PROPER FORCING, CARDINAL ARITHMETIC, AND UNCOUNTABLE LINEAR ORDERS

JUSTIN TATCH MOORE

Abstract. In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of ℝ which is Σ_{1}-definable in (H(ω_{2}), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω_{1}, ω_{1}^{∗}, C, C^{∗} where X is any suborder of the reals of size ω_{1} and C is any Countryman line. Third, the Proper Forcing Axiom always fails in generic extensions by Prikry forcing. It is also shown that PFA implies the Singular Cardinals Hypothesis at κ unless stationary subsets of S_{κ^{+}}^{ω} reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω_{2}).

1. Introduction

The purpose of this note is to communicate the following results.

Theorem 1.1. [10] (BPFA) There is a well ordering of P(ω_{1})/NS (and hence of ℝ) which is Σ_{1}-definable in (H(ω_{2}), ∈).

Theorem 1.2. [11] (PFA) Suppose that X is a subset of ℝ of size ℵ_{1} and C is a Countryman line. Then any uncountable linear order contains an isomorphic copy of one of the following five orders: X, ω_{1}, ω_{1}^{∗}, C, or C^{∗}.^{1}

Theorem 1.3. [12] The Proper Forcing Axiom fails in any generic extension by Prikry forcing.

Theorem 1.4. [12] (PFA) If κ^{+} < κ^{ω} for an uncountable cardinal κ then stationary subsets of

S_{κ^{+}}^{ω} = {α < κ^{+} : cf(α) = ω}

reflect.

The research presented in this paper was supported in part by NSF grant DMS-0401893.

1If L is a linear order then L^{∗} denotes the converse of L.

The first of these theorems answers a question from the folklore which is made explicit in [25]. The second confirms a well known conjecture of Shelah [13] (see also [14], [20], [21]). The third and forth address an approach to proving the consistency of the Proper Forcing Axiom and the negation of the Singular Cardinals Hypothesis. It strongly suggests that PFA implies SCH.

The first of these theorems resulted in the development of a new reflection principle which follows from the Proper Forcing Axiom. An analysis of this new principle then gave rise to the remaining theorems.

This note is organized as follows. In the first section we will discuss the proof of the first theorem. In the second section we will introduce the notion of trace reflection for sets of ordinals and discuss how it relates to the Singular Cardinals Problem. In the third section we will look at a new reflection principle which was isolated from the proof and mention some of its basic consequences and how they relate to Theorems 1.1, 1.3, and 1.4. In the forth section we will briefly discuss the techniques involved in proving the more involved Theorem 1.2 as well as some historical discussion of the theorem. In the final section I will offer some open questions and future directions.

This paper is not intended to provide any detailed proofs of the claims it makes. The details will be in the forthcoming [11], [10], and [12]. The intention is to announce the results and to give the reader an idea of the arguments involved. The reader is referred to [7], [25], [26], and [29] for some further information on results related to forcing axioms and well orderings of the continuum. For further reading on the basis problem for uncountable linear orders see [13], [20], and [21]. See [7] and [28] for more information of the Singular Cardinals Hypothesis and forcing axioms.

2. A Σ_{1}-definable well ordering of ℝ from BPFA

Recall the following axiom of Goldstern and Shelah [8] (stated here in an equivalent formulation due to Bagaria [3]):

BPFA: For every proper forcing Q the structure (H(ω_{2}), ∈) is Σ_{1}- elementary in V^{Q}.

In this section we will see that the Bounded Proper Forcing Axiom can be used to code reals in a way which is reminiscent of the SRP style coding methods (e.g. ϕ_{AC}, ψ_{AC} of [29]).

Theorem 2.1. [10] (BPFA) There is a well ordering of P(ω_{1})/NS which is Σ_{1}-definable in (H(ω_{2}), ∈).

In order to discuss the idea behind the proof, we first need to introduce some notation. Fix a sequence ⟨C_{ξ} : ξ ∈ lim(ω_{1})⟩ such that C_{ξ} is a cofinal subset of ξ of ordertype ω for each limit ξ < ω_{1}. Let N, M be countable sets of ordinals such that N ⊆ M, otp(M) is a limit, and sup(N) < sup(M). Define

w(N, M) = | sup(N) ∩ π_{M}^{−1}[C_{α}]|

where α is the ordertype of M and π_{M} : M → α is the collapsing map.

