∃ listing w/ EF + p’wise am. 4  1  2  4  1
∃ listing w/ EF + fin. am.  3  1  2  2  1

Figure 1. Versions of Theorem 6.4

• A 1 means that the principle is true in RCA_{0}.

• A 2 means that the principle is equivalent to IΣ_{2}^{0} over RCA_{0}.

• A 3 means that the principle is provable from Π_{1}^{0}GA over RCA_{0} and is equivalent to IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0} (and hence not provable in RCA_{0}, or even RCA_{0} + BΣ_{2}^{0}).

• A 4 means that the principle is provable from Π_{1}^{0}GA over RCA_{0}, but its strength is otherwise unknown.

• A 5 means that the principle is provable from IΣ_{2}^{0} over RCA_{0},

but its strength is otherwise unknown. We now arrange our justifications into two charts on the next two

pages, the first showing in which systems our principles are provable, and the second showing what they imply.

p’wise finite 1-pt. p’wise finite
free free full full full
amalg. amalg. amalg. amalg. amalg.
X has pairwise amalg.  √  √  2 E  √  √
X has finite amalg.  2 E  √  2 E 2 E  √
∃ listing w/ finite amalg.  3 K  √  2 E 2 E  √
∃ model  3 J  1 C 2 F 3 H 1 A
∃ 1-pt. homogeneous model  X  X  2 F 3 H 1 A
∃ 1-homogeneous model  X  X  2 F 3 H 1 A
∃ homogeneous model  X  X  2 F  2 F 1 A
∃ listing w/ EF  4 L 1 D 5 G  4 I  1 B
∃ listing w/ EF + p’wise am. 4 L 1 D 2 G  4 I  1 B
∃ listing w/ EF + fin. am.  3 L 1 D 2 G 2 G 1 B

Figure 3. Justifications of Figure 2: where provable

A: Provable in RCA_{0} by part (1) of Theorem 6.10. B: Provable in RCA_{0} by part (1) of Theorem 6.10 and Proposition 6.6. C: Provable in RCA_{0} by part (2) of Theorem 6.10. D: Provable in RCA_{0} by part (2) of Theorem 6.10 and Proposition 6.6. E: Provable in RCA_{0} + IΣ_{2}^{0} by Theorem 5.11. F: Provable in RCA_{0} + IΣ_{2}^{0} by part (1) of Theorem 6.10 and Theorem 5.11. (We do not really need Theorem 6.10 here, since the finite injury construction in the usual proof of Theorem 6.4 can be carried out in RCA_{0} + IΣ_{2}^{0}.) G: Provable in RCA_{0} + IΣ_{2}^{0} by part (1) of Theorem 6.10, Proposition 6.6, and Theorem 5.11. (As above, we do not really need Theorem 6.10 here.) H: Provable in RCA_{0} + Π_{1}^{0}GA by part (3) of Theorem 6.10. I: Provable in RCA_{0} + Π_{1}^{0}GA by part (3) of Theorem 6.10 and Proposition 6.6. J: Provable in RCA_{0} + Π_{1}^{0}GA by part (4) of Theorem 6.10. K: Provable in RCA_{0} + Π_{1}^{0}GA by part (4) of Theorem 6.10 and Proposition 5.13. L: Provable in RCA_{0} + Π_{1}^{0}GA by part (4) of Theorem 6.10 and Proposition 6.6.

p’wise finite 1-pt. p’wise finite
free free full full full
amalg. amalg. amalg. amalg. amalg.
X has pairwise amalg.  √  √  2 P  √  √
X has finite amalg.  2 M  √  2 N 2 N  √
∃ listing w/ finite amalg.  3 S  √  2 N 2 N  √
∃ model  3 T  1  2 R 3 T  1
∃ 1-pt. homogeneous model  X  X  2 R 3 T  1
∃ 1-homogeneous model  X  X  2 R 3 T  1
∃ homogeneous model  X  X  2 R 2 O  1
∃ listing w/ EF  4  1  5  4  1
∃ listing w/ EF + p’wise am. 4  1  2 Q  4  1
∃ listing w/ EF + fin. am.  3 S  1  2 N 2 N  1

Figure 4. Justifications of Figure 2: implications

M: Implies IΣ_{2}^{0} over RCA_{0} by Theorem 6.11, applied with hypothesis (3)(a). N: Implies IΣ_{2}^{0} over RCA_{0} by Theorem 6.11, applied with hypothesis (3)(b). O: Implies IΣ_{2}^{0} over RCA_{0} by Theorem 6.11, applied with hypothesis (3)(b), and Proposition 5.13. P: Implies IΣ_{2}^{0} over RCA_{0} by Theorem 6.12. Q: Implies IΣ_{2}^{0} over RCA_{0} by Theorem 6.12 and Proposition 5.10. R: Implies IΣ_{2}^{0} over RCA_{0} by Theorem 6.12 and Propositions 5.10 and 5.13. S: Implies IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0} by Theorem 6.11, applied with hypothesis (3)(c). T: Implies IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0} by Theorem 6.11, applied with hypothesis (3)(c), and Proposition 5.13. □

7. The reverse mathematics of model existence theorems

In this section, we compare the principles discussed in Sections 3 and 5.5. We can think of each of HMT, 1-HMT, 1-PHMT, WMT, and SMT as the combination of a principle asserting the existence of the desired model given the existence of an extension function approximation, and a principle asserting the existence of an extension function approximation given the stated hypothesis. The first kind of principle was analyzed in the previous section. We now examine the latter. For a set C of closure conditions, define the following principle.

EFAE_{C} (Extension Function Approximation Existence Under the Closure Conditions C): Every list of types satisfying the closure conditions C has an extension function approximation.

Of course, for this principle to hold, C must contain at least the extension condition, i.e., item (4) in Theorems 5.1 and 5.7.

The five sets of closure conditions considered above are the pairwise free, finite free, 1-point full, pairwise full, and finite full amalgamation closure conditions, which correspond to SMT, WMT, 1-PHMT, 1-HMT, and HMT, respectively.

We show in Theorem 7.5 that if C denotes the finite full or finite free amalgamation closure conditions, then EFAE_{C} is equivalent to AMT over RCA_{0}; while if C denotes the pairwise full, pairwise free, or 1point full amalgamation closure conditions, then EFAE_{C} is somewhere between AMT and ATT in strength.

Let P be one of SMT, WMT, 1-PHMT, 1-HMT, or HMT, and let C_{P} be the corresponding set of closure conditions. By Proposition 6.6, P implies EFAE_{C_{P}} over RCA_{0}.

By the first two parts of Theorem 6.10, it is also provable in RCA_{0} that if a list of types X satisfies the finite free (finite full) amalgamation closure conditions and has an extension function approximation, then there is a (homogeneous) model A such that X is an enumeration of the type spectrum of A. Thus WMT and HMT are each equivalent to EFAE_{C_{P}} for the corresponding set of closure conditions, and hence to AMT.

By the last two parts of Theorem 6.10, it is provable in RCA_{0} + Π_{1}^{0}GA that if a list of types X satisfies the pairwise free (pairwise full) amalgamation closure conditions and has an extension function approximation, then there is a (1-homogeneous) model A such that X is an enumeration of the type spectrum of A. Thus SMT and 1-HMT follow from EFAE_{C_{P}} for the corresponding set of closure conditions over RCA_{0} + Π_{1}^{0}GA, and hence are somewhere between AMT and ATT + Π_{1}^{0}GA in strength.

We also show, in Theorem 7.7, that 1-PHMT is equivalent to AMT+ IΣ_{2}^{0}.

7.1. Extension function approximations, AMT, and ATT. The following theorems compare our EFAE_{C} principles with AMT and ATT.

Theorem 7.1. (1) Let C denote the finite free amalgamation closure

conditions. Then AMT implies EFAE_{C} over RCA_{0}.

(2) Let C be any set of closure conditions containing the extension condition. Then ATT implies EFAE_{C} over RCA_{0}.

Proof. The proofs of the two parts of the theorem are almost identical. We first prove part 1, then indicate the changes necessary to prove part 2. As mentioned in Section 3, we can take AMT as stating that for every strongly atomic extendible tree T , there is a listing of the paths of T . Let T be a theory and X = {p_{i}}_{i∈ℕ} a list of elements of S(T) satisfying the finite free amalgamation closure conditions. Arguing in RCA_{0}, we define a family of trees T_{i,j}, and put them together to form a strongly atomic extendible tree T so that any enumeration of the isolated paths of T can be used to define an extension function approximation for X.

Given an n-type p_{i}, an (n + l)-ary formula ψ consistent with p_{i}, and an s, we say that the (n + l)-type p_{j} amalgamates p_{i} and ψ at stage s if the conjunction of ψ and the formulas in p_{j} ↾ s is consistent with p_{i}. Such a p_{j} always exists, since X satisfies the extension condition.

Let θ_{0}, θ_{1}, … be the formulas in the language of T. If p_{i} and θ_{j} are inconsistent, or not of the form p_{i}(x̄) and θ_{j}(x̄ȳ), then let T_{i,j} = {1^{k} : k ∈ ℕ}. Otherwise, let T_{i,j} = ⋃_{s∈ℕ} T_{i,j}^{s}, where T_{i,j}^{s} is defined in stages as follows.

