FORCING AXIOMS AND THE CONTINUUM HYPOTHESIS

DAVID ASPERÓ, PAUL LARSON, AND JUSTIN TATCH MOORE

1. Introduction

One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natural to consider strengthenings of this statement in which one replaces countably many with ℵ_{1}-many. Even taking the compact space to be the unit interval, this already implies the failure of the Continuum Hypothesis and therefore is a statement not provable in ZFC. Additionally, there are ZFC examples of compact spaces which can be covered by ℵ_{1} many nowhere dense sets. For instance if K is the one point compactification of an uncountable discrete set, then K^{ω} can be covered by ℵ_{1} many nowhere dense sets. Hence some restriction must be placed on the class of compact spaces in order to obtain even a consistent statement.

Still, there are natural classes of compact spaces for which the corresponding statement about Baire Category — commonly known as a forcing axiom — is consistent. The first and best known example is Martin’s Axiom for ℵ_{1} dense sets (MA_{ℵ_{1}}) whose consistency was isolated from solution of Souslin’s problem [21]. This is the forcing axiom for compact spaces which do not contain uncountable families of pairwise disjoint open sets. For broader classes of spaces, it is much more natural to formulate the class and state the corresponding forcing axiom in terms of the equivalent language of forcing notions.

Date: August 26, 2012.

2000 Mathematics Subject Classification. 03E35, 03E50, 03E57.

Key words and phrases. Continuum Hypothesis, iterated forcing, forcing axiom, Martin’s Maximum, Π_{2} maximality, Proper Forcing Axiom.

This research was completed while all three authors were visiting the Mittag-Leffler Institute in September, 2009. An early draft of this paper appeared as Report No. 38 in the Institute’s preprint series (under the title “On Π_{2}-maximality and the Continuum Hypothesis”). We would like to thank the staff of the institute for their generous hospitality. The research of the first author was supported by Projects MTM2008–03389 (Spain) and 2009SGR–00187 (Catalonia). The research of the second author is supported by NSF grant DMS-0801009. The research of the third author was supported by NSF grant DMS–0757507.

Foreman, Magidor, and Shelah have isolated the broadest class of forcings for which a forcing axiom is relatively consistent — those forcings which preserve stationary subsets of ω_{1} [10]. The corresponding forcing axiom is known as Martin’s Maximum (MM) and has a vast wealth of consequences which are still being developed (many are in fact consequences of the weaker Proper Forcing Axiom (PFA) — see [17] for a recent surgery).

Many consequences of MM (and in fact MA_{ℵ_{1}} itself) are examples of Π_{2} sentences concerning the structure H(ℵ_{2}) = (H(ℵ_{2}), ∈, ω_{1}, NS_{ω_{1}}). Woodin has produced a forcing extension of L(ℝ), under an appropriate large cardinal assumption, which is provably optimal in terms of the Π_{2} sentences which its H(ℵ_{2}) satisfies [22]. Not surprisingly, the theory of the H(ℵ_{2}) of this model largely coincides with the consequences of MM which concern H(ℵ_{2}).

What will concern us in the present paper is the extent to which there is a corresponding strongest forcing axiom which is consistent with the Continuum Hypothesis (CH). More specifically, Woodin has posed the following problem.

Problem 1.1. [22] Are there two Π_{2}-sentences ψ_{1} and ψ_{2} in the language of (H(ℵ_{2}), ∈, ω_{1}, NS_{ω_{1}}) such that ψ_{1} and ψ_{2} are each individually Ω-consistent with CH but such that ψ_{1} ∧ ψ_{2} Ω-implies ¬CH?

For the present discussion, it is sufficient to know that “Ω-consistent” means something weaker than “provably forcible from large cardinals” and “Ω-implies” means something weaker than just “implies.”

Even though CH implies that [0, 1] can be covered by ℵ_{1}-many nowhere dense sets, some forcing axioms are in fact compatible with CH. Early on in the development of iterated forcing, Jensen established that Souslin’s Hypothesis was consistent with CH (see [5]). Shelah then developed a general framework for establishing consistency results with CH by iterated forcing [20]. The result was a largely successful but ad hoc method which Shelah and others used to prove that many consequences of MM are consistent with CH (see [2], [8], [7], [14], [19], [20]). Moreover, with a few exceptions, it was known that starting from a ground model with a supercompact cardinal, these consequences of MM could all be made to hold in a single forcing extension which satisfies CH.

The purpose of the present paper is to prove the following theorem, which shows that that Problem 1.1 has a positive answer if it is consistent that there is an inaccessible limit of measurable cardinals (usually this question is discussed in the context of much stronger large cardinal hypotheses).

Theorem 1.2. There exist sentences ψ_{1} and ψ_{2} which are Π_{2} over the structure (H(ω_{2}), ∈, ω_{1}) such that

• ψ_{2} can be forced by a proper forcing not adding ω-sequences of ordinals;

• if there exists a strongly inaccessible limit of measurable cardinals, then ψ_{1} can be forced by a proper forcing which does not add ω-sequences of ordinals;

• the conjunction of ψ_{1} and ψ_{2} implies that 2^{ℵ_{0}} = 2^{ℵ_{1}}.

Note that neither of ψ_{1} and ψ_{2} requires the use of the non-stationary ideal on ω_{1} as a predicate. The third conclusion of Theorem 1.2 is proved in Proposition 2.3. The first conclusion follows from Theorem 3.3 and Lemmas 3.5 and 3.6. The first conclusion follows from Theorem 3.10 and Lemmas 4.2 and 4.3.

The relative consistency of these sentences with CH is obtained by adapting Eisworth and Roitman’s preservation theorems for not adding reals [8] (which are closely based on Shelah’s framework noted above) in two different — and necessarily incompatible — ways. Traditionally, the two ingredients in any preservation theorem of this sort are completeness and some form of (< ω_{1})-properness. For the preservation theorem for one of our sentences (which is essentially proved in [8]), the completeness condition is weakened while maintaining the other requirement. In the other preservation theorem the completeness condition is strengthened slightly from the condition in [8], but (<ω_{1})- properness is replaced by the weaker combination of properness and (<ω_{1})-semiproperness.

The paper is organized as follows. In Section 2 we formulate the two Π_{2} sentences and outline the tasks which must be completed to prove the main theorem. Section 3 contains a discussion of the preservation theorems which will be needed for the main result, including the proof of a new preservation theorem for not adding reals. Section 4 is devoted to the analysis of the single step forcings associated to one of the Π_{2}- sentences. Section 5 contains some concluding remarks.

The reader is assumed to have familiarity with proper forcing and with countable support iterated forcing constructions. While we aim to keep the present paper relatively self contained, readers will benefit from familiarizing themselves with the arguments of [3] and [8]. We will also deal with revised countable support and will use [15] as a reference. The notation is mostly standard for set theory and we will generally follow the conventions of [12] and [13]. We will now take the time to fix some notational conventions which are not entirely standard. If A is a set of ordinals, ot(A) will denote the ordertype of A. If θ is a regular cardinal, then H(θ) will denote the collection of all sets of hereditary cardinality less than θ. Unless explicitly stated otherwise, θ will always denote an uncountable regular cardinal. If X is an uncountable set, we will let [X]^{ℵ_{0}} denote the collection of all countable subsets of X. If X has cardinality ω_{1}, then an ω_{1}-club in [X]^{ℵ_{0}} is a cofinal subset which is closed under taking countable unions and is well ordered in type ω_{1} by containment. At certain points we will need to code hereditarily countable sets as elements of 2^{ω}. If r ∈ 2^{ω} and A is in H(ℵ_{1}), then we say that r codes A if (tc(A), ∈, A) is isomorphic to (ω, R_{1}, R_{2}) where R_{1} ⊆ ω^{2} and R_{2} ⊆ ω are defined by

((i, j) ∈ R_{1}) ↔ (r(2^{i+1}(2j + 1)) = 1)
i ∈ R_{2} ↔ r(2i + 1) = 1

(Here tc denotes the transitive closure operation.) While not every r in 2^{ω} codes an element of H(ℵ_{1}), every element of H(ℵ_{1}) has a code in 2^{ω}. Also, if f is a finite-to-one function from a set of ordinals of ordertype ω into 2^{<ω}, then we will say that f codes A ∈ H(ℵ_{1}) if, for some cofinite subset X of the domain of f, ⋃ f[X] is a single infinite length sequence which codes A in the sense above. Finally, if r and s are elements of 2^{≤ω}, we will let Δ(r, s) denote the least i such that r(i) ≠ s(i) (if no such i exists, we define Δ(r, s) = min(|r|, |s|)).