Consider the following assertion for a subset A of ω_{1}:

υ_{AC}(A): There is an uncountable δ < ω_{2} and an increasing sequence ⟨N_{ξ} : ξ < ω_{1}⟩ which is club in [δ]^{ω} such that for all limit ν < ω_{1} there is a ν_{0} < ν such that if ν_{0} < ξ < ν then N_{ν} ∩ ω_{1} ∈ A is equivalent to w(N_{ξ} ∩ ω_{1}, N_{ν} ∩ ω_{1}) < w(N_{ξ}, N_{ν}).

The sentence υ_{AC} (which takes ⟨C_{ξ} : ξ ∈ lim(ω_{1})⟩ as a parameter) is the assertion that υ_{AC}(A) holds for all A ⊆ ω_{1}. It is routine to show that υ_{AC} implies that there is a Σ_{1}-definable well order of P(ω_{1})/NS.

What is actually proved in [10] is that for any A ⊆ ω_{1}, υ_{AC}(A) can be forced by a proper partial order. It follows immediately that the Bounded Proper Forcing Axiom implies υ_{AC} which in turn generates the requisite well ordering. David Aspero pointed out that these techniques can also be used to force a well order of P(ω_{1})/NS which is Σ_{1}-definable in (H(ω_{2}), ∈) over any model with an inaccessible cardinal. The sentences ϕ_{AC} and ψ_{AC} are both known to have considerable large cardinal strength. While the strength of θ_{AC} is unknown, the best upper bound also large.

3. Trace reflection and singular cardinals

In [7], Foreman, Magidor, and Shelah proved the following theorem.

Theorem 3.1. [7] (MM) If κ ≥ ω_{2} is a regular cardinal and F is a collection of ℵ_{1} many stationary subsets of S_{κ}^{ω} then there is a δ < κ such that cf(δ) = ω_{1} and X ∩ δ is stationary for all X in F.

Since S_{κ}^{ω} can be partitioned into κ many disjoint stationary sets, it easily follows that, assuming MM, [κ]^{ω_{1}} has cofinality κ. By Silver’s theorem [17] this in turn shows that Martin’s Maximum implies the Singular Cardinals Hypothesis [7].

In this section, I will discuss a new notion of simultaneous reflection which can be used to prove Theorems 1.3 and 1.4. First, we need to recall Todorčević’s notion of a minimal walk [22]. A C-sequence on an ordinal θ is a sequence C⃗ = ⟨C_{α} : α < θ⟩ such that C_{α} ⊆ α is closed and unbounded for every limit α ∈ θ and C_{α+1} = {α}. If Ω ⊆ θ, then we say that C⃗ avoids Ω if C_{α} is disjoint from Ω for limit α. Observe that if C⃗ avoids Ω, then Ω ∩ α is non-stationary for all limit α. Conversely if Ω ⊆ θ is such that Ω ∩ α is non-stationary for all limit α, then there is a C-sequence which avoids it.

The trace function for C⃗ is defined recursively as follows:^{2}

tr(α, α) = ∅,
tr(α, β) = tr(α, min(C_{β} \ α)) ∪ {β}.

A key property of the trace function is the following. Proposition 3.2. If δ < β < θ and δ is a limit ordinal then there is a δ_{0} < δ such that

tr(δ, β) ⊆ tr(α, β)

whenever α is in (δ_{0}, δ). If C⃗ avoids {δ} then it can be further arranged that δ is in tr(α, β) as well.

If C⃗ is given, let H(C⃗) be all X ⊆ θ such that if Z ⊆ θ is closed and unbounded then

X ∩ ⋃_{α,β∈Z} tr(α, β) ≠ ∅

The following is an immediate consequence of a theorem of Todorčević (see the section on the square bracket operation in [27]).

Proposition 3.3. If C⃗ avoids Ω then every stationary subset of Ω is in H(C⃗).

We are now ready to define the new notion of reflection.

Definition 3.4. An element X of H(C⃗) is said to trace reflect at δ < θ if X ∩ δ is in H(C⃗ ↾ δ).