Stage 0. Let ψ_{0} = θ_{j}. Let T_{i,j}^{0} = {1^{k} : k ∈ ℕ}. We associate with each node τ = 1^{k} in T_{i,j}^{0} a pair s_{τ} = k and t_{τ} = 0, and a formula ψ_{τ} = θ_{j}. We label all the nodes τ in T_{i,j}^{0} as active.

Stage s + 1. We are given the tree T_{i,j}^{s}, and for each τ ∈ T_{i,j}^{s}, we have defined

(1) s_{τ} (the guide amalgamator being followed),

(2) t_{τ} (the length of time s_{τ} has been followed), and

(3) ψ_{τ} consistent with p_{i} ∪ {θ_{j}} (the guide formula), where p_{s_{τ}} amalgamates p_{i} and ψ_{τ} at stage t_{τ}.

We obtain T_{i,j}^{s+1} by adding nodes to T_{i,j}^{s}. For every active τ ∈ T_{i,j}^{s}, check whether p_{s_{τ}} amalgamates p_{i} and ψ_{τ} at stage t_{τ} + 1. If so, add τ0 to T_{i,j}^{s+1}, let s_{τ0} = s_{τ}, let t_{τ0} = t_{τ} + 1, let ψ_{τ0} = ψ_{τ}, deactivate τ, and activate τ0.

If p_{s_{τ}} does not amalgamate p_{i} and ψ_{τ} at stage t_{τ} + 1, add τ1^{k+1} to T_{i,j}^{s+1} for all k ∈ ℕ. For each τ′ = τ1^{k+1}, let s_{τ′} = k and t_{τ′} = 0, and let ψ_{τ′} be the conjunction of the formulas in p_{s_{τ}} ↾ t_{τ} and ψ_{τ}. Deactivate τ, and activate each such τ′.

Let T = {0^{n} : n ∈ ℕ} ∪ {0^{⟨i,j⟩}1σ : σ ∈ T_{i,j}}. It is easy to see that the existence of T can be established in RCA_{0}.

Lemma 7.2. The tree T is extendible and strongly atomic.

Proof. Extendibility is clear from the definition. Now let ρ_{0}, …, ρ_{n} ∈ T . By extending ρ_{k} if necessary, we may assume that each ρ_{k} is of the form 0^{⟨i_{k},j_{k}⟩}1τ_{k} for some τ_{k} ∈ T_{i_{k},j_{k}}.

Let P be the property that holds of k ⩽ n if and only if p_{i_{k}} and θ_{j_{k}} are inconsistent or not of the form p_{i}(x̄) and θ_{j}(x̄ȳ), and let Q be the property that holds of k ⩽ n if and only if p_{s_{τ_{k}}} amalgamates p_{i_{k}} and ψ_{τ_{k}}. These properties are Δ_{0}^{0} and Π_{1}^{0}, respectively, so by bounded Π_{1}^{0}-comprehension, we can split our sequence of ρ’s into three subsequences, one consisting of the ρ_{k}’s such that P (k) holds, another of the ρ_{k}’s such that Q(k) holds, and the last of all other ρ_{k}’s. It is enough to find atoms extending the ρ’s in each of these sequences independently.

If P (k) holds then ρ_{k} is itself an atom of T . If Q(k) holds then it follows easily by induction that the extensions of τ_{k} in T_{i_{k},j_{k}} are exactly those of the form τ_{k}0^{l}, so again ρ_{k} is an atom of T .

Thus we may assume that we have a sequence ρ_{0}, …, ρ_{n} as above such that for each k ⩽ n, the type p_{s_{τ_{k}}} does not amalgamate p_{i_{k}} and ψ_{τ_{k}}. Let r_{k} be the stage at which τ_{k} is declared active and let t_{k} ⩾ r_{k} be least such that p_{s_{τ_{k}}} does not amalgamate p_{i_{k}} and ψ_{τ_{k}} at stage t_{k} +1. Then ν_{k} = τ_{k}0^{t_{k}−r_{k}} ∈ T_{i_{k},j_{k}}, and we have s_{ν_{k}} = s_{τ_{k}}, t_{ν_{k}} = t_{k}, and ψ_{ν_{k}} = ψ_{τ_{k}}. Furthermore, for each l, we have σ_{k,l} = ν_{k}1^{l+1} ∈ T_{i_{k},j_{k}}. All the formulas ψ_{σ_{k,l}} are the same formula φ_{k}, which by induction is consistent with p_{i_{k}}. As noted in Remark 5.9, there are m_{0}, …, m_{n} such that p_{m_{k}} amalgamates p_{i_{k}} and φ_{k}. Then 0^{⟨i_{k},j_{k}⟩}1σ_{k,m_{k}} is an atom of T extending ρ_{k} for each k ⩽ n. □

Lemma 7.3. An extension function approximation for X can be constructed from any enumeration of the isolated paths of T .

Proof. Let f_{0}, f_{1}, … be an enumeration of the isolated paths of T . Let i and j be such that θ_{j} is consistent with p_{i}. Since T is atomic, there is a least n such that f_{n} extends 0^{⟨i,j⟩}1. Let f(k) = f_{n}(k + ⟨i, j⟩ + 1). Then f is an isolated path of T_{i,j}. By the construction of T_{i,j}, any τ ∈ T such that τ(|τ| − 1) = 1 has more than one extension in T , so f = τ0^{ℕ} for some τ ∈ 2^{<ℕ}.

Let m be such that f extends 1^{m}0. Let τ_{0} = 1^{m}0 and let g(i, j, 0) = m. Given g(i, j, s) and τ_{s}, proceed as follows. If f extends τ_{s}0 then let τ_{s+1} = τ_{s}0 and g(i, j, s + 1) = s_{τ_{s}} = s_{τ_{s+1}}. Otherwise, let τ_{s+1} be the shortest string extended by f of the form ν01^{k}0^{s+1}, where k ⩾ 1 and |ν| ⩾ |τ_{s}| − 2. It is easy to see that such a string must exist, and that s_{τ_{s+1}} = k. Let g(i, j, s + 1) = k = s_{τ_{s+1}}.

There are only finitely many s such that f extends τ_{s}1, and hence lim_{s} g(i, j, s) exists. It follows from the construction of T_{i,j} that g is an extension function approximation. □

This concludes the proof of part 1 of the theorem. The only place in this proof where any property of C other than containing the extension condition is used is at the end of the proof of Lemma 7.2, to justify the existence of m_{0}, …, m_{n}. Thus, if we restrict ourselves to the n = 0 case of this lemma, all we need is for C to contain the extension condition. This case is enough to conclude that T is atomic, which gives us part 2 of the theorem. □

Theorem 7.4. Let C denote the finite full amalgamation closure conditions. Then EFAE_{C} implies AMT over RCA_{0}.

Proof. We argue in RCA_{0}. Let T be a strongly atomic extendible tree. We define a set of axioms A whose deductive closure Δ is a theory. Then we define a list X of types of Δ satisfying the finite full amalgamation closure conditions, so that from an extension function approximation for X, we can construct a listing of the isolated paths of T .

Axioms. We work in the language with unary relation symbols U, R_{i} for i ∈ ℕ, and D_{σ} for σ ∈ T ; and binary relation symbols P and E. We first define the following set A of axioms and then discuss the intuition behind them.

A1. P (x, y) → ¬U(x) ∧ U(y).

A2. ¬U(x) → (∃^{=1}y)P (x, y).

A3. (∀y)[U(y) → (∃^{⩾k}x)P (x, y)] for all k ∈ ℕ.

A4. E(x_{0}, x_{1}) ↔ (∃y)[P (x_{0}, y) ∧ P (x_{1}, y)].

A5. D_{σ}(x) → ¬U(x) for all σ ∈ T .

A6. D_{σ}(x) → D_{τ}(x) for all τ ≺ σ ∈ T .

A7. D_{σ}(x) → ¬D_{τ}(x) for all σ, τ ∈ T such that |σ| = |τ| and σ ≠ τ.

A8. R_{i}(y) → U(y) for all i ∈ ℕ.

A9. R_{i}(y) → R_{i+1}(y) for all i ∈ ℕ.

A10. (∃^{⩾k}y)R_{0}(y) for all k ∈ ℕ.

A11. (∃^{⩾k}y)[R_{i+1}(y) ∧ ¬R_{i}(y)] for all i, k ∈ ℕ.

A12. ¬R_{n}(y) → (∃^{⩾k}x)[D_{σ}(x)∧P (x, y)] for all n, k ∈ ℕ and all σ ∈ T such that |σ| = n.

A13. [¬U(x) ∧ ⋀_{σ∈S_{n}} ¬D_{σ}(x)] ↔ (∃y)[R_{n}(y) ∧ P (x, y)] for all n ∈ ℕ, where S_{n} is the set of elements of T of length n.

Axiom groups A1, A2, and A3 tell us that we can divide our universe into two disjoint parts, D = {x : ¬U(x)} and R = {x : U(x)}, and that P determines a function with domain D and range R such that the preimage of any y ∈ R is infinite. Axiom A4 tells us that E is the equivalence relation on D wherein two elements are equivalent if they have the same image under P. (Having a symbol for this relation is needed in establishing effective quantifier elimination.)