2. Two Π_{2}-sentences

In this section we will present the two Π_{2}-sentences which are used to resolve Problem 1.1 and will prove that their conjunction implies 2^{ℵ_{0}} = 2^{ℵ_{1}}. This will be done by appealing to the following theorem of Devlin and Shelah.

Theorem 2.1. [4] The equality 2^{ℵ_{0}} = 2^{ℵ_{1}} is equivalent to the following statement: There is an F : H(ℵ_{1}) → 2 such that for every g : ω_{1} → 2, there is an X ∈ H(ℵ_{2}) such that whenever M is a countable elementary submodel of (H(ℵ_{2}), ∈, X),

F (\overline{X}) = g(δ).

where \overline{X} and δ are the images of X and ω_{1} respectively under the transitive collapse of M.

Let us also note the following equivalent formulation of ♢.

Proposition 2.2. ♢ holds if and only if there is a sequence ⟨X_{α} : α < ω_{1}⟩ of elements of H(ℵ_{1}) such that whenever Y ∈ H(ℵ_{2}), there is a countable elementary submodel M of (H(ℵ_{2}), ∈, Y) such that X_{δ} = \overline{Y} where \overline{Y} and δ are the images of X and ω_{1} respectively under the transitive collapse of M.

The first of our Π_{2}-sentences is essentially the same as one used by Caicedo and Veličković in [3] in order to prove that BPFA implies there is a Δ_{1}-well ordering of the H(ℵ_{2}), definable from a parameter in H(ℵ_{2}). (This coding has its roots in work of Gitik [11].) We will now take some time to recall the definitions associated to this coding. Given x ⊆ ω, let ∼_{x} be the equivalence relation on ω \ x defined by letting m ∼_{x} n iff [m, n] ∩ x = ∅. Given two further subsets y and z of ω, let (I_{k})_{k<t} (for some t ≤ ω) be the increasing enumeration of the set of ∼_{x}–equivalence classes intersecting both y and z, and let the oscillation of x, y and z be the function o(x, y, z) : t → 2 defined by

o(x, y, z) = 0 if and only if min(I_{k} ∩ y) ≤ min(I_{k} ∩ z)

Let C⃗ = ⟨C_{δ} : δ ∈ Lim(ω_{1})⟩ be a ladder system on ω_{1} (so that each C_{δ} is a cofinal subset of δ ordertype ω), and let α < β < γ be limit ordinals greater than ω_{1}. Let N ⊆ M be countable subsets of γ with {ω_{1}, α, β} ⊆ N such that, for all ξ ∈ {ω_{1}, α, β, γ}, sup(N ∩ ξ) < sup(M ∩ ξ) and sup(M ∩ ξ) is a limit ordinal. We are going to specify a way of decoding a finite binary sequence from C⃗, N, M, α and β. This decoding will be a very minor variation of the one defined in [3].

Let \overline{M} be the transitive collapse of M, and let π : M ⟶ \overline{M} be the corresponding collapsing function. Let ω\overline{N} ; 1 and ω\overline{M} ; 1 denote the respective ordertypes of N ∩ω_{1} and M ∩ω_{1}. Let α_{M} = π(α), β_{M} = π(β) and γ_{M} = ot(M). The height of N in M with respect to C⃗ is defined

as n(N, M) = |ω\overline{N} ; 1 ∩ C_{ω\overline{M} ; 1} |. Set

x = {|π(ξ) ∩ C_{α_{M}} | : ξ ∈ α ∩ N},
y = {|π(ξ) ∩ C_{β_{M}} | : ξ ∈ β ∩ N},
z = {|π(ξ) ∩ C_{γ_{M}} | : ξ ∈ N}.

If the length of o(x, y, z) is at least n(N, M), then we define

s(N, M) = sC⃗ ; α,β(N, M) = o(x, y, z)

Otherwise we leave s(N, M) undefined. If s is a finite length binary sequence, we define s̄ to be the sequence of the same length l with its digits reversed: s̄(i) = s(l − i).

If α < β < γ are ordinals of cofinality ℵ_{1} in the interval (ω_{1}, ω_{2}), and f is a function from ω_{1} to 2^{ω}, then we say that (α, β, γ) codes f (relative to C⃗) if there is an ω_{1}-club ⟨N_{ξ} : ξ < ω_{1}⟩ in [γ]^{ℵ_{0}} such that

• {ω_{1}, α, β} ⊆ N_{0};

• for all ν < ω_{1} and all ξ ∈ {ω_{1}, α, β, γ}, sup(N_{ν} ∩ ξ) is a limit ordinal; • for all ν_{0} < ν_{1} < ω_{1} and all ξ ∈ {ω_{1}, α, β, γ}, sup(N_{ν_{0}} ∩ ξ) <

sup(N_{ν_{1}} ∩ ξ);

• for every limit ν < ω_{1}, there is a ν_{0} < ν such that if ν_{0} < ξ < ν, then

Δ(s̄(N_{ξ}, N_{ν}), f(N_{ν} ∩ ω_{1})) ≥ n(N_{ξ}, N_{ν}),

where the functions s and n are computed using the parameters C⃗, α, β, and γ.

It is not difficult to show that if (α, β, γ) codes both f and g with respect to some C⃗, then there is a closed unbounded set of δ such that f(δ) = g(δ).

Let us pause for a moment to note that the assertion for some C⃗, every f is coded by some triple (α, β, γ) implies that 2^{ℵ_{0}} = 2^{ℵ_{1}}. To see this, define F by

• F (\overline{N} , ᾱ, β̄) = 1 whenever there exist an ω_{1}-club N in [γ]^{ℵ_{0}}, ordinals α < β < γ < ω_{2} as above, and a countable elementary submodel M of H(ℵ_{2}) containing {N , α, β}, such that s(N_{ξ}, N_{ν})(0) = 1 for a cobounded set of ξ < ν = M ∩ ω_{1}, and (\overline{M}, ∈, \overline{N} , ᾱ, β̄) is the collapse of (M, ∈, N , α, β);

• F (X) = 0 for all other X in H(ℵ_{1}). Now let g: ω_{1} → 2 be given and define f: ω_{1} → 2^{ω} by letting f(δ) be the real which takes the constant value g(δ). If N witnesses that (α, β, γ) codes f, and M is a countable elementary submodel of H(ℵ_{2}) containing \{N , α, β\}, then F (\overline{N} , ᾱ, β̄) = g(δ). By Theorem 2.1, this implies 2^{ℵ_{0}} = 2^{ℵ_{1}}.

Definition 2.1. ψ_{1} is the assertion that for every A : ω_{1} → 2 and for every ladder system C⃗, there is a triple (α, β, γ) and a function f : ω_{1} → 2^{ω} such that (α, β, γ) codes f relative to C⃗ and for each δ < ω_{1}, f(δ) is a code for A ↾ δ.

We will prove in Section 4 that the conjunction of ψ_{1} and CH can be forced over any model in which there is an inaccessible limit of measurable cardinals.