The following theorem is then used to deduce Theorems 1.3 and 1.4 as corollaries. Theorem 3.5. [12] (PFA) Suppose that θ ≥ ω_{2} is a regular cardinal and ⟨C_{α} : α < θ⟩ is a C-sequence which avoids some stationary Ω ⊆ S_{θ}^{ω}. If F is a collection of ℵ_{1} many elements of H(C⃗) which are contained in Ω then there is a δ < θ such that cf(δ) = ω_{1} and every element of F trace reflects at δ.

It should be noted that while Theorem 1.4 is easily deduced from the comments made above, Theorem 1.3 requires some further argument — see [12].

2The convention that tr(α, α) be equal to ∅ rather than {α} is not entirely standard but has little effect on the theory of tr. Our convention is needed for Proposition 3.2.

4. Set Mapping Reflection

In this section I will define a reflection principle which was isolated from the proof in the previous section. First we will need some definitions. Recall that for an uncountable set X, [X]^{ω} is the collection of all countable subsets of X.

Definition 4.1. Let X be an uncountable set and M be a countable elementary submodel of H(θ) for some regular θ such that [X]^{ω} is in M. A subset Σ of [X]^{ω} is M-stationary if whenever E ⊆ [X]^{ω} is a club in M there is an N in E ∩ Σ ∩ M.

The set [X]^{ω} is equipped with the Ellentuck topology obtained by declaring the intervals

[x, N] = {Y ∈ [X]^{ω} : x ⊆ Y ⊆ N}

to be open for all N in [X]^{ω} and finite x ⊆ N. For ease of reading I will also make the following definition.

Definition 4.2. A set mapping Σ is said to be open stationary if, for some uncountable set X_{Σ} and regular cardinal θ_{Σ} with X_{Σ} in H(θ_{Σ}), it is the case that elements of the domain of Σ are elementary submodels of H(θ_{Σ}) which contain X and Σ(M) ⊆ [X_{Σ}]^{ω} is open and M-stationary for all M in the domain of Σ.

Among the simplest examples of open stationary set mappings are the following.

Example. Let r : ω_{1} → ω_{1} be regressive on the limit ordinals. If ⟨P_{ξ} : ξ < ω_{1}⟩ is a continuous ∈-chain of elementary submodels of H(ω_{2}) then define Σ(P_{ξ}) = (r(δ_{ξ}), δ_{ξ}) where δ_{ξ} = P ∩ ω_{1}.

The Mapping Reflection Principle asserts that any open stationary set mapping contains a copy of such an example.

MRP: If Σ is an open stationary set mapping whose domain is a club then there is a continuous ∈-chain ⟨N_{ν} : ν < ω_{1}⟩ in the domain of Σ such that for all limit 0 < ν < ω_{1} there is a ν_{0} < ν such that N_{ξ} ∩ X_{Σ} ∈ Σ(N_{ν}) whenever ξ ∈ (ν_{0}, ν).

Now consider the following examples which are motivated by the previous section.

Example. If M is a countable elementary submodel of H(2^{ω_{1}}^{+}) then the sets

Σ_{<}(M) = {N ∈ M ∩ [ω_{2}]^{ω} : w(N ∩ ω_{1}, M ∩ ω_{1}) < w(N, M ∩ ω_{2})}

Σ_{≥}(M) = {N ∈ M ∩ [ω_{2}]^{ω} : w(N ∩ ω_{1}, M ∩ ω_{1}) ≥ w(N, M ∩ ω_{2})} are open. With a little effort, they are also seen to be M-stationary.

Example. Let ⟨C_{α} : α < θ⟩ be a C-sequence which avoids a stationary set Ω ⊆ S_{θ}^{ω}. If M is a countable elementary submodel of H(2^{θ}^{+}) which contains C⃗ and some stationary Ω_{0} ⊆ Ω then put

Σ_{Ω_{0}}(M) = {N ∈ M ∩ [θ]^{ω} : Ω_{0} ∩ tr(sup(N), sup(M ∩ θ)) ≠ ∅}.

This mapping is open by Proposition 3.2 and M-stationary by Proposition 3.3.

It follows from the comments made in the example above that MRP implies υ_{AC} Theorem 3.5 on trace reflection.

The connection to PFA is that the following is true.