Axiom groups A5, A6, and A7 tell us that the D_{σ} hold only of elements of D, and that for each x ∈ D, the set of σ such that D_{σ}(x) holds is either empty, of the form {σ : σ ≼ τ} for some τ ∈ T , or of the form {σ : σ ≺ f} for some f ∈ [T ].

Axiom groups A8, A9, A10, and A11 tell us that the R_{i} hold only of elements of R; that for each y ∈ R, the set of i such that R_{i}(x) holds is either empty or consists of all i ⩾ j for some j; and that for each j there are infinitely many y such that this set is of the latter form.

Finally, axiom groups A12 and A13 connect the set of σ such that D_{σ}(x) holds with the set of i such that R_{i}(y) holds, for x and y such that P (x, y) holds. They say that if R_{n}(y) holds then there is no σ ∈ T of length n and no x in the P -preimage of y such that D_{σ}(x) holds, while if R_{n}(y) fails, then for every x in the P -preimage of y, some D_{σ}(x) with σ of length n holds (because in this case, by A2, the right hand side of A13 fails, and hence so does the left hand side), and for every σ ∈ T of length n there are infinitely many x in the P -preimage of y such that D_{σ}(x) holds.

We show that A is consistent and admits effective quantifier elimination, so that its deductive closure Δ exists and is a theory. We then give a list of types X of Δ satisfying the finite full amalgamation closure conditions, and use the existence of an extension function approximation for X to obtain a listing of the isolated paths of T . The idea is to ensure the following two properties. Every 1-type in X containing ¬U(x) contains D_{σ}(x) either for only finitely many σ or for all σ ≺ f for an isolated path f of T ; and there is a 1-type p(y) in X containing U(y) and ¬R_{n}(y) for all n. Then for any τ ∈ T , we can use the given extension function approximation for X to obtain an amalgamator q(x, y) ∈ X for p(y) and the formula D_{τ}(x) ∧ P (x, y). By the axioms and the properties of X, the set of σ such that D_{σ}(x) ∈ q(x, y) must determine an isolated path of T extending τ.

Quantifier elimination. We begin by showing that A admits effective quantifier elimination, as discussed at the end of Section 2.2. Fix an existential formula (∃y)φ(x̄y), where φ(x̄y) is a conjunction of literals. Let m be such that no R_{i} with i > m or D_{σ} with |σ| > m occurs in φ, and let L′ be the language consisting of U, P, E, all R_{i} with i ⩽ m, and all D_{σ} with σ ∈ T and |σ| ⩽ m. As mentioned in Remark 2.3, we may assume that φ is an L′-atomic diagram that includes ¬u = v for all distinct u, v ∈ x̄y. By a similar argument to the one mentioned at the end of that section, we may also assume that φ is consistent with the assertion that E is an equivalence relation on the set of u such that ¬U(u) holds.

Suppose that at least one of the following conditions holds.

(1) P (u, v) occurs in φ, and either U(u) or ¬U(v) occurs in φ.

(2) P (u, v) and P (u, w) both occur in φ, where v and w are distinct.

(3) P (u, w), P (v, w), and ¬E(u, v) all occur in φ.

(4) P (u, w), ¬P (v, w), and E(u, v) all occur in φ.

(5) E(u, v) and U(u) both occur in φ.

(6) ¬E(u, u) and ¬U(u) both occur in φ.

(7) D_{σ}(u) and U(u) both occur in φ.

(8) D_{σ}(u) and ¬D_{τ}(u) for τ ≺ σ both occur in φ.

(9) D_{σ}(u) and D_{τ}(u), where |σ| = |τ| and σ ≠ τ, both occur in φ.

(10) R_{i}(u) and ¬U(u) both occur in φ.

(11) R_{i}(u) and ¬R_{j}(u) with i < j both occur in φ.

(12) D_{σ}(u), P (u, v), and R_{|σ|}(v) all occur in φ.

(13) E(u, v), D_{σ}(u), and ¬D_{τ}(v) for all τ such that |τ| = |σ|, all occur in φ.

Then A ⊢ (∃y)φ(x̄y) ↔ F. (For (1)–(12), this fact is easy to see from the axioms. For (13), assume that E(u, v) and D_{σ}(u) occurs in φ. We may assume that ¬U(u) and ¬U(v) both occur in φ, as otherwise φ satisfies (5). Then T ∪ {φ} implies the negation of the left hand side of A13 for x = u and n = |σ|, and hence the negation of the right hand side of A13 for x = u and n = |σ|. By A4, T ∪{φ} implies the negation of the right hand side of A13 for x = v and n = |σ|, hence the negation of the left hand side of A13 for x = v and n = |σ|, and hence D_{τ}(v) for some τ such that |τ| = |σ|.)

Otherwise, let D be the set of variables u such that ¬U(u) occurs in φ and let R be the set of variables v such that U(v) occurs in φ. For each v ∈ R, let D_{v} be the set of variables u such that P (u, v) occurs in φ. Note that the sets D_{v} are pairwise disjoint. Let D_{∅} be the set of all elements of D that are not in any D_{v}. Let θ(x̄) be the result of removing all literals mentioning y from φ.

If y ∈ R then the only positive literals in φ involving y and another variable are the ones of the form P (x, y) for x ∈ D_{y}. If x, x′ ∈ D_{y}, then E(x, x′) occurs in φ, and if x ∈ D_{y}, then R_{i}(y) occurs in φ if and only if ¬D_{σ}(x) occurs in φ for all σ ∈ T of length i. Combining A2, A4, and A13, we see that A ⊢ (∃y)φ(x̄y) ↔ θ(x̄).

If y ∈ D_{v}, where v is either a variable or ∅, then the only positive literals in φ involving y and another variable are ones of the form E(x, y) or E(y, x) with x ∈ D_{v}, and P (y, v) if v is a variable. Let x ∈ D_{v} and i ⩽ m. Then D_{σ}(x) occurs in φ for some σ of length i if and only if D_{τ}(y) occurs in φ for some τ of length i. Thus, by A2, A3, A4, A12, and A13, we again have A ⊢ (∃y)φ(x̄y) ↔ θ(x̄).

We have completed the quantifier elimination process, and hence conclude that the deductive closure Δ of A exists and is a theory.

Consistency. We now establish the consistency of Δ by providing a model M. By effective quantifier elimination, to define M, it suffices to specify the interpretations of the symbols of our language in M. Let σ_{0}, σ_{1}, … be a listing of the elements of T .

(1) Let M = ℕ.

(2) Let U^{M} be the odd numbers.

(3) Let D_{τ}^{M} = {2⟨i, j, m⟩ : τ ≼ σ_{i}}.

(4) Let R_{n}^{M} = {2⟨k, j⟩ + 1 : n ⩾ k}.

(5) Let P^{M} = {(2⟨i, j, m⟩, 2⟨k, j⟩ + 1) : |σ_{i}| = k − 1}.

(6) Let E^{M} = {(2⟨i, j, m⟩, 2⟨i, j, n⟩) : m, n ∈ ℕ}.

It is easy to check that A1–A13 hold in this model. (Indeed, M is an atomic model of the theory induced by these axioms, though this fact is not needed in our proof.)

Defining the list of types X. We now proceed to define a list X of types of Δ. By effective quantifier elimination, we can identify types of Δ with atomic types. Notice that the 0-type Δ corresponds to the empty atomic 0-type. From now on, we refer to an atomic type as simply a type.

We define a few particular 1-types in a given variable x. Each 1-type in x includes x = x and ¬P (x, x). If it includes ¬U(x), then it includes E(x, x) and ¬R_{i}(x) for all i ∈ ℕ. If it includes U(x), then it includes ¬E(x, x) and ¬D_{σ}(x) for all σ ∈ T .

Let i ∈ ℕ. Let r_{i}(x) be the principal type generated by ¬R_{i−1}(x) ∧ R_{i}(x), or just R_{0}(x) if i = 0. Let s(x) be the nonprincipal type that includes U(x) and ¬R_{j}(x) for all j ∈ ℕ.

Let σ ∈ T . If σ is not the empty string λ, let σ^{−} be σ ↾ |σ| − 1. Let d_{σ}(x) be the principal type generated by D_{σ^{−}}(x) ∧ ¬D_{σ}(x). Let d_{λ}(x) be the principal type generated by ¬U(x) ∧ ¬D_{λ}(x).

Let e_{σ}(x) be the type defined as follows. First, e_{σ}(x) includes ¬U(x), D_{τ}(x) for every τ ≼ σ, and ¬D_{τ}(x) for every τ ∈ T incompatible with σ. Next, for each τ ∈ T extending σ, if τ is the only extension of σ of length |τ| in T , then e_{σ}(x) includes D_{τ}(x). Otherwise, e_{σ}(x) includes ¬D_{ρ}(x) for every extension ρ of σ of length |τ| in T , including τ itself. Notice that if σ is not an atom, then e_{σ}(x) is equal to d_{τ}(x) for some τ. Otherwise, the set of τ such that D_{τ}(x) ∈ e_{σ}(x) determines an isolated path of T .

Let B be the set of 1-types described above. These types (where x ranges over all variables) are exactly the 1-types in X. Notice that they include every principal 1-type of Δ, and every 1-type of Δ extending U(x). To describe the n-types in X for n > 1, we first define a few operators. It is straightforward to check from the axioms that operators with the stated properties exist.