Now we will turn to the task of defining a Π_{2}-sentence ψ_{2} which, together with ψ_{1}, provides a solution to Problem 1.1. Suppose for a moment that ⟨N_{ξ} : ξ < ω_{1}⟩ witnesses that (α, β, γ) codes A : ω_{1} → 2 relative to C⃗. If X_{i} (i < ω) is an infinite increasing sequence in {N_{ξ} : ξ < ω_{1}}, define the height of {X_{i}}_{i<ω} to be δ = ω_{1} ∩ ⋃_{i<ω} X_{i}. Observe that, together with α, β and C⃗, {X_{i}}_{i<ω} uniquely determines A ↾ δ. Moreover, A ↾ δ can be recovered from just the isomorphism type of the structure

(N, ∈, ω_{1}, α, β; X_{i} : i < ω),

where N = ⋃_{i<ω} X_{i}. We will refer to a structure arising in this way as a ψ_{1}-structure and say that this structure codes A ↾ δ.

Definition 2.2. ψ_{2} is the assertion that for every ladder system C⃗, every triple α < β < γ of ordinals strictly between ω_{1} and ω_{2} and every ω_{1}-club N in [γ]^{ℵ_{0}}, there is a function f : ω_{1} → 2^{<ω} such that for every limit δ < ω_{1}, f ↾ C_{δ} codes (in the sense discussed at the end of the introduction) the transitive collapse of a structure

(N, ∈, ω_{1}, α, β; X_{i} : i < ω),

where {X_{i}}_{i<ω} is an increasing sequence in N of height greater than δ and N = ⋃_{i<ω} X_{i}.

In Section 3, we will prove that ψ_{2} is relatively consistent with CH. We now have the following proposition.

Proposition 2.3. ψ_{1} ∧ψ_{2} implies 2^{ℵ_{0}} = 2^{ℵ_{1}}. In fact, 2^{ℵ_{0}} = 2^{ℵ_{1}} follows from the existence of a ladder system C⃗ for which the conjunction of ψ_{1} and ψ_{2}, both relative to C⃗, holds.

Proof. Fix a ladder system C⃗ and suppose that ψ_{1} and ψ_{2} are true. If t : δ → 2^{<ω} for some countable limit ordinal δ and if t ↾ C_{δ} codes a ψ_{1}-structure which in turn codes g ↾ δ^{∗} for some δ^{∗} > δ and g : ω_{1} → 2, then define F (t) = g(δ). Now if (α, β, γ) codes g : ω_{1} → 2 relative to C⃗ as witnessed by N and f : ω_{1} → 2^{<ω} witnesses the corresponding instance of ψ_{2}, then F (f ↾ δ) = g(δ) for every limit ordinal δ. By Theorem 2.1, 2^{ℵ_{0}} = 2^{ℵ_{1}}. □

We will finish this section by showing that both ψ_{1} and ψ_{2} imply that ♢ fails. Let us say that an ω_{1}-club of [γ]^{ω} (for some γ < ω_{2} of uncountable cofinality) is typical in case for all ν_{0} < ν_{1} < ω_{1}, N_{ν_{0}} ∩ ω_{1} and sup(N_{ν_{0}}) are limit ordinals, N_{ν_{0}} ∩ ω_{1} < N_{ν_{1}} ∩ ω_{1}, and sup(N_{ν_{0}}) < sup(N_{ν_{1}}). The following fact shows that our methods do not extend to show nonexistence of a Π_{2}-maximal model for ♢.

Fact 2.4. ♢ implies the failure of ψ_{1}. In fact, ♢ implies that there is a ladder system C⃗ with the property that for every ordinal γ in ω_{2} of uncountable cofinality and every typical ω_{1}-club ⟨N_{ν} : ν < ω_{1}⟩ of [γ]^{ω} there are stationary many ν < ω_{1} such that for unboundedly many ξ < ν, |C_{N_{ν}∩ω_{1}} ∩ N_{ξ}| > |C_{ot(N_{ν})} ∩ sup(π“N_{ξ})|, where π is the collapsing function of N_{ν}.

Proof. It is easy to fix a natural notion of coding in such a way that for every γ < ω_{2} and every ω_{1}-club ⟨N_{ν} : ν < ω_{1}⟩ of [γ]^{ω} there is a set X ⊆ ω_{1} and there is a closed unbounded set of δ < ω_{1} such that X ∩ δ codes a directed system S = ⟨δ_{ν}, i_{ν,ν′} : ν ≤ ν′ < δ⟩ where, for all ν ≤ ν′ < δ, δ_{ν} = ot(N_{ν}) and i_{ν,ν′} = π_{N_{ν′}} ∘ π_{N_{ν}}^{−1} (where π_{N_{ν}} denotes the collapsing function of N_{ν}). Let us fix such a notion of coding. Let X⃗ = (X_{ν})_{ν<ω_{1}} be a ♢-sequence. We recursively define from X⃗ a ladder system C⃗ = ⟨C_{δ} : δ ∈ Lim(ω_{1})⟩ in the following way.

Let δ ∈ Lim(ω_{1}) and suppose X_{δ} codes a directed system S = ⟨δ_{ν}, i_{ν,ν′} : ν ≤ ν′ < δ⟩ with well-founded direct limit, where the δ_{ν}’s are countable limit ordinals, and each i_{ν,ν′} is an order-preserving map from δ_{ν} to δ_{ν′}, i_{ν,ν′} ≠ id. Suppose that for all ν < δ, crit(i_{ν,ν+1}) is a limit ordinal and ν ≤ crit(i_{ν,ν+1}) < crit(i_{ν+1,ν+2}), where crit(i_{ν,ν′}) is the least ordinal moved by i_{ν,ν′}, and that sup(range(i_{ν,ν+1})) < δ_{ν+1}. Let η_{δ} be the direct limit of S and let i_{ν,δ} : δ_{ν} → η_{δ} be the corresponding limit map for each ν < δ. We identity η_{δ} with an ordinal. Suppose that δ > η_{δ′} for all limit ordinals δ′ < δ. Then we pick C_{δ} and C_{η_{δ}} in such a way that for unboundedly many ν below δ, |C_{δ} ∩ crit(i_{ν,δ})| is bigger than |C_{η_{δ}} ∩ sup(range(i_{ν,δ}))|. Now, using the fact that X⃗ is a ♢-sequence it is not difficult to check that C⃗ is a ladder system as required. □

It is also easy to see that ♢ — and in fact ♣ — implies the failure of ψ_{2}. To see this, let ⟨C_{δ} : δ ∈ Lim(ω_{1})⟩ be a ♣-sequence. Suppose that f : ω_{1} → 2^{<ω} is such that for all limit δ < ω_{1} there is a co-finite set X ⊆ C_{δ} such that ⋃ f[X] is a member of 2^{ω}. There is then some n < ω such that S = {ν ∈ ω_{1} : |f(ν)| = n} is unbounded in ω_{1}. But if δ is such that C_{δ} ⊆ S, then ⋃ f[C_{δ}] is finite, which is a contradiction.

3. Iteration theorems

In this section we will review and adapt Eisworth and Roitman’s general framework for verifying that an iteration of forcings does not add new reals. We will need two preservation results, one which is essentially established in [8] and one which is an adaptation of the result in [8] to iterations of totally proper α-semiproper forcings. In the course of the section, we will also establish that ψ_{2} is relatively consistent with CH.

Before we begin, we will review some of the definitions which we will need in this section. A forcing ℚ is a partial order with a greatest element 1_{ℚ}. A cardinal θ is sufficiently large for a forcing ℚ if P(P(ℚ)) is an element of H(θ). We will say that M is a suitable model for ℚ if ℚ is in M and M is a countable elementary submodel of H(θ) for some θ which is sufficiently large for ℚ. If M is a suitable model for ℚ and q is in ℚ, then we will say that q is (M, ℚ)-generic if whenever r ≤ q and D ∈ M is a dense subset of ℚ, r is compatible with an element of D ∩ M. If, moreover, {p ∈ ℚ ∩ M : q ≤ p} is an (M, ℚ)-generic filter, then we say that q is totally (M, ℚ)-generic. ℚ is (totally) proper if whenever M is a suitable model for ℚ and q is in ℚ ∩ M, q has a (totally) (M, ℚ)-generic extension. It is easily verified that a forcing is totally proper if and only if it is proper and does not add any new reals.