Theorem 4.3. The Proper Forcing Axiom implies MRP.

The forcing which is used is the collection of all countable successor length approximations to a reflecting sequence, ordered by extension. The open stationarity of the set mapping, together with the fact that the domain is a club, seems to be exactly what is needed to make the partial order proper.

It is worth noting that MRP and Todorčević’s Strong Reflection Principle (SRP) [5] have no formal relationship to one another. Of course MRP can’t imply SRP (or even many of its typical consequences such as Chang’s Conjecture and the saturation of the non-stationary ideal) since MRP follows from PFA. On the other hand, it can be shown that counterexamples to MRP are typically preserved by ω-semiproper forcing.^{3} Since SRP can be forced by an ω-semiproper forcing, it follows that SRP does not imply MRP.

5. A five element basis for the uncountable linear orders

In this section we will discuss the proof of Shelah’s conjecture for uncountable linear orders from the Proper Forcing Axiom. The following statement was conjectured to be consistent by Shelah in [13].

Shelah’s Conjecture. If X is any set of reals of size ℵ_{1} and C is any Countryman order then any uncountable linear order contains an isomorphic copy of one of the following five orders: X, ω_{1}, ω_{1}^{∗}, C, or C^{∗}.

It was inspired in part by the following result of Baumgartner (cast here in modern language).

Theorem 5.1. [4] The Bounded Proper Forcing Axiom implies that every two ℵ_{1}-dense^{4} sets of reals are isomorphic.

3A ♣-sequence which guesses closed unbounded sets, for instance.

4Here a set of reals is ℵ_{1}-dense if it meets every interval in a set of size ℵ_{1}.

The conclusion of the theorem implies the existence of a single element basis for the uncountable separable linear orders.

An important class of orders isolated long ago and natural to consider in this context are the Aronszajn lines — those uncountable linear orders in which every suborder which is separable, a well order, or a converse well order, is countable. Such orders necessarily have cardinality ℵ_{1}. Analysis of these orders tends to be necessarily set theoretic in nature and even their existence (proved by Aronszajn) is non-trivial to establish.

For some time it was open whether the Aronszajn lines could consistently have a single element basis and that all uncountable linear orders could have a four element basis. Shelah refuted this in [13]. He proved the existence of the following object whose definition is due to Countryman.

Definition 5.2. A linear order C is Countryman if C^{2} is the union of countably many chains in the coordinate-wise partial order.

Observe that if L is a linear order and f, f^{∗} are order embeddings from L into C and C^{∗} respectively than the set

A = {(f(l), f^{∗}(l)) : l ∈ L}

intersects every chain in C^{2} in a singleton and, since C is Countryman, L must be countable. It follows that any basis for the uncountable linear orders must contain both C and C^{∗}.

Todorčević has given the following canonical example of a Countryman line.

Theorem 5.3. [22] If ⟨e_{α} : α < ω_{1}⟩ is a sequence of functions such that e_{α} : α → ω is finite-1 for all α < ω_{1} and e_{α} =^{∗} e_{β} ↾ α for all α < β then {e_{α} : α < ω_{1}} is Countryman when given the lexicographical ordering.

Early on the following reduction was made (see the final section of [21] for a complete proof).

Theorem 5.4. [1] (BPFA) Suppose that there is an Aronszajn tree T such that whenever K ⊆ T there is an uncountable antichain A ⊆ T such that ∧(A) = {s∧t : s, t ∈ A} is either contained in or disjoint from K. Then X, ω_{1}, ω_{1}^{∗}, C, and C^{∗} forms a five element basis whenever X is a subset of ℝ of size ℵ_{1} and C is a Countryman order.

My contribution has been to prove the next result, thereby proving that Shelah’s conjecture is an consequence of the Proper Forcing Axiom.

Theorem 5.5. [11] (MRP + BPFA) There is an Aronszajn tree T such that whenever K ⊆ T there is an uncountable antichain A ⊆ T such that ∧(A) = {s ∧ t : s, t ∈ A} is either contained in or disjoint from K.