Our first operator gives us a way to extend a formula to a type in a safe way, i.e., without introducing any unwanted 1-types. For a conjunction of literals θ consistent with Δ, let J (θ) be a type containing the literals in θ such that for each variable x, there are at most finitely many σ for which D_{σ}(x) ∈ J (θ). Notice that the 1-subtypes of J (θ) must be of the form d_{σ}, r_{i}, or s.

Our second operator gives us a way to transform a possibly inconsistent complete set of literals into a type. Let q(x̄) be a set of literals that, for every atomic formula φ with free variables in x̄, includes either φ or ¬φ (and possibly both). Let φ_{0}, φ_{1}, … be the elements of q (ordered according to a standard fixed ordering of literals). If there is an n such that {φ_{0}, …, φ_{n}} is not consistent with Δ, then for the least such n, let P(q) = J (⋀_{i<n} φ_{i}). Otherwise, let P(q) = q. Note that this operator is effective, since to know whether a given literal φ is in P(q), we need only find an m such that φ_{m} is either φ or its negation. If {φ_{0}, …, φ_{m}} is consistent with Δ then φ ∈ P(q) if and only in φ_{m} ≡ φ. Otherwise we check whether φ is in J (⋀_{i<n} φ_{i}) for the least n such that {φ_{0}, …, φ_{n}} is not consistent with Δ, which must exist.

Our third operator performs amalgamation. Let q_{0}(x̄_{0}), …, q_{n}(x̄_{n}) be types, let x̄ be a tuple such that x̄ = x̄_{0} ∪ ⋯ ∪ x̄_{n}, and let q = q_{0} ∪ ⋯ ∪ q_{n}. Let θ(x̄) be a conjunction of literals such that for each pair of distinct variables u, v ∈ x̄, either P (u, v) or ¬P (u, v) is in θ, either E(u, v) or ¬E(u, v) is in θ, and either u = v or ¬u = v is in θ. Let r be the union of q with the literals in θ. Note that if r is consistent with Δ, then it is in fact a type, and amalgamates q_{0}, …, q_{n}. Let A(q_{0}, …, q_{n}, θ) = P(r).

Our fourth operator extracts subtypes and performs variable substitutions. Let V be the set of variables in our language. By a finite injective partial function on variables we mean a finite subset f of V^{2} that satisfies the properties of an injective function. We use the usual functional notation when discussing f, and in particular write f(x)↓ to mean that x is in the domain of f. Let q(x_{0}, …, x_{n}) be a type and let f be an finite injective partial function on variables. Let S be the set of x_{i} with i ⩽ n such that f(x_{i})↓. If x_{i} ∈ S then let g(x_{i}) = f(x_{i}). Define g on all other x_{i} with i ⩽ n so that g is a finite injective partial function on variables. Let S(q, f) = q[g(x_{0}), …, g(x_{n})] ↾ S (using the notation introduced in Remark 5.9).

We define X recursively. Even if we can ensure that we do not introduce any unwanted 1-types at each step, we do not have sufficient induction to conclude that no unwanted 1-types ever appear. Our final operator allows us to ensure that this is indeed the case by transforming an arbitrary type into one whose 1-subtypes are all of the desired form. Let q(x_{0}, …, x_{k−1}) be a type and σ_{0}, …, σ_{k−1} ∈ T . Let φ_{0}, φ_{1}, … be the elements of q (ordered according to a standard fixed ordering of literals). If there are an n and a j < k such that φ_{n} ≡ D_{τ}(x_{j}) and either τ is incompatible with σ_{j} or there is more than one extension of σ_{j} of length τ in T , then for the least such n, let N (q, σ_{0}, …, σ_{k−1}) = J (⋀_{i<n} φ_{i}). Otherwise, let N (q, σ_{0}, …, σ_{k−1}) = q. Notice that, in either case, the 1-subtypes of N (q, σ_{0}, …, σ_{k−1}) are all in B.

We are now ready to define X = {p_{i}}_{i∈ℕ}. Let p_{⟨0,k⟩} = ∅ for all k ∈ ℕ. Let t_{0}, t_{1}, … be a listing of B, let σ_{0}, σ_{1}, … be a listing of T , let x_{0}, x_{1}, … be the variables in our language, and let p_{⟨1,k⟩} = N (t_{i}(x_{j}), σ_{m}) where k = ⟨i, j, m⟩. (While we do not need to apply the N operator here, it is helpful for uniformity of definition. Notice that for each t_{i}, there is a σ such that N (t_{i}, σ) = t_{i}.)

Having defined the types p_{⟨m,k⟩} for m < n, we define the types of the form p_{⟨n,k⟩} to consist of all possible amalgamations of previously defined types, together with all types obtained from these amalgamations by permuting variables or taking subtypes. We use the N operator to ensure that we do not create any unwanted 1-types.

Let F be the set of 4-tuples consisting of a finite set G of numbers of the form ⟨m, k⟩ with m < n, a conjunction of literals in the variables x̄ appearing in the union of the types p_{c} for c ∈ G, a sequence σ_{0}, …, σ_{|x̄|−1} of elements of T , and a finite injective partial function on variables. Fix a listing of F, and let G_{0}, G_{1}, … be the corresponding finite sets, θ_{0}, θ_{1}, … the corresponding formulas, s_{0}, s_{1}, … the corresponding sequences of strings, and f_{0}, f_{1}, … the corresponding partial functions. For each i, let q_{0}, …, q_{l} be the types p_{c} such that c ∈ G_{i}, and let p_{⟨n,i⟩} = S(N (A(q_{0}, …, q_{l}, θ_{i}), s_{i}), f_{i}).

Let X = {p_{i}}_{i∈ℕ}. Then X is a list of types of Δ, and, because of our application of the N operator, every 1-type in X is in B. We claim that X satisfies the finite full amalgamation closure conditions.

First, ∅ ∈ X. Closure under sequences of variable substitutions, and hence closure under variable substitutions and under subtypes, follows from the fact that each time a type is added to X, so are all the types obtained from it by applications of the S operator, in an effective way.

It is also easy to see that for a finite set of elements q_{0}, …, q_{n} of X, if there is a type of Δ containing q_{0} ∪ ⋯ ∪ q_{n}, then there is a conjunction of literals θ such that A(q_{0}, …, q_{n}, θ) amalgamates these types. Furthermore, since the 1-subtypes of q_{0} ∪ ⋯ ∪ q_{n} are all in B, the fact that T is strongly atomic means that there is a sequence t of elements of T such that N (A(q_{0}, …, q_{n}, θ), t) = A(q_{0}, …, q_{n}, θ). Thus finite full type amalgamation holds.

To verify that the extension condition is satisfied, it is enough to fix a p(x̄) ∈ X and a conjunction of literals θ(x̄ȳ) consistent with p, and show that there is a q(x̄ȳ) ∈ X containing both. We do so by defining a type t(u) for each u ∈ ȳ, and then applying the A operator to p, the types t(u), and θ.

By extending θ if needed, we may assume that p ∪ {θ} decides every atomic formula in its free variables involving P, E, or U (i.e., it includes either the formula or its negation). We may also assume that if it includes R_{i}(u) then it decides R_{j}(u) for all j ⩽ i, and that if it includes ¬D_{σ}(u), then it decides D_{τ}(u) for all τ ≼ σ. We may also assume that u = v does not occur in p ∪ {θ} for any distinct variables u and v, and hence that ¬u = v occurs in p ∪ {θ} for all distinct u and v among its free variables.

For each u ∈ ȳ, let σ_{u} be the longest string such that D_{σ}(u) occurs in θ, if one exists, and let σ_{u} = λ otherwise. Since T is strongly atomic, for each u ∈ ȳ there is an atom ρ_{u} of T extending σ_{u}.

Let u ∈ ȳ. If R_{i}(u) occurs in θ for some i, then, for the least such i, let t(u) = r_{i}(u). If ¬D_{τ}(u) and ¬U(u) both occur in θ for some τ compatible with ρ_{u}, then for the shortest such τ, let t(u) = d_{τ}(u).

If P (v, u) occurs in θ for v ∈ x̄, and t(u) has not yet been defined, then let t(u) be the 1-type that includes U(u) and includes R_{i}(u) if and only if p includes ¬D_{σ}(v) for all σ ∈ T of length i. Notice that t(u) either is r_{j}(u) for some j or is s.

If P (u, v) occurs in θ for v ∈ x̄, and t(u) has not yet been defined, then let t(u) be the 1-type defined as follows. The type t(u) includes ¬U(u). For each n such that ¬R_{n}(v) ∈ p, it includes D_{τ}(u) for the unique τ ∈ T of length n that is compatible with ρ_{u}. If R_{n}(v) ∈ p, then t(u) includes ¬D_{τ}(u) for all τ of length n. Notice that t(u) either is d_{τ}(u) for some τ ≽ σ_{u} or is d_{ρ_{u}}(u).

If P (u, v) occurs in θ for some v ∈ ȳ for which we have defined t(v), and t(u) has not yet been defined, then define t(u) as in the previous paragraph, with t(v) in place of p.

If P (v, u) occurs in θ for some v ∈ ȳ for which we have defined t(v), and t(u) has not yet been defined, then we must have defined t(v) = r_{i}(v) for some i. In this case, let t(u) be some d_{σ}(u) such that |σ| = i and d_{σ}(u) is consistent with the literals of the form D_{τ}(u) and ¬D_{τ}(u) occurring in θ.