Remark 3.1. It is important to note that if ℚ is totally proper and M is suitable for ℚ, it need not be the case that every (M, ℚ)-generic condition is totally (M, ℚ)-generic. It is true that every (M, ℚ)-generic condition can be extended to a totally (M, ℚ)-generic condition. This distinction is very important in the discussion of when an iteration of forcings adds new reals.

A suitable tower (in H(θ)) for ℚ is a set N = {N_{ξ} : ξ < η} (for some ordinal η) such that for some θ which is sufficiently large for ℚ:

• each N_{ξ} is a countable elementary submodel of H(θ) having ℚ as a member;

• if ν < η is a limit ordinal, then N_{ν} = ⋃_{ξ<ν} N_{ξ};

• if ν < η is a successor ordinal, then {N_{ξ} : ξ < ν} is in N_{ν}. Since a tower is naturally ordered by ∈, we notationally identify it with the corresponding sequence. A condition q is (N , ℚ)-generic if it is (N, ℚ)-generic for each N in N. A partial order ℚ is η-proper if whenever N = ⟨N_{ξ} : ξ < η⟩ is a suitable tower for ℚ and q is in N_{0}, then q has a (N , ℚ)-generic extension. If a forcing is η-proper for every η < ω_{1}, we will say that it is (<ω_{1})-proper.

Now we will return to our discussion of iterated totally proper forcing.

Definition 3.2. Suppose that η is a countable ordinal and ℙ ∗ ℚ̇ is a two-step forcing iteration such that ℙ is η-proper. The iteration ℙ ∗ ℚ̇ is η-complete if whenever

(1) ⟨N_{ξ} : ξ < 1 + η⟩ is a suitable tower of models for ℙ ∗ ℚ̇,

(2) G ⊆ ℙ ∩ N_{0} is (N_{0}, ℙ)-generic, and

(3) (p, q̇) is in ℙ ∗ ℚ̇ ∩ N_{0} with p in G, then there is a G^{∗} ⊆ ℙ ∗ ℚ̇ ∩ N_{0} extending G with (p, q̇) ∈ G^{∗} such that whenever r is a lower bound for G which is (⟨N_{ξ} : ξ < 1+η⟩, ℙ)-generic, then r forces G^{∗}/G has a lower bound in ℚ̇.

Notice that if η < ζ and ℙ∗ℚ̇ is η-complete, then ℙ∗ℚ̇ is ζ-complete. By routine adaptations to the proof of Theorem 4 of [8], we obtain the following iteration theorem.

Theorem 3.3. Let η, γ be ordinals, with η < ω_{1}, and let

⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩

be a countable support iteration with countable support limit ℙ_{γ}. Suppose that for all α < γ,

• ⊩_{α} ℚ̇_{α} is (<ω_{1})–proper,

• the iteration ℙ_{α} ∗ ℚ̇_{α} is η–complete.

Then ℙ_{γ} is totally proper.

We will now argue that this theorem is sufficient to prove that the conjunction of ψ_{2} and CH can be forced over any model of ZFC. First we will recall a general fact which we will use repeatedly below.

Lemma 3.4. Suppose that X ⊆ H(ℵ_{1}) and ℚ ⊆ X^{<ω_{1}} is a partial order, ordered by extension, with the following properties:

• ℚ is closed under initial segments;

• for every α < ω_{1}, {q ∈ ℚ : |q| ≥ α} is dense;

• if q is in ℚ with |q| = α, p : α → X and

{ξ < α : q(ξ) ≠ p(ξ)}

is finite, then p is in ℚ. Then if

• M is a suitable model for ℚ,

• q is in ℚ ∩ M;

• C ⊆ (M ∩ ω_{1}) \ |q| is cofinal in M ∩ ω_{1} with ordertype ω;

• f is a function from C into X ∩ M; then there is a q′ : M ∩ ω_{1} → X extending q such that q′(ξ) = f(ξ) for all ξ ∈ C and {q′ ↾ ξ : ξ ∈ M ∩ ω_{1}} is an M-generic filter for ℚ.

Proof. It is sufficient to prove that if ℚ, M, q, C and f are as in the statement of the lemma and D ⊆ ℚ is dense and in M, then there is a q′ ≤ q in M ∩ D such that q′(ξ) = f(ξ) for all ξ in |q′| ∩ C. By the elementarity of M, there is a countable elementary N ≺ H(ℵ_{1}) in M such that q is in N, D ∩N is dense in ℚ∩N, and {p ∈ ℚ∩N : ξ ≤ |p|} is dense in N for every ξ ∈ N ∩ ω_{1}. Let ν = N ∩ ω_{1} and let C′ = C ∩ ν. Since ν is a limit ordinal and C′ is finite, there is a q_{0} ≤ q in N such that (f ↾ C′) ⊆ q_{0}. By the density of D ∩N in ℚ∩N, there is a q′ ≤ q_{0} in D ∩ N. Since |q′| ∩ C = |q_{0}| ∩ C, we are done. □

By performing a preliminary forcing if necessary, we may assume that our ground model satisfies 2^{ℵ_{0}} = ℵ_{1} and 2^{ℵ_{1}} = ℵ_{2}. Suppose that C⃗, α, β, γ, and N represent an instance of ψ_{2}, i.e., C⃗ is a ladder system on ω_{1}, ω_{1} < α < β < γ < ω_{2} and N is an ω_{1}-club contained in [γ]^{ℵ_{0}}. Define ℚ = ℚ_{C⃗,α,β,N} to be the collection of all q such that the domain of q is η for some countable limit ordinal η, q maps into 2^{<ω}, and q satisfies the conclusion of ψ_{2} for δ ≤ η. Note that ℚ has cardinality 2^{ℵ_{0}} = ℵ_{1}.

Lemma 3.5. ℚ is totally α-proper for each α < ω_{1}.

Proof. First observe that ℚ satisfies the hypothesis of Lemma 3.4. (We leave it to the reader to verify that if ξ < ω_{1}, then {q ∈ ℚ : |q| ≥ ξ} is dense in ℚ.) We will prove by induction on α that:

If M = ⟨M_{ξ} : ξ ≤ α⟩ is a suitable tower for ℚ, q_{0} ∈

M_{0} ∩ ℚ and f_{0} is a finite partial function from ω_{1}^{M_{α}} \ |q_{0}|
to 2^{<ω}, then there is a totally (M, ℚ)-generic q̄ ≤ q_{0}
with f_{0} ⊂ q̄.

If α = 0, this is vacuously true. If α = β + 1, then by our inductive assumption there is a q′ ≤ q_{0} such that q′ is totally (M_{ξ}, ℚ)-generic for all ξ < β and such that q̄(ξ) = f_{0}(ξ) whenever ξ ∈ dom(f_{0}) ∩ dom(q̄). By elementarity, such a q′ can be moreover found in M_{β}. Define δ = M_{β} ∩ ω_{1} and let f : C_{δ} → 2^{<ω} be such that for some {X_{i}}_{i<ω} ⊆ N of height greater than δ, f codes the ψ_{1}-structure corresponding to {X_{i}}_{i<ω}. By modifying f if necessary, we may assume that q′ ∪ f ∪ f_{0} is a function. By Lemma 3.4, there is a q̄ : δ → 2^{<ω} such that q̄ extends q′, {q̄ ↾ ξ : ξ ∈ N_{β} ∩ ω_{1}} is an (⟨N_{ξ} : ξ ≤ β⟩, ℚ)-generic filter, and q̄∪ f ∪ f_{0} is a function. Notice that this implies that q̄ is in ℚ and is therefore as desired.