The analysis is rather lengthy and a multitude of definitions are required just to state the main lemmas. In short, the techniques used are a modification of Todorčević’s method of building forcings with elementary submodels as side conditions (see [23], [24]). The proof has an interesting new feature, however. In the past, applications of PFA have involved defining a forcing and proving, in ZFC, that it is proper. The Proper Forcing Axiom is then applied to supply the requisite generic absoluteness. The necessary objects — such as embeddings, homogeneous sets, isomorphisms, and uncountable discrete sets — then exist in V where the analysis is finished. In the proof of Theorem 1.2, however, the Proper Forcing Axiom is needed just to prove that the relevant forcing is proper. Hence, while the conclusion of Theorem 5.5 is a conjunction of Σ_{1}-sentences in the language of (H(ω_{2}), ∈), it is possible that the Bounded Proper Forcing Axiom does not imply imply Shelah’s conjecture on grounds of consistency strength alone.

6. Questions and concluding remarks

Recently Larson, Veličković, and myself showed that PFA(ω_{2}) is sufficient to imply Shelah’s conjecture, thus greatly reducing the consistency strength to something less than the existence of O^{♯}. The following two questions, however, remain open.

Question 6.1. Does BPFA imply Shelah’s Conjecture?

Question 6.2. Does Shelah’s Conjecture imply that ω_{2} is weakly compact in L?

It is not known that Shelah’s Conjecture has any large cardinal strength at all.

One question left conspicuously open in [12] is:

Question 6.3. Does the Proper Forcing Axiom imply the Singular Cardinals Hypothesis?

In light of the results mentioned above and a recent result of Shelah that stationary reflection in [θ]^{ω} to sets of size ℵ_{1} implies SCH [15] it seems likely that the answer is yes (see also the earlier results in [28]). It should be noted, however, that Assaf Sharon has recently shown that SCH can fail at κ (even for κ = ℵ_{ω}) and yet all stationary subsets of κ^{+} reflect. Hence it is not possible to supplement Theorem 3.5 with a ZFC theorem and resolve this question.

Another question I have become interested in is:

Question 6.4. Does the Proper Forcing Axiom imply that there is a generic almost huge embedding with critical point ω_{1}?

One approach is to show that PFA implies there is a completion of P(ω_{1})/NS which preserves ω_{2}. Giving a negative answer to this question would be rather tricky since a Woodin cardinal is already sufficient to give such a completion (the stationary tower forcing) [29]. On the other hand, presently all of PFA’s consistency strength at the level of a Mahlo or higher that I am aware of has come from the fact that PFA implies the failure of the combinatorial principle □(κ) at all levels [19].

From an old result of Sierpiñski’s [16] one can readily deduce that Shelah’s conjecture implies that |ℝ| > ℵ_{1}.

Question 6.5. Does Shelah’s conjecture imply |ℝ| = ℵ_{2}?

The current methods require that the continuum is ℵ_{2}. On the other hand, it is known [2] that a one element basis for the separable linear orders is consistent with a large continuum. If this question has a positive answer, it would be a remarkable development as it would draw a rather unexpected connection between the theory of linear orders and the analysis of the Continuum Problem.

Finally, it seems plausible that the methods for proving Shelah’s Conjecture from PFA may give some new insight into problems such as whether ω_{1} → (ω_{1}, (ω_{1}, ω_{1}))^{2} is consistent [18] and Fremlin’s problem on perfectly normal compacta (see Question 2.1 in [9]). The only use of MRP in [11] can be isolated to a single lemma. With a stronger assumption, this lemma can be stated in the following abstract form.

0-1 Law for Open Set Mappings. (SMRP) Suppose that Σ is an open set mapping defined on a club and that Σ has the following properties:

(1) If N is in the domain of Σ, then Σ(N) is closed under end extensions.^{5}

(2) If N and \overline{N} are in the domain of Σ and \overline{N} is an end extension

of N then Σ(N) = Σ(\overline{N}) ∩ N. Then for a closed unbounded set of N in the domain of Σ, there is a club E ⊆ [X_{Σ}]^{ω} in N such that E ∩ N is either contained in or disjoint from Σ(N).

5Here we define \overline{N} end extends N as meaning that N ∩ω_{1} = \overline{N} ∩ω_{1} and N ⊆ \overline{N}.