If we have not yet defined t(u) for u ∈ ȳ, then no R_{i}(u) or ¬D_{τ}(u) for τ compatible with ρ_{u} occurs in θ. In this case, if U(u) occurs in θ then let t(u) = s, and otherwise let t(u) = d_{ρ_{u}}(u).

It is straightforward to check that applying the A operator to p, the types t(u) with u ∈ ȳ, and θ yields a type q containing both p and θ, and that there is a sequence of strings t such that N (q, t) = q. Thus q ∈ X.

Applying EFAE_{C}. By EFAE_{C}, we have an extension function approximation g for X. Recall the type s defined above. Let i be such that p_{i} = s(x_{0}). Given σ ∈ T , let j be such that θ_{j} = D_{σ}(x_{1}) ∧ P (x_{1}, x_{0}). Applying g to i and j, we obtain a 2-type q(x_{0}, x_{1}) of Δ amalgamating p_{i} and θ_{j}. Note that this type is obtained effectively from i and j, as q ↾ s = p_{g(i,j,s)} ↾ s for all s. Since lim_{s} g(i, j, s) exists, q ∈ X, so q ↾ x_{1} ∈ X. The only 1-types in X are the ones of the form d_{ρ}, e_{ρ}, r_{i}, or s. Since ¬R_{j}(x_{0}) ∈ q for all j, for each m there must be some τ of length m such that D_{τ}(x_{1}) ∈ q. Thus we must have q ↾ x_{1} = e_{ρ} for some atom ρ of Δ. Then f_{σ} = ⋃_{D_{τ} (x_{1})∈q} τ is an isolated path of T .

The function σ ↦ f_{σ} exists. Let σ_{0}, σ_{1}, … be a listing of the elements of T and let g_{i} = f_{σ_{i}}. Then {g_{i}}_{i∈ℕ} is a listing of the isolated paths of T . □

Theorems 7.1 and 7.4 together give us the following results.

Theorem 7.5. (1) Let C denote the finite full or finite free amalgamation closure conditions. Then EFAE_{C} is equivalent to AMT over RCA_{0}.

(2) Let C be any set of closure conditions that includes the extension condition, in particular the pairwise full, pairwise free, or 1-point full amalgamation closure conditions. Then EFAE_{C} implies AMT and is implied by ATT over RCA_{0}.

Assuming AMT does not imply ATT, the second part of the above theorem has a gap, which remains open.

7.2. Comparing model existence theorems. As mentioned above, HMT and WMT are each equivalent to the corresponding EFAE_{C} principle over RCA_{0}, so Theorem 7.5 gives us the following result.

Theorem 7.6. Both HMT and WMT are equivalent to AMT over RCA_{0}.

Now consider the statement that every list of types that satisfies the 1-point full amalgamation closure conditions and has an extension function approximation is an enumeration of the type spectrum of some 1-point homogeneous model. By Theorem 6.12, this statement implies IΣ_{2}^{0} over RCA_{0}. Since 1-PHMT implies this statement, it also implies IΣ_{2}^{0} over RCA_{0}. Since 1-PHMT also implies EFAE_{C}, where C denotes the 1-point full amalgamation closure conditions, it follows from Theorem 7.5 that 1-PHMT also implies AMT over RCA_{0}. On the other hand, by Theorem 5.11, under RCA_{0} + IΣ_{2}^{0}, satisfying the 1-point full amalgamation closure conditions implies satisfying the finite full amalgamation closure conditions. Thus we have the following result.

Theorem 7.7. 1-PHMT is equivalent to AMT + IΣ_{2}^{0} over RCA_{0}.

The situation for 1-HMT and SMT is not quite as clear. Since they each imply the corresponding EFAE_{C} principle over RCA_{0}, Theorem 7.5 gives us the following result.

Theorem 7.8. Both 1-HMT and SMT imply AMT over RCA_{0}.

By Theorems 6.10 and 7.5, we also have the following result.

Theorem 7.9. Both 1-HMT and SMT are provable in RCA_{0} +ATT+ Π_{1}^{0}GA.

Let P be one of 1-HMT or SMT, and let C_{P} be the set of closure conditions mentioned in the statement of P. Consider the statement that if X is a list of types satisfying C_{P} and X has an extension function approximation, then there is a (1-homogeneous, if P is 1-HMT) model A such that X enumerates the type spectrum of A. By Theorem 6.13 (in particular, item T in the charts justifying that theorem), this statement implies IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0}. Since the statement clearly follows from P over RCA_{0}, we see that P implies IΣ_{2}^{0} over RCA_{0}+BΣ_{2}^{0}. But ATT is Π_{1}^{1}-conservative over RCA_{0} +BΣ_{2}^{0}, as mentioned in Section 3, so we have the following result, which in particular implies that Theorem 7.8 cannot be reversed, even over BΣ_{2}^{0}.

Theorem 7.10. Neither 1-HMT nor SMT are provable in RCA_{0} + ATT + BΣ_{2}^{0}.

We do not know the exact strengths of 1-HMT and SMT.

7.3. Computability theoretic equivalences. Theorem 7.6 has a strong computability theoretic analog. In the computability theoretic setting, where we work over the standard model of first order arithmetic, there is no difference between our versions of HMT. There is also no difference between SMT and WMT, so in this subsection we refer to this principle as MT. Similarly, in this setting we have only two versions of EFAE_{C}, one for full amalgamation, which we call EFAE_{full}, and one for free amalgamation, which we call EFAE_{free}.

We may think of a principle of the form ∀X [Θ(X) → ∃Y Ψ(X, Y)], where Θ and Ψ are arithmetic, as a problem. An instance of this problem is an X such that Θ(X) holds and a solution to this instance is a Y such that Ψ(X, Y) holds. For example, an instance of HMT consists of a theory T together with a list of types X of T satisfying the full amalgamation closure conditions, and a solution to this instance is the elementary diagram of a homogeneous model A of T such that X is an enumeration of 𝕋(A).

Let P and Q be problems. We say that P is uniformly reducible to Q, and write P ⩽_{u} Q, if there are Turing functionals Φ and Ψ such that, for every instance X of P, the set X̂ = Φ^{X} is an instance of Q, and for every solution Ŷ to X̂, the set Y = Ψ^{X⊕Ŷ} is a solution to X. We say that P and Q are uniformly equivalent, and write P ≡_{u} Q, if P ⩽_{u} Q and Q ⩽_{u} P. (As noted by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [10], uniform reducibility is equivalent to a special case of the notion of Weihrauch reducibility that has been widely studied in computable analysis. This notion and other related computability theoretic reducibilities between problems, many with strong relations to reverse mathematics, are presented and analyzed in papers such as [10] and Hirschfeldt and Jockusch [21].)

Theorem 7.11. HMT ≡_{u} MT ≡_{u} AMT.

Proof. It is easy to check that the translations between trees and theories in Section 3 are uniform, so AMT ≡_{u} ATT. (In this setting, there is no distinction between atomic and strongly atomic trees.)

The proof of Proposition 6.6 shows that EFAE_{full} ⩽_{u} HMT and EFAE_{free} ⩽_{u} MT. The proofs of Lemma 6.9 and Theorem 6.10 are also both uniform, so in fact HMT ≡_{u} EFAE_{full} and MT ≡_{u} EFAE_{free}. (These facts can also be obtained from the original proofs of Theorem 6.4 by Goncharov [15] and Peretyat’kin [41].)

The construction of the tree T from the theory T and list of types X in the proof of Theorem 7.1 is uniform, as is the construction of an extension function approximation for X from an enumeration of the isolated paths of T . Thus ATT ⩽_{u} EFAE_{free}.

In the proof of Theorem 7.4, the construction of the theory Δ and the list of types X from the tree T is uniform. The construction of a listing of the isolated paths of T from an extension function approximation for X is also uniform, except for the choice of an i such that p_{i} = s(x_{0}). We can find such an i if we know an atom σ of T , since then, given our listings σ_{0}, σ_{1}, … of T and t_{0}, t_{1}, … of B (where B is as in the proof of Theorem 7.4), we can fix m and i such that σ_{m} = σ and t_{i} = s, and have that p_{⟨1,⟨i,0,m⟩⟩} = s_{x_{0}}. So if we let T̂ be the tree consisting of all strings of the form 0^{n} and all strings of the form 1τ for τ ∈ T , and apply our construction to T̂ instead of T, we get a fully uniform construction. From a listing of the isolated paths of T̂, we can uniformly obtain one of the isolated paths of T , so EFAE_{full} ⩽_{u} ATT.

The above results, together with the obvious fact that EFAE_{full} ⩽_{u} EFAE_{free}, yield the theorem. □

For a problem P, a degree a is P -bounding if every computable instance of P has an a-computable solution. (The AMT-bounding degrees have been referred to in the literature as atomic bounding or prime bounding.) More generally, a is P -bounding relative to b if every b-computable instance of P has an a-computable solution.

Corollary 7.12. For any degree b, the HMT-bounding degrees relative to b, the MT-bounding degrees relative to b, and the AMT-bounding degrees relative to b all coincide.

8. Open questions

In this section, we gather some open questions raised in previous sections.