If α is a limit ordinal, let α_{n} (n < ω) be an increasing sequence of ordinals converging to α with α_{0} = 0. Define δ = M_{α} ∩ω_{1} and as above let f : C_{δ} → 2^{<ω} be such that for some {X_{i}}_{i<ω} ⊆ N of height greater than δ, f codes the ψ_{1}-structure corresponding to {X_{i}}_{i<ω}. Let q_{0} be a given element of M_{0} ∩ ℚ and let f_{0} be a given finite partial function from ω_{1} \ |q_{0}|. By modifying f if necessary, we may assume that f ∪ f_{0} is a function. Construct a ≤-descending sequence q_{n} (n < ω) such that:

• q_{n+1} is totally (⟨M_{ξ} : ξ ≤ α_{n}⟩, ℚ)-generic;

• q_{n+1} is in M_{α_{n}+1} and has domain M_{α_{n}} ∩ ω_{1};
• q_{n+1} extends f_{0} ∪ f ∩ M_{α_{n}}.

Given q_{n}, q_{n+1} can be found in H(θ) by applying our induction hypothesis to q_{n} and to (f ∪ f_{0}) ∩ M_{α_{n}}. Such a q_{n+1} moreover exists in M_{α_{n}+1} by elementarity, completing the inductive construction. It now follows that q̄ = ⋃_{n<ω} q_{n} is a totally (⟨M_{ξ} : ξ ≤ α⟩, ℚ)-generic extension of q_{0} as desired. □

Under CH, length-ω_{2} countable support iterations of proper forcings which are forced to have cardinality at most ℵ_{1} are ℵ_{2}-c.c. (see for instance Theorem 2.10 of [1]). Standard book-keeping arguments then reduce our task to verifying that an iteration of forcings of the form ℚ_{C⃗,α,β,N} is ω-complete.

Lemma 3.6. Suppose that

• ℙ is a totally proper forcing;

• for each δ ∈ Lim(ω_{1}), Ċ_{δ} is a ℙ-name for a cofinal subset of δ of ordertype ω;

• Ċ is a ℙ-name for the ladder system on ω_{1} induced by the names Ċ_{δ} (δ ∈ Lim(ω_{1}));

• α̇, β̇, and γ̇ are ℙ-names for an increasing sequence of ordinals between ω_{1} and ω_{2};

• Ṅ is a ℙ-name for an ω_{1}-club contained in [γ̇]^{ℵ_{0}};

• ℚ̇ is a ℙ-name for the partial order ℚ_{˙C,α̇,β̇,Ṅ}.

Then ℙ ∗ ℚ̇ is an ω-complete iteration.

Proof. Let ⟨N_{k} : k < ω⟩ be a tower of models with ℙ ∗ ℚ̇ in N_{0}, G ⊆ ℙ ∩ N_{0} be an (N_{0}, ℙ)-generic filter, and (p, q̇) be in ℙ ∗ ℚ̇ ∩ N_{0} such that p is in G. Notice that some condition in G decides q̇, α̇, β̇, and γ̇ to be some q, α, β and γ, respectively. Let r be a real which codes the transitive collapse of

( ⋃_{k<ω} N_{k} ∩ γ, ∈, α, β; N_{k} ∩ γ : k < ω).

The key point is that if p̄ is (⟨N_{k} : k < ω⟩, ℙ)-generic, then p̄ forces N_{k} ∩ γ is in Ṅ for all k < ω.

Set δ = N_{0} ∩ ω_{1}. Notice that there is a ladder Ĉ_{δ} on δ such that whenever C′ is a ladder on δ which is in N_{1}, then C′ \ Ĉ_{δ} is finite and Ĉ_{δ} consists only of ordinals not in the domain of q̇ as decided by G. In particular if p̄ is (N_{1}, ℙ)-generic, then p̄ forces that Ċ_{δ} is contained in Ĉ_{δ} except for a finite set. Let f_{δ} be a bijection between Ĉ_{δ} and {r ↾ n : n < ω}. Lemma 3.4 now allows us to build a G^{∗} ⊆ ℙ ∗ ℚ̇ such that (p, q̇) is in G^{∗} and if

g = ∪{s ∈ H(ω)^{<δ} : ∃p ∈ G(p, š) ∈ G^{∗}}

then f_{δ} is a restriction of g. It follows that whenever r is a lower bound for G which is (⟨N_{k} : k < ω⟩, ℙ)-generic, then r forces G^{∗}/G has a lower bound. □

Putting together Theorem 3.3 with Lemmas 3.5 and 3.6, we have (in ZFC) that there exists a partial order forcing ψ_{2} + CH. Corresponding results for ψ_{1} are proved in Section 4.

Unlike ψ_{2}, it is generally not possible to force an instance of ψ_{1} with an ω-proper forcing. Fortunately, assuming the existence of three measurable cardinals, there is a forcing to force an instance of ψ_{1} which is (< ω_{1})-semiproper. In the remainder of this section, we formulate and prove a version of Theorem 4 of [8] which applies to iterations of totally proper (< ω_{1})-semiproper iterands. This seems to provide the first example of a forcing which is proper and (< ω_{1})-semiproper, but not (<ω_{1})-proper.

In order to state this definition we will borrow the following pieces of notation from [8] (originating in [20]): Given a set N and a forcing notion ℙ ∈ N, Gen(N, ℙ) denotes the set of all (N, ℙ)-generic filters G ⊆ N ∩ ℙ. Furthermore, if p ∈ N ∩ ℙ, Gen(N, ℙ, p) = {G ∈ Gen(N, ℙ) : p ∈ G} and Gen^{+}(N, ℙ, p) denotes the set of all G ∈ Gen(N, ℙ, p) such that G has a lower bound in ℙ.

Given a partial order ℙ, a regular cardinal θ which is sufficiently large for ℙ, and a countable N ≺ H(θ) with ℙ ∈ N, we say that a condition p ∈ ℙ is (ℙ, N)-semi-generic if p ⊩ τ ∈ (ω̌_{1} ∩ Ň) for all ℙ-names τ in N for countable ordinals. Given a countable ordinal η, ℙ is said to be η-semiproper if for every suitable tower ⟨N_{ξ} : ξ < η⟩ with ℙ ∈ N_{0}, and every p ∈ ℙ ∩ N_{0}, there is a condition q ≤ p in ℙ which is (⟨N_{ξ} : ξ < η⟩, ℙ)-semi-generic, i.e., which is (N_{ξ}, ℙ)-semi-generic for all ξ < η.

Given a countable elementary substructure N of H(θ) with ℙ ∈ N, and given G ∈ Gen(N, ℙ), we let N[G] denote the set of Ginterpretations of ℙ-names which are in N (see section 3 from [8] for details).

In the following definition, we have replaced the condition that r be (⟨N_{ξ} : ξ < 1 + η⟩, ℙ)-generic from Definition 3.2 with the condition that it be merely (⟨N_{ξ} : ξ < 1 + η⟩, ℙ)-semi-generic. We call the corresponding notion η-semi-completeness, and note that it is a stronger condition than η-completeness.

Definition 3.7. Suppose that η is a countable ordinal and ℙ ∗ ℚ̇ is a two-step forcing iteration such that ℙ is η-semiproper. The iteration ℙ ∗ ℚ̇ is η-semi-complete if whenever (1) ⟨N_{ξ} : ξ < 1 + η⟩ is a suitable tower of models for ℙ ∗ ℚ̇,

(2) G ⊆ ℙ ∩ N_{0} is (N_{0}, ℙ)-generic, and

(3) (p, q̇) is in ℙ ∗ ℚ̇ ∩ N_{0} with p in G,

then there is a G^{∗} ⊆ ℙ ∗ ℚ̇ ∩ N_{0} extending G with (p, q̇) ∈ G^{∗} such that whenever r is a lower bound for G which is (⟨N_{ξ} : ξ < 1 + η⟩, ℙ)-semi-generic, then r forces G^{∗}/G has a lower bound in ℚ̇.