Here SMRP is the Strong Mapping Reflection Principle obtained by replacing “club” in the statement of MRP with “projective stationary” (see [6]). It follows from Martin’s Maximum by a routine modification of the proof that PFA implies MRP [10]. While at the present it does not seem that the answers to these problems will simply “fall out” of the new methods, the methods of [11] reduce these problem to new, unfamiliar difficulties.

References

[1] U. Abraham and S. Shelah. Isomorphism types of Aronszajn trees. Israel J. Math., 50(1-2):75–113, 1985.

[2] Uri Abraham, Matatyahu Rubin, and Saharon Shelah. On the consistency of some partition theorems for continuous colorings, and the structure of ℵ_{1}-dense real order types. Ann. Pure Appl. Logic, 29(2):123–206, 1985.

[3] Joan Bagaria. Bounded forcing axioms as principles of generic absoluteness. Arch. Math. Logic, 39(6):393–401, 2000.

[4] James E. Baumgartner. All ℵ_{1}-dense sets of reals can be isomorphic. Fund. Math., 79(2):101–106, 1973.

[5] M. Bekkali. Topics in set theory. Springer-Verlag, Berlin, 1991. Lebesgue measurability, large cardinals, forcing axioms, ρ-functions, Notes on lectures by Stevo Todorčević.

[6] Qi Feng and Thomas Jech. Projective stationary sets and a strong reflection principle. J. London Math. Soc. (2), 58(2):271–283, 1998.

[7] Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s Maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1–47, 1988.

[8] Martin Goldstern and Saharon Shelah. The Bounded Proper Forcing Axiom. J. Symbolic Logic, 60(1):58–73, 1995.

[9] Gary Gruenhage. Perfectly normal compacta, cosmic spaces, and some partition problems. In Open problems in topology, pages 85–95. North-Holland, Amsterdam, 1990.

[10] Justin Tatch Moore. Set mapping reflection. submitted to JML in Nov. 2003.

[11] Justin Tatch Moore. A five element basis for the uncountable linear orders. Preprint, February 2004.

[12] Justin Tatch Moore. The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis. Preprint, July 2004.

[13] Saharon Shelah. Decomposing uncountable squares to countably many chains. J. Combinatorial Theory Ser. A, 21(1):110–114, 1976.

[14] Saharon Shelah. On what I do not understand (and have something to say). I. Fund. Math., 166(1-2):1–82, 2000. Saharon Shelah’s anniversary issue.

[15] Saharon Shelah. Reflection implies SCH. Preprint, July 2004.

[16] W. Sierpiñski. Sur un problème concernant les types de dimensions. Fundamenta Mathematicae, 19:65–71, 1932.

[17] Jack Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pages 265–268. Canad. Math. Congress, Montreal, Que., 1975.

[18] Stevo Todorčević. Forcing positive partition relations. Trans. Amer. Math. Soc., 280(2):703–720, 1983.

[19] Stevo Todorčević. A note on the proper forcing axiom. In Axiomatic set theory (Boulder, Colo., 1983), volume 31 of Contemp. Math., pages 209–218. Amer. Math. Soc., Providence, RI, 1984.

[20] Stevo Todorčević. Basis problems in combinatorial set theory. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, pages 43–52, 1998.

[21] Stevo Todorčević. Lipszhitz maps on trees. report 2000/01 number 13, Institut Mittag-Leffler.

[22] Stevo Todorčević. Partitioning pairs of countable ordinals. Acta Math., 159(3– 4):261–294, 1987.

[23] Stevo Todorčević. Partition Problems In Topology. Amer. Math. Soc., 1989.

[24] Stevo Todorčević. A classification of transitive relations on ω_{1}. Proc. London Math. Soc. (3), 73(3):501–533, 1996.

[25] Stevo Todorčević. Localized reflection and fragments of PFA. In Logic and scientific methods, volume 259 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 145–155. AMS, 1997.

[26] Stevo Todorčević. Generic absoluteness and the continuum. Mathematical Research Letters, 9:465–472, 2002.

[27] Stevo Todorčević. Coherent sequences. In Handbook of Set Theory. North-Holland, (in preparation).

[28] Boban Veličković. Forcing axioms and stationary sets. Adv. Math., 94(2):256– 284, 1992.

[29] W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. Logic and its Applications. de Gruyter, 1999.