As mentioned in Section 3, Π_{1}^{0}G implies AMT+Π_{1}^{0}GA, but, although Π_{1}^{0}GA can be seen as an attempt to capture the first order difference between AMT and Π_{1}^{0}G, the reverse implication remains open. Question 8.1. Does AMT + Π_{1}^{0}GA imply Π_{1}^{0}G over RCA_{0}?

In connection with both this question and the following one, recall from Corollary 3.4 that AMT does not imply Π_{1}^{0}GA.

As noted following Proposition 3.6, AMT is equivalent to the statement that if T is a strongly atomic extendible tree, then there is a listing of the isolated paths of T . Weakening the hypothesis of T being strongly atomic to that of T being atomic yields ATT. ATT implies AMT, and Π_{1}^{0}G implies ATT. Furthermore, as noted following the definition of ATT in Section 3, AMT implies ATT over BΣ_{2}^{0}. Question 8.2. Does AMT imply ATT over RCA_{0}? Does AMT+Π_{1}^{0}GA imply ATT over RCA_{0}? What if we replace ATT by the principle ATT^{+} defined at the end of Appendix B below?

Note that a positive answer to Question 8.1 implies a positive answer to the second part of Question 8.2.

By Proposition 4.5, it is provable in RCA_{0} that every saturated model is 1-homogeneous, and that every strongly saturated model is homogeneous. As noted following that proposition, the statement that every saturated model is homogeneous is provable in both BΣ_{2}^{0} and WKL_{0} (and hence does not imply BΣ_{2}^{0}). Question 8.3. What is the exact strength of the statement that every saturated model is homogeneous?

Proposition 4.6 says that the statement that every atomic model is strongly 1-homogeneous is equivalent to ACA_{0}. Combining Proposition 4.5 with the fact that ACA_{0} suffices to show that all of our definitions of homogeneity are equivalent, we see that ACA_{0} proves that every prime model is strongly 1-homogeneous, and so is every saturated model. Question 8.4. What are the exact strengths of the statements that every prime model is strongly 1-homogeneous, and that every (strongly) saturated model is strongly 1-homogeneous?

As noted in Section 6.1, the statement that if X has a weak extension function approximation then it has an extension function approximation follows from IΣ_{2}^{0}, and Theorem 7.1 implies that it also follows from ATT (assuming X in fact satisfies the extension condition).

Question 8.5. What more can be said about the strength of the above statement? What about the version of Theorem 6.10 where extension function approximations are replaced by weak extension function approximations?

Recall that, in Figure 2, a 3 means that the principle is provable from Π_{1}^{0}GA over RCA_{0} and is equivalent to IΣ_{2}^{0} over RCA_{0} + BΣ_{2}^{0}; a 4 means that the principle is provable from Π_{1}^{0}GA over RCA_{0}, but its strength is otherwise unknown; and a 5 means that the principle is provable from IΣ_{2}^{0} over RCA_{0}, but its strength is otherwise unknown.

Question 8.6. Are the principles marked with a 3 in Figure 2 equivalent to each other? Are they equivalent to Π_{1}^{0}GA? What more can be said about the items marked 4 or 5 in Figure 2? Are they equivalent to Π_{1}^{0}GA?

Theorem 7.5 shows that if C denotes the pairwise full, pairwise free, or 1-point full amalgamation closure conditions, or indeed any set of closure conditions containing the extension condition, then EFAE_{C} implies AMT and is implied by ATT.

Question 8.7. What is the exact strength of EFAE_{C} in the above cases?

Note that if the answer to the first part of Question 8.2 is positive then Theorem 7.5 answers this question.

By Theorem 7.8, both 1-HMT and SMT imply AMT. By Theorems 7.9 and 7.10, both of these principles are implied by ATT + Π_{1}^{0}GA, but not by ATT + BΣ_{2}^{0}.

Question 8.8. What are the exact strengths of 1-HMT and SMT?

In Appendix B below, we define the principle FATT, which follows from BΣ_{2}^{0} and from ATT, but is not provable in RCA_{0}.

Question 8.9. Does Π_{1}^{0}GA imply FATT over RCA_{0}?

Appendix A. Approximating generics

We can define analogs of Π_{1}^{0}G for higher levels of the arithmetic hierarchy. As in the Π_{1}^{0} case, when we say that D is a Π_{n}^{0} set of strings, we mean that D is a Π_{n}^{0} predicate on 2^{<ℕ}, and write σ ∈ D to mean that D holds of σ.

Π_{n}^{0}G: For any uniformly Π_{n}^{0} collection of sets of strings D_{0}, D_{1}, …, each of which is dense in 2^{<ℕ}, there is a G ∈ 2^{ℕ} such that (∀i)(∃m)[G ↾ m ∈ D_{i}].

In Section 3, we introduced the Π_{1}^{0}-generic approximation principle Π_{1}^{0}GA, and showed that it is equivalent to IΣ_{2}^{0} over BΣ_{2}^{0}. In the spirit of the strong form of the limit lemma, we have the following analogous generic approximation principles. (We use (∀^{∞}x) as an abbreviation of (∃y)(∀x > y).)

Π_{n}^{0}GA: For any uniformly Π_{n}^{0} collection of sets of strings D_{0}, D_{1}, …, each of which is dense in 2^{<ℕ}, there are g_{k_{0},...,k_{n−1}} ∈ 2^{ℕ} for k_{0}, …, k_{n−1} ∈ ℕ such that

(∀i)(∃σ ∈ D_{i})(∀^{∞}k_{0}) ⋯ (∀^{∞}k_{n−1})[g_{k_{0},...,k_{n−1}} ≻ σ].

Clearly, Π_{n}^{0}GA follows from Π_{n}^{0}G. Let M be a countable model of RCA_{0}. For a subset G of the domain of M, let M[G] be the structure obtained by adding to the second order part of M all sets definable over M by Δ_{1}^{0} formulas with G as an added set parameter. If G is Cohen 1-generic, then adding G to M preserves IΣ_{1}^{0}, which implies that M[G] is a model of RCA_{0}. (See [47]; for more on the general structure of arguments establishing conservativity results of the kind we are discussing, see [20, 22, 47].) It is easy to see that, if G is sufficiently Cohen generic, then M[G] contains solutions to all instances of Π_{n}^{0}G in M. By iterating this construction, we obtain an extension N of M that is a model of RCA_{0} +Π_{n}^{0}G, which is enough to conclude that Π_{n}^{0}G is Π_{1}^{1}-conservative over RCA_{0}. In particular, Π_{n}^{0}G (and hence Π_{n}^{0}GA) does not imply IΣ_{n+1}^{0}, even over IΣ_{n}^{0}. (In fact, the same argument as in [23] for Π_{1}^{0}G shows that Π_{n}^{0}G is restricted Π_{2}^{1}-conservative over RCA_{0}.)

We have the following analogs to Theorems 3.2 and 3.3.

Theorem A.1. RCA_{0} + IΣ_{n+1}^{0} ⊢ Π_{n}^{0}GA.

Proof. Let D_{0}, D_{1}, … be uniformly Π_{n}^{0} sets of strings, each of which is dense in 2^{<ℕ}. Let Θ be a Δ_{0}^{0} predicate such that σ ∈ D_{i} if and only if (∀x_{0})(∃x_{1}) ⋯ (Qx_{n−1})Θ(i, σ, x_{0}, …, x_{n−1}), where Q is either ∃ or ∀, depending on the parity of n. Let

D_{i}[k_{0}, …, k_{n−1}] = {σ : |σ| > k_{n−1} ∨ (∀x_{0} < k_{0})(∃x_{1} < k_{1}) ⋯ (Qx_{n−1} < k_{n−1})Θ(i, σ, x_{0}, …, x_{n−1})}.

Let ⩽_{ll} be the length-lexicographic ordering on 2^{<ℕ}. For each tuple (k_{0}, …, k_{n−1}), define σ_{k_{0},...,k_{n−1},i} for i ⩽ k_{n−1} by recursion as follows. Let σ_{k_{0},...,k_{n−1},0} be the ⩽_{ll}-least string in D_{0}[k_{0}, …, k_{n−1}]. Given σ_{k_{0},...,k_{n−1},i−1}, let σ_{k_{0},...,k_{n−1},i} be the ⩽_{ll}-least extension of σ_{k_{0},...,k_{n−1},i−1} in D_{i}[k_{0}, …, k_{n−1}]. Let g_{k_{0},...,k_{n−1}} = σ_{k_{0},...,k_{n−1},k_{n−1}}0^{ℕ}.

Let φ(i, σ, τ) be a Π_{n}^{0} formula stating that τ ≽ σ and τ ∈ D_{i}. Let ψ(i, σ, τ) be a formula stating that φ(i, σ, τ) and ¬φ(i, σ, ρ) for all ρ <_{ll} τ. Note that for each i and σ there is a τ such that ψ(i, σ, τ), by Π_{n}^{0}-induction. Consider the statement

(∃⟨σ_{0}, …, σ_{i}⟩)[ψ(0, λ, σ_{0}) ∧ (∀j < i)ψ(j + 1, σ_{j}, σ_{j+1})].

This statement is Σ_{n+1}^{0}, and if it holds of i, then it holds of i + 1, so by Σ_{n+1}^{0}-induction, it holds of all i. Fix i and let σ_{0}, …, σ_{i} be as in the statement. We claim that

(∀^{∞}k_{0}) ⋯ (∀^{∞}k_{n−1})[g_{k_{0},...,k_{n−1}} ≻ σ_{i}],

which is enough to establish the theorem, as σ_{i} ∈ D_{i}.