Even though we will be working exclusively with iterations of proper forcings in this paper, we will use the terminology of revised countable support iterations in order to prove the analogue of Theorem 3.3 for η-semi-complete iterations. By revised countable support (RCS) we mean either the original presentation of RCS due to Shelah [20], or the later reformulation due to Miyamoto [15]. Theorem 3.8 and Fact 3.9 below are proved in [15] but are already implicit in [20]. In [14] it is claimed, erroneously, that these facts apply to the presentation of RCS due to Donder and Fuchs [6]. Under the Donder-Fuchs presentation of RCS, an RCS iteration of proper forcings it identical to the corresponding countable support iteration, for which Theorem 3.8 fails. For the Shelah version and the Miyamoto versions, an RCS limit of proper forcings and the corresponding countable support limit are merely isomorphic on a dense set. It follows, in the end, that Theorem 3.10 is true when one uses countable support in place of revised countable support, though again our proof of this fact requires RCS. A similar situation holds in [14].

To facilitate the statements below, we let “⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩ has RCS limit ℙ_{γ}” include the case that γ = β + 1 and ℙ_{γ} = ℙ_{β} ∗ ℚ̇_{β} (and similarly for countable support).

Theorem 3.8. ([15], Corollary 4.12) Suppose that γ is an ordinal and that ⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩ is an RCS iteration with RCS limit ℙ_{γ}. Fix β < γ and p ∈ ℙ_{β}. Suppose that τ is a ℙ_{β}-name for a condition in ℙ_{γ} for which p forces that τ ↾ β ∈ G_{β}. Then there is a condition p′ in ℙ_{γ} such that p′ ↾ β = p and p forces that p′ ↾ [β, γ) = τ ↾ [β, γ).

The following fact is extracted from pages 7-10 of [15].

Fact 3.9. Suppose that γ is a limit ordinal and that ⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩ is an RCS iteration with RCS limit ℙ_{γ}. Then for each p ∈ ℙ_{γ} there exists a sequence of ℙ_{γ}-names τ_{i} (i ∈ ω) for elements of γ +1 such that

• for any condition q, for any i ∈ ω and any α ≤ γ, if q ⊩ τ_{i} = α̌,

then (q ↾ α)^{⌢}1_{ℙ_{γ}/ℙ_{α}} forces τ_{i} = α̌;
• for all i ∈ ω, p ⊩ τ_{i} < γ̌;

• the empty condition in ℙ_{γ} forces that for every

β ≥ sup{τ_{i} : i ∈ ω},

p(β) = 1Q̇_{β}.

The following is our extension of Theorem 3.3 to η-semi-complete iterations. We will introduce two more useful facts before we start the proof.

Theorem 3.10. Let η, γ be ordinals, with η < ω_{1}, and let

⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩

be an RCS iteration with RCS limit ℙ_{γ}. Suppose that for all α < γ, • ⊩_{α} ℚ̇_{α} is (<ω_{1})–semiproper,

• the iteration ℙ_{α} ∗ ℚ̇_{α} is η–semi-complete,
• ⊩_{α+1} |ℙ_{α}| ≤ ℵ_{1}.

Then ℙ_{γ} is totally proper.

A proof of the following fact appears in [14].

Fact 3.11. Let η be a countable ordinal, let γ be an ordinal, and let

⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩

be a revised countable support iteration with RCS limit ℙ_{γ}. Suppose that for all α < γ,

• ⊩_{α} ℚ̇_{α} is η–semiproper, and

• ⊩_{α+1} |ℙ_{α}| ≤ ℵ_{1}. Then ℙ_{γ} is η–semiproper.

The proof of Theorem 3.10 uses the following lemma, a simplified (and ostensibly weaker) version of Lemma 4.10 in [14] which is used in the course of proving Fact 3.11 above.

Lemma 3.12. Let γ be an ordinal, and let η be a countable ordinal. Suppose that ℙ_{γ} is the RCS limit of an RCS iteration ⟨ℙ_{α}, ℚ̇_{α} : α < γ⟩ such that for each α < γ,

• 1_{ℙ_{α}} forces ℚ̇_{α} to be η-semiproper, and

• 1_{ℙ_{α+1}} forces ℙ_{α} to have cardinality ℵ_{1}.

Let θ be sufficiently large for ℙ_{γ}. Fix α ≤ β ≤ γ, and fix a suitable tower ⟨N_{ξ} : ξ < η⟩ for ℙ_{γ} with α, β ∈ N_{0}. Let s ∈ ℙ_{γ} and r ∈ ℙ_{α} be such that

• r is (N_{ξ}, ℙ_{α})-semi-generic for all ξ < η,

• s ↾ α ≥ r,
• r forces that for some t ∈ ℙ_{γ} ∩ N_{0}, s ↾ [α, γ) = t ↾ [α, γ).

Then there exists r^{†} ∈ ℙ_{β} such that • r^{†} is (N_{ξ}, ℙ_{β})-semi-generic for all ξ < η,

• r^{†} ≤ s ↾ β,

• r^{†} ↾ α = r.

To prove Theorem 3.10, let θ be a regular cardinal which is sufficiently large for ℙ_{γ}. Let N ≺ H(θ) be countable with ℙ and η in N, and let p ∈ ℙ_{γ} ∩ N be an arbitrary condition. We must produce a totally (N, ℙ_{γ})-generic condition q ≤ p. For each α ∈ N ∩ (γ + 1), let α^{∗} denote the ordertype of N ∩α. Fix a suitable tower \overline{N} = ⟨N_{ξ} : ξ ≤ ηγ^{∗}⟩ with N_{0} = N. The following claim is a variation of Claim 6.2 of [8]. In order to facilitate the statement of the claim, we let N_{ηγ^{∗}+1} stand for H(θ).

Claim 3.13. Given α < β in N_{0} ∩ (γ + 1), p ∈ ℙ_{β} and

G ∈ Gen^{+}(N_{0}, ℙ_{α}, p ↾ α) ∩ N_{ηα^{∗}+1},

there is a G^{†} ∈ Gen(N_{0}, ℙ_{β}, p) ∩ N_{ηβ^{∗}+1} such that whenever r ∈ ℙ_{α} is a lower bound for G that is (N_{ξ}, ℙ_{α})-semi-generic for all ξ ∈ (ηα^{∗}, ηγ^{∗}], there is an r^{†} ∈ ℙ_{β} such that

(1) r^{†} is a lower bound for G^{†},

(2) r^{†} ↾ α = r,

(3) r^{†} is (N_{ξ}, ℙ_{β})-semi-generic for every ξ ∈ (ηβ^{∗}, ηγ^{∗}].

Theorem 3.10 follows from taking α = 0 and β = γ in Claim 3.13. Inducting primarily on γ, we assume that the claim holds for all γ′ < γ in place of γ, for this fixed sequence of N_{ξ}’s. This will be useful in the limit case below.

Remark 3.14. Item (1) above implies that {q ↾ α | q ∈ G^{†}} = G, since otherwise, these two generic filters could not have the same lower bound r in common.

Since ℙ_{0} is the trivial forcing, the case α = 0, β = 1 follows from the assumption that ℚ̇_{0} is totally proper and (< ω_{1})-semiproper.

Now consider the case where β = β_{0} + 1. We are given a G ∈ Gen^{+}(N_{0}, ℙ, p ↾ α) ∩ N_{ηα^{∗}+1}, and, applying the induction hypothesis we may fix a G_{0}^{†} ∈ Gen(N_{0}, ℙ_{β_{0}}, p ↾ β_{0}) ∩ N_{ηβ_{0}^{∗}+1} satisfying the claim with β_{0} in the role of β. Since ℙ_{α} is (<ω_{1})-semiproper, the conclusion of the claim implies that G_{0}^{†} ∈ Gen^{+}(N_{0}, ℙ_{β_{0}}, p ↾ β_{0}). We apply the definition of “ℚ̇_{β_{0}} is η-semi-complete for ℙ_{β_{0}}” in N_{ηβ^{∗}+1} with

{N_{0}} ∪ {N_{ξ} : ηβ_{0}^{∗} + 1 ≤ ξ ≤ ηβ^{∗}},

G_{0}^{†} and p(β_{0}) in place of ⟨N_{ξ} : ξ < 1 + η⟩, \overline{G} and q̇ there. This gives us a G^{†} ∈ Gen(N_{0}, ℙ_{β}, p) ∩ N_{ηβ^{∗}+1} extending G_{0}^{†} such that whenever r_{0}^{†} is a lower bound for G_{0}^{†} which is ({N_{0}} ∪ {N_{ξ} : ηβ_{0}^{∗} + 1 ≤ ξ ≤ ηβ^{∗}}, ℙ_{β_{0}})- semi-generic, then r_{0}^{†} forces that G^{†}/G_{0}^{†} has a lower bound in ℚ_{β_{0}}.

Now, whenever r ∈ ℙ_{α} is a lower bound for G that is (N_{ξ}, ℙ_{α})-semi-generic for all ξ ∈ (ηα^{∗}, ηγ^{∗}], there is by the choice of G_{0}^{†} a condition r_{0}^{†} ∈ ℙ_{β_{0}} such that

• r_{0}^{†} is a lower bound for G_{0}^{†},

• r_{0}^{†} ↾ α = r,
• r_{0}^{†} is (N_{ξ}, ℙ_{α})-semi-generic for every ξ ∈ (ηβ_{0}^{∗}, ηγ^{∗}].

By Theorem 3.8, there is a condition s ∈ ℙ_{β} ∩ N_{ηβ^{∗}+1} such that s ↾ β_{0} is 1_{ℙ_{β_{0}}} and 1_{ℙ_{β_{0}}} forces that s(β_{0}) is a lower bound for G^{†}/G_{0}^{†} if such a lower bound exists. By Lemma 3.12, then, there is an r^{†} as desired, with r^{†} ↾ β_{0} = r_{0}^{†} and s ≥ r^{†}. This takes care of the case where β is a successor ordinal.

Finally, suppose that β is a limit ordinal. Fix a strictly increasing sequence ⟨α_{n} : n ∈ ω⟩ ∈ N_{ηβ^{∗}+1} which is cofinal in N_{0} ∩β, with α_{0} = α, and let ⟨D_{n} : n ∈ ω⟩ ∈ N_{ηβ^{∗}+1} be a listing of the dense open subsets of ℙ_{β} in N_{0}.

Subclaim 3.1. There exist sequences ⟨p_{n} : n ∈ ω⟩, ⟨G_{n} : n ∈ ω⟩ in N_{ηβ^{∗}+1} such that p_{0} = p, G_{0} = G and, for all n ∈ ω,

• p_{n+1} ∈ N_{0} ∩ D_{n};

• p_{n+1} ≤ p_{n};

• p_{n+1} ↾ α_{n} ∈ G_{n};

• G_{n} ∈ Gen(N_{0}, ℙ_{α_{n}}, p_{n} ↾ α_{n}) ∩ N_{ηα_{n}^{∗}+1};

• whenever r ∈ ℙ_{α_{n}} is a lower bound for G_{n} that is (N_{ξ}, ℙ_{α_{n}})- semi-generic for all ξ ∈ (ηα_{n}^{∗}, ηγ^{∗}], there is an r^{+} ∈ ℙ_{α_{n+1}} such that

∘ r^{+} is a lower bound for G_{n+1};

∘ r^{+} ↾ α_{n} = r;

∘ r^{+} is (N_{ξ}, ℙ_{α_{n+1}})-semi-generic whenever

ξ ∈ (ηα_{n+1}^{∗}, ηγ^{∗}].

Given n ∈ ω, r ∈ ℙ_{α_{n}} and δ ∈ (α_{n}, γ] ∩ N_{0}, let A(r, n, δ^{∗}) denote the statement that r is a lower bound for G_{n} and r is (N_{ξ}, ℙ_{α_{n}})-semi-generic for all ξ ∈ (ηα_{n}^{∗}, ηδ^{∗}] (this is just for notational convenience, and we will use it only when the G_{n} in question has already been established). Then the last item of the subclaim says that for all r ∈ ℙ_{α_{n}} satisfying A(r, n, γ^{∗}), there exists an r^{+} ∈ ℙ_{α_{n+1}} such that

• r^{+} ↾ α_{n} = r;

• r^{+} satisfies A(r^{+}, n + 1, γ^{∗}) (again, for G_{n+1} as chosen).

To verify the subclaim, suppose that p_{n} and G_{n} are given. We will verify that p_{n+1} and G_{n+1} exist as described in the subclaim. First note that E = {t ↾ α_{n} | t ∈ D_{n}, t ≤ p_{n}} is dense in ℙ_{α_{n}} below p_{n} ↾ α_{n}, and that E ∈ N_{0}. Since p_{n} ↾ α_{n} ∈ G_{n}, there exists a t ∈ E ∩ G_{n}. Applying the definition of E inside N_{0}, we get a p_{n+1} ∈ N_{0} ∩ D_{n} with p_{n+1} ≤ p_{n} and p_{n+1} ↾ α_{n} ∈ G_{n}, as desired.

Applying the induction hypothesis inside of N_{ηβ^{∗}+1}, with α_{n}, α_{n+1} and β in place of α, β and γ, we can find a filter

G_{n+1} ∈ Gen(N_{0}, ℙ_{α_{n+1}}, p_{n+1} ↾ α_{n+1})

such that for any condition r ∈ ℙ_{α_{n}} satisfying A(r, n, β^{∗}) there is an r′ ∈ ℙ_{α_{n+1}} satisfying A(r′, n + 1, β^{∗}) such that r′ ↾ α_{n} = r. We need to see that for this G_{n+1}, for any condition r ∈ ℙ_{α_{n}} satisfying A(r, n, γ^{∗}) there is an r^{+} ∈ ℙ_{α_{n+1}} satisfying A(r′, n + 1, γ^{∗}) such that r^{+} ↾ α_{n} = r.

Fix such an r. Since r satisfies A(r, n, γ^{∗}), it satisfies A(r, n, β^{∗}). Fix r′ ∈ ℙ_{α_{n+1}} such that r′ ↾ α_{n} = r and r′ satisfies A(r, n + 1, β^{∗}). In order to apply Lemma 3.12, we want to see that there is an r′′ ∈ ℙ_{α_{n+1}} satisfying A(r′′, n + 1, β^{∗}) such that r′′ ↾ α_{n} = r and such that

r′′ ⊩_{ℙ_{αn}} r′′/G_{α_{n}} ∈ N_{ηβ^{∗}+1}[G_{α_{n}}],

that is, that r forces (in ℙ_{α_{n}}) that there is a ℙ_{α_{n}}-name t in N_{ηβ^{∗}+1} such that r′′/G_{α_{n}} = t_{G_{αn}}.

If we force with ℙ_{α_{n}} below r ↾ α_{n}, in V [G_{α_{n}}], r′/G_{α_{n}} ∈ ℙ_{α_{n+1}}/G_{α_{n}} satisfies condition (∗∗), i.e.,

• is a lower bound for {s/G_{α_{n}} : s ∈ G_{n+1}};

• is semi-generic for (N_{ξ}[G_{α_{n}}], ℙ_{α_{n+1}}/G_{α_{n}}), for all ξ ∈ (ηα_{n+1}^{∗}, ηβ^{∗}].

So there exists a condition satisfying (∗∗) in N_{ηβ^{∗}+1}[G_{α_{n}}].

Let τ be a ℙ_{α_{n}} ↾ r-name for an element of ℙ_{α_{n+1}}/G_{n} in N_{ηβ^{∗}+1}[G_{α_{n}}] satisfying (∗∗). Viewing ℙ_{α_{n+1}} as ℙ_{α_{n}} ∗ Q̇_{α_{n},α_{n+1}} (so that Q̇_{α_{n},α_{n+1}} is a ℙ_{α_{n}}-name for the rest of the iteration ℙ_{α_{n+1}}), let r′′ = (r, τ).