For each j ⩽ i and each ρ <_{ll} σ_{j} such that ρ ≽ σ_{j−1} (or j = 0), there is an m_{ρ,j} such that

¬(∃x_{1}) ⋯ (Qx_{n−1})Θ(j, ρ, m_{ρ,j}, x_{1}, …, x_{n−1}),

so by Π_{n−1}^{0}-bounding, there is an m_{0} bounding such an m_{j,ρ} for each such j and ρ. Fix k_{0} > m_{0}. It is enough to show that

(∀^{∞}k_{1}) ⋯ (∀^{∞}k_{n−1})[g_{k_{0},...,k_{n−1}} ≻ σ_{i}].

For each j ⩽ i,

(∀x_{0} < k_{0})(∃x_{1})(∀x_{2}) ⋯ (Qx_{n−1})Θ(j, σ_{j}, x_{0}, x_{1}, …, x_{n−1}),

so by Π_{n−2}^{0}-bounding, there is an m_{1} such that for all j ⩽ i,

(∀x_{0} < k_{0})(∃x_{1} < m_{1})(∀x_{2}) ⋯ (Qx_{n−1})Θ(j, σ_{j}, x_{0}, x_{1}, …, x_{n−1}).

Fix k_{1} ⩾ m_{1}. It is enough to show that

(∀^{∞}k_{2}) ⋯ (∀^{∞}k_{n−1})[g_{k_{0},...,k_{n−1}} ≻ σ_{i}].

For each j ⩽ i and each ρ <_{ll} σ_{j} such that ρ ≽ σ_{j−1} (or j = 0),

¬(∃x_{1} < k_{1})(∀x_{2})(∃x_{3}) ⋯ (Qx_{n−1})Θ(j, ρ, m_{j,ρ}, x_{1}, …, x_{n−1}),

where m_{j,ρ} is as above, so for each x_{1} < k_{1}, there is an m_{j,ρ,x_{1}} such that

¬(∀x_{2} < m_{j,ρ,x_{1}})(∃x_{3}) ⋯ (Qx_{n−1})Θ(j, ρ, m_{j,ρ}, x_{1}, …, x_{n−1}).

Let m_{2} bound all such m_{j,ρ,x_{1}}. Fix k_{2} > m_{2}. It is enough to show that

(∀^{∞}k_{3}) ⋯ (∀^{∞}k_{n−1})[g_{k_{0},...,k_{n−1}} ≻ σ_{i}].

We continue in this manner until we have defined k_{0}, …, k_{n−1}, and established that it is enough to show that g_{k_{0},...,k_{n−1}} ≻ σ_{i}. (We make sure to have k_{n−1} ⩾ i.) Our definition ensures that for each j ⩽ i, we have σ_{j} ∈ D_{j}[k_{0}, …, k_{n−1}] and ρ ∉ D_{j}[k_{0}, …, k_{n−1}] for all ρ <_{ll} σ_{j} such that ρ ≽ σ_{j−1} (or j = 0), whence σ_{k_{0},...,k_{n−1},j} = σ_{j}. Thus g_{k_{0},...,k_{n−1}} ≻ σ_{i}, as desired. □

Theorem A.2. RCA_{0} + BΣ_{n+1}^{0} + Π_{n}^{0}GA ⊢ IΣ_{n+1}^{0}, and hence RCA_{0} + BΣ_{n+1}^{0} ⊢ Π_{n}^{0}GA ↔ IΣ_{n+1}^{0}.

Proof. Fix a model of RCA_{0} + BΣ_{n+1}^{0} + Π_{n}^{0}GA with first order part M and second order part S. Suppose for the sake of a contradiction that IΣ_{n+1}^{0} fails in this model. As in the proof of Theorem 3.3, there is a Σ_{n+1}^{0} cut I in M. As in that proof, let D_{i} be the set of all σ ∈ 2^{<ℕ} such that, thinking of σ as a finite set, the last two elements of σ, in order, are w and ⟨i, x, y⟩, and

(∃y′ < y)(∀z)[φ(x, y′, z) ∧ (∀y′ ⩽ w)(∃z′ < y)¬φ(x, y′, z′)].

The D_{i} are uniformly Π_{n}^{0}. There is no bound on the least witnesses y such that (∀z)φ(x, y, z) for x ∈ I, as otherwise I would be a Π_{n}^{0} cut, which is not possible since IΠ_{n}^{0} holds in our model. Now the same argument as in the proof of Theorem 3.3 shows that each D_{i} is dense.

Let g_{k_{0},...,k_{n−1}} be as in Π_{n}^{0}GA and let a be any number above all those in I. Let

E_{i} = {⟨σ, m⟩ : σ ∈ D_{i} ∧ (∀k_{0} > m)(∃^{∞}k_{1})(∀^{∞}k_{2}) ⋯ (Q^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ σ},

where Q is either ∃ or ∀ depending on the parity of n. Note that the E_{i} are uniformly Π_{n}^{0}. For each i < a there is a ⟨σ, m⟩ ∈ E_{i}, so by the finite axiom of choice for Π_{n}^{0} properties, which follows from BΣ_{n+1}^{0}, there is a function f with domain a that is M-finite and such that for each i < a there is a ⟨σ, m⟩ ∈ E_{i} with the last element of σ of the form ⟨i, f(i), y⟩. Since ⟨σ, m⟩ ∈ E_{i} ⇒ σ ∈ D_{i}, every f(i) is in I.

We claim that f is one-to-one, which gives us a contradiction as in the proof of Theorem 3.3. So assume for a contradiction that f(i) = f(j) = x for i ≠ j. Then there are ⟨σ, m⟩ ∈ E_{i} and ⟨σ′, m′⟩ ∈ E_{j}, and there are v, w, y_{i}, y_{j}, such that the next to last and last elements of σ are v and ⟨i, x, y_{i}⟩, and the next to last and last elements of σ′ are w and ⟨j, x, y_{j}⟩. Let l = max(|σ|, |σ′|), and let τ ∈ D_{l} be such that

(∀^{∞}k_{0})(∀^{∞}k_{1}) ⋯ (∀^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ τ.

Note that |τ| > l. Let m_{0} be such that

(∀k_{0} > m_{0})(∀^{∞}k_{1}) ⋯ (∀^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ τ,

and let k_{0} > m, m_{0}. Let m_{1} be such that

(∀k_{1} > m_{1})(∀^{∞}k_{2}) ⋯ (∀^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ τ,

and let k_{1} > m_{1} be such that

(∀^{∞}k_{2})(∃^{∞}k_{3}) ⋯ (Q^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ σ.

Let m_{2} be such that

(∀k_{2} > m_{2})(∀^{∞}k_{3}) ⋯ (∀^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ τ,

and let k_{2} > m_{2} be such that

(∃^{∞}k_{3})(∀^{∞}k_{3}) ⋯ (Q^{∞}k_{n−1})g_{k_{0},...,k_{n−1}} ≻ σ.

Keep defining m_{l} and k_{l} this way, to obtain k_{0}, …, k_{n−1}. Then g_{k_{0},...,k_{n−1}} ≻ τ and g_{k_{0},...,k_{n−1}} ≻ σ, so σ ≺ τ. The same argument shows that σ′ ≺ τ. Thus σ and σ′ are compatible. We cannot have σ = σ′, since σ and σ′ have different last elements, so, without loss of generality, we may assume that σ ≺ σ′ and so v < ⟨i, x, y_{i}⟩ ⩽ w < ⟨j, x, y_{j}⟩. We now have a contradiction, as the definitions of D_{i} and D_{j} then say that (∃y′ < y_{i})(∀z)φ(x, y′, z) and (∀y′ ⩽ w)(∃z′ < y)¬φ(x, y′, z′), respectively, but y_{i} ⩽ w. □

Appendix B. Atomic trees

In this appendix, we discuss a couple of results on atomic trees mentioned above. We begin with the following fact.

Proposition B.1. The statement that every atomic extendible tree is strongly atomic is equivalent to BΣ_{2}^{0} over RCA_{0}.

Proof. Since being an atom of a tree is a Π_{1}^{0} property, the statement follows from the finite axiom of choice for Π_{1}^{0} properties. In the other direction, we assume that every atomic extendible tree is strongly atomic and argue in RCA_{0} as follows. Let P be a Π_{1}^{0} property and m be such that (∀i < m)(∃u)P (i, u). Let R be a Δ_{1}^{0} property such that P (x, y) if and only if (∀z)R(x, y, z), and for i < m, let w(i, s) be the least u such that (∀z < s)R(i, u, s). We define a tree T in stages. At stage 0 put into T [0] all strings of the forms 0^{i} and 0^{i}1 for i < m. At stage s + 1, for each i < m and each leaf σ of T [s] extending 0^{i}1, if w(i, s + 1) = w(i, s) then put σ0 into T [s + 1], and otherwise put both σ0 and σ1 into T [s + 1].

It is easy to check that T is a well-defined extendible tree. Let τ ∈ T . By extending τ if needed, we may assume that τ extends 0^{i}1 for some i < m. Since P (i, u) holds for some u, there is an s ⩾ |τ| such that w(i, t) = w(i, s) for all t > s. Let σ ≻ τ be a leaf of T [s]. Then σ is an atom of T extending τ. Thus T is atomic, and hence by hypothesis T is strongly atomic. Let τ_{0}, …, τ_{m−1} be atoms of T extending 01, 001, …, 0^{m−1}1, respectively, and let s be greater than the length of each τ_{i}. Then w(i, t) = w(i, s) for all t > s, so P (i, w(i, s)) holds for all i < m. Letting v be greater than all w(i, s) for i < m, we have (∀i < m)(∃u < v)P (i, u). Thus BΠ_{1}^{0} holds. □

Next, we define the “miniaturization” of ATT discussed at the end of Section 3.

FATT (Finite Atomic Tree Theorem): For each atomic extendible tree T and each finite set of nodes σ_{0}, …, σ_{k} ∈ T , there is a set of isolated paths P_{0}, …, P_{k} such that P_{i} extends σ_{i}.

It is easy to see that FATT follows from BΣ_{2}^{0}. It also follows from ATT, and hence from Π_{1}^{0}G, so it is strictly weaker than BΣ_{2}^{0}. We do not know whether FATT follows from Π_{1}^{0}GA, but we can show it is not provable in RCA_{0}.

Theorem B.2. RCA_{0} ⊬ FATT.

Proof. Define a k-path list to be a sequence L = (P_{0}, …, P_{k}) ∈ (2^{ℕ})^{k+1} such that P_{i} extends 0^{i}1. We write L(i) for P_{i}. We assume that BΣ_{2}^{0} fails and argue in RCA_{0} to build a computable (i.e., Δ_{1}^{0}-definable) atomic extendible tree T such that, for some k, we have 0^{i}1 ∈ T for all i ⩽ k, but there is no computable k-path list L such that L(i) is an isolated path of T for all i ⩽ k. The reason such a construction suffices to establish the theorem is the following: Let N be a model of Σ_{1}^{0}- PA (i.e., Peano Arithmetic with induction restricted to Σ_{1}^{0} formulas) in which BΣ_{2}^{0} fails. Let M be the structure in the language of second order arithmetic with first order part N and second order part consisting of the subsets of the domain of N that are Δ_{1}^{0}-definable over N. Then M is a model of RCA_{0} (see [47]), and our construction shows that M is not a model of FATT.

As shown by Hirst [24], BΣ_{2}^{0} is equivalent over RCA_{0} to RT_{<∞}^{1}, the statement that for each k and each partition A_{0}, …, A_{k} of ℕ, some A_{i} is infinite. Thus we may assume there is a k ∈ ℕ and a partition A_{0}, …, A_{k} of ℕ such that each A_{k} is bounded.

Let L_{0}, L_{1}, … list all partial computable k-path lists (i.e., partial computable sets that could be extended to k-path lists). We construct T to ensure that if L_{j} is total and j ∈ A_{i}, then L_{j}(i) is not isolated. Our construction is a version of the one used by Goncharov and Nurtazin [16] to build an atomic extendible tree for which there is no computable listing of the isolated paths.

Begin by putting 0^{i}1 in T for all i ⩽ k. Every other node in T extends one of these nodes, so it suffices to describe how T is built above a fixed 0^{i}1 with i ⩽ k. We proceed in stages. At each stage, certain strings will be declared to be active. The construction will ensure that if σ is active at the end of stage s, then there is an n such that the extensions of σ put into T by the end of stage s are exactly those of the form σ0^{j} for j ⩽ n.

At stage 0, declare 0^{i}1 to be active.

At stage s + 1, proceed as follows for each active σ. Let n be such that the extensions of σ put into T so far are exactly those of the form σ0^{j} for j ⩽ n. If by this stage we have not yet found a j ∈ A_{i} such that σ ≺ L_{j}(i), then put σ0^{j+1} into T . Otherwise, put both σ0^{j+1} and σ0^{j}1 into T , declare both these strings to be active, and declare σ to no longer be active.

It is clear that T is an extendible tree. Now suppose that j ∈ A_{i} and L_{j}(i) is an infinite path of T . Let S be the set of all σ ≺ L_{j}(i) that are active at some point in the construction. If σ ∈ S then, at some point in the construction, σ is declared to no longer be active, and strings σ0^{j+1} and σ0^{j}1 are declared to be active. These strings are the only extensions of σ of length |σ| + j + 1 in T , so L_{j}(i) must extend one of them. Thus we see that S is infinite, which implies that L_{j}(i) is not isolated, since for every τl ∈ S other than 0^{i}1, we also have τ(1 − l) ∈ T . So we are left with showing that T is atomic.

Fix a ρ ∈ T . We may assume that ρ extends some 0^{i}1. Let P be any path of T extending ρ (say the leftmost one). If P is isolated then we are done, so assume otherwise. Then P must go through infinitely many nodes that are active at some point in the construction, and we can find a sequence ρ ≺ σ_{0} ≺ σ_{1} ≺ ⋯ of such nodes. Each σ_{l} is of the form τ_{l}j_{l} for some τ_{l} and j_{l} ⩽ 1. If τ_{l}(1 − j_{l}) is permanently active then there is a unique path of T extending τ_{l}(1 − j_{l}). Since τ_{l}(1 − j_{l}) extends ρ, in this case we are done. So assume that for all l, the node τ_{l}(1 − j_{l}) is not permanently active. Then there is an n_{l} ∈ A_{i} such that τ_{l}(1 − j_{l}) ∈ L_{n_{l}}(i). But if l < l′ then τ_{l}(1 − j_{l}) and τ_{l′}(1 − j_{l′}) are incompatible, so n_{l} ≠ n_{l′}. Thus we get a computable injective map from ℕ into the bounded set A_{i}, which is a contradiction. □

The statement of FATT suggests the idea of strengthening the hypothesis of ATT by replacing it with the conclusion of FATT, yielding a principle intermediate between ATT and AMT (see Question 8.2 above):

ATT^{+}: Let T be an extendible tree with the property that for each finite set of nodes σ_{0}, …, σ_{k} ∈ T , there is a set of isolated paths P_{0}, …, P_{k} such that P_{i} extends σ_{i}. Then there is a listing of the isolated paths of T .

Appendix C. Saturated models

Recall that A is saturated if for all ā ⊆ A, each complete type of Th(A, ā) is realized in A; and strongly saturated if for all ā ⊆ A, each partial type of Th(A, ā) is realized in A. In this appendix, we prove the following result, mentioned in Section 2.2.

Theorem C.1. The statement that every saturated model is strongly saturated is equivalent to WKL_{0} over RCA_{0}.

Proof. To show that the statement holds in WKL_{0}, it is enough to fix ā ⊆ A and show in WKL_{0} that every partial type of Th(A, ā) is contained in a complete type of Th(A, ā). Let p(x̄) be a partial n-type of Th(A, ā). Add an n-tuple of constant symbols c̄ to the language of (A, ā), and let S = Th(A, ā) ∪ {φ(c̄) : φ(x̄) ∈ p}. It is well-known that Lindenbaum’s Lemma, which implies that S has a completion T, is provable in WKL_{0} (see for instance [47]). Then {φ(x̄) : φ(c̄) ∈ T } is a complete type containing p.

In the other direction, we argue in RCA_{0} as follows. Assume that every saturated model is strongly saturated. Let f and g be functions with disjoint ranges, and let T be a tree such that P is an infinite path of T if and only if n ∈ rng f ⇒ P (n) = 1 and n ∈ rng g ⇒ P (n) = 0. To show that WKL_{0} holds, it is enough to show that a tree of this form has an infinite path (see [47]). Let S be the set of terminal nodes of T , and let T̂ = T ∪ {σ0^{n} : σ ∈ S, n ∈ ℕ}. Let T = T (T̂) be as in Definition 3.5, and let A be a model of T (which exists because the Completeness Theorem for theories holds in RCA_{0}). If T has an infinite path then we are done, so assume otherwise. Let ā ⊂ A and let p(x_{0}, …, x_{n−1}) be a complete n-type of Th(A, ā). For each i < n, the induced 1-type on x_{i} corresponds to an infinite path of T̂, which must be of the form σ_{i}0^{ℕ} with σ_{i} ∈ S. Since S is a set, we can form the tuple (σ_{0}, …, σ_{n−1}), from which we can define a formula generating p. So p is principal, and hence realized in A.

Thus A is saturated and hence, by hypothesis, strongly saturated. Using the notation of Definition 3.5, let φ_{n,s}(x) ≡ P_{n}(x) ∧ P_{n}(x) ∧ ⋯ ∧ P_{n}(x), where P_{n}(x) is repeated s + 1 many times, and let ψ_{n,s}(x) ≡ ¬P_{n}(x) ∧ ¬P_{n}(x) ∧ ⋯ ∧ ¬P_{n}(x), where ¬P_{n}(x) is repeated s + 1 many times. Let Γ = {φ_{n,s} : f(s) = n} ∪ {ψ_{n,s} : g(s) = n}. Then Γ is a partial 1-type of T, and hence is realized by some a ∈ A. It follows that {n : A ⊨ P_{n}(a)} is a path on T . □

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Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, drh@math.uchicago.edu

Department of Mathematics, Wellesley College, 106 Central St.,

Wellesley, MA 02481, klange2@wellesley.edu

Department of Mathematics, Malott Hall, Cornell University,

Ithaca, NY 14853, shore@math.cornell.edu