Now apply Lemma 3.12. We have that

• r is (N_{ξ}, ℙ_{α_{n}})-semi-generic for all ξ ∈ (ηβ^{∗}, ηγ^{∗}];

• r′′ ↾ α_{n} = r;

• r forces that there is a t ∈ ℙ_{α_{n+1}} ∩ N_{ηβ^{∗}+1} such that r′′ ↾

[α_{n}, α_{n+1}) = t ↾ [α_{n}, α_{n+1}).

Then by the lemma, there exists an r^{+} which is (N_{ξ}, ℙ_{α_{n+1}})-semi-generic for all ξ ∈ (ηβ^{∗}, ηγ^{∗}], such that r^{+} ≤ r′′ and r^{+} ↾ α_{n} = r. This verifies the subclaim.

Let G^{†} = {t ∈ N_{0} ∩ ℙ_{β} | ∃n t ≥ p_{n}}. Then

G^{†} ∈ Gen(N_{0}, ℙ_{β}, p) ∩ N_{ηβ^{∗}+1}.

Subclaim 3.2. G^{†} has a lower bound.

To see this, let r be a lower bound for G that is (N_{ξ}, ℙ_{α})-semi-generic for all ξ ∈ (ηα^{∗}, ηγ^{∗}]. The properties of the sequence ⟨G_{n} : n ∈ ω⟩ allow us to build a sequence ⟨r_{n} : n ∈ ω⟩ satisfying:

• r_{0} = r;

• r_{n} is a lower bound for G_{n} in ℙ_{α^{∗}};
• r_{n} is (N_{ξ}, ℙ_{α_{n}})-semi-generic for all ξ ∈ (ηα_{n}^{∗}, ηγ^{∗}];
• r_{n+1} ↾ α_{n} = r_{n}.

Finally let r^{+} = ⋃_{n∈ω} r_{n} ∈ ℙ_{β}. Let us check that r^{+} is a lower bound for G^{†}. First note that by the argument presented in Remark 3.14, {q ↾ α_{n} | q ∈ G_{m}} = G_{n}, whenever n ≤ m. When m ≤ n, p_{m} ≥ p_{n}, so p_{m} ↾ ℙ_{α_{n}} ≥ p_{n} ↾ ℙ_{α_{n}}. Since for each n ∈ ω we have p_{n} ↾ α_{n} ∈ G_{n}, we get that for each such n, {p_{m} ↾ α_{n} : m ∈ ω} ⊆ G_{n}.

For each n ∈ ω, let τ_{i}^{n} be the names as in Fact 3.9 corresponding to p_{n}. Since the p_{m}’s collectively meet all dense subsets of ℙ_{β} in N_{0}, a value for each τ_{i}^{n} is decided by some p_{m}, and since p_{n} and p_{m} are compatible this value is decided to be some value in N_{0} ∩ β. Since for each m ∈ ω, r ↾ ℙ_{α_{m}} is a lower bound for G_{m}, we have that r ↾ α_{m} ≤ p_{n} ↾ α_{m} for all m ∈ ω, and thus that r ≤ p. It follows that r is a lower bound for G^{†}. This proves the subclaim, and thereby the limit case of Claim 3.13 and thereby Theorem 3.10.

4. The single step forcing for ψ_{1}

In this section we examine the single step forcings associated with ψ_{1}. Before proceeding, we will recall some terminology from [18]. Let X be an uncountable set and let θ be a regular cardinal with P([X]^{ℵ_{0}}) in H(θ). [X]^{ℵ_{0}} is topologized by declaring sets of the form

[a, M] = {N ∈ [X]^{ℵ_{0}} : a ⊆ N ⊆ M}

to be open whenever M is in [X]^{ℵ_{0}} and a is a finite subset of M. If M is a countable elementary submodel of H(θ) with X in M, then Σ ⊆ [X]^{ℵ_{0}} is M-stationary if M ∩ E ∩ Σ is non empty whenever E ⊆ [X]^{ℵ_{0}} is a club in M. If Σ is a function whose domain is a club of countable elementary submodels of H(θ), then we say that Σ is an open stationary set mapping if Σ(M) is open and M-stationary whenever M is in the domain of Σ. If N = ⟨N_{ξ} : ξ < ω_{1}⟩ is a continuous ⊆-chain where ⟨N_{ξ} : ξ ≤ ν⟩ is in N_{ν+1} for each ν, then we say that N is a reflecting sequence for Σ if whenever ν < ω_{1} is a limit ordinal, there is a ν_{0} < ν such that

N_{ξ} ∩ X ∈ Σ(N_{ν})

whenever ν_{0} < ξ < ν. If N = ⟨N_{ξ} : ξ ≤ δ⟩ is a sequence of countable successor length which has the above properties for all limit ν ≤ δ, then we will say that N is a partial reflecting sequence for Σ. In [18] it is shown that PFA implies all open stationary set mappings admit reflecting sequences and that the forcing ℙ_{Σ} of all countable partial reflecting sequences for an open stationary set mapping Σ is always totally proper.

Except for trivial cases ℙ_{Σ} is not ω-proper. Moreover it will be (< ω_{1})-semiproper only under rather special circumstances. The following lemma gives a useful sufficient condition for when we can build generic conditions in ℙ_{Σ} for a given suitable tower of models.

Lemma 4.1. Let Σ be an open stationary set mapping whose domain consists of elements of H(θ) and let λ be sufficiently large for ℙ_{Σ}. Suppose that M = ⟨M_{δ} : δ ≤ α⟩ is a tower of countable elementary submodels of H(λ) which is suitable for ℙ_{Σ} and such that ⟨M_{δ} ∩ H(θ) : δ ≤ α⟩ is a partial reflecting sequence for Σ. Then every condition in M_{0} can be extended to a totally (M, ℙ_{Σ})-generic condition.

Proof. This follows from the properness of ℙ_{Σ} when α = 0, and by the induction hypothesis, elementarity and the total properness of ℙ_{Σ} when α is a successor ordinal. When α is a limit ordinal, choose an increasing sequence ⟨β_{i} : i < ω⟩ converging to α, such that for all δ in the interval [β_{0}, α), M_{δ} ∩ X ∈ Σ(M_{α} ∩ H(θ)). Note that any condition in ℙ_{Σ} which is (M_{δ}, ℙ_{Σ})-generic for all δ < α will be (M_{α}, ℙ_{Σ})-generic. The difficulty in what follows is in ensuring that a tail of the generic sequence we build falls inside of Σ(M_{α} ∩ H(θ)). We will ensure that this happens for all members of the sequence containing M_{β_{0}}. We have that for each δ in the interval [β_{0}, α) there is a finite set a_{δ} ⊂ M_{δ} ∩ X such that [a_{δ}, M_{δ} ∩ X] ⊂ Σ(M_{α} ∩ H(θ)).

By elementarity and the induction hypothesis, we may assume first that s_{0} is a condition in M_{β_{0}+1} which is (M_{δ}, ℙ_{Σ})-generic for all δ ≤ β_{0}, and which extends any given condition s ∈ M_{0}. We may assume that the last member of s_{0} is M_{β_{0}} ∩ X, and we have then that a tail of s_{0} is contained in Σ(M_{α} ∩ H(θ)). Suppose now that i ∈ ω, that s_{i} is a condition in M_{β_{i}+1} which is (M_{δ}, ℙ_{Σ})-generic for all δ ≤ β_{i}, which extends s_{0}, whose last member is M_{β_{i}} ∩X, and such that every member of s_{i} containing M_{β_{0}} ∩ X is in Σ(M_{α} ∩ H(θ)). We show how to choose s_{i+1} satisfying these conditions for i + 1. If a_{β_{i+1}} ⊆ M_{β_{i}+1}, then we let s′_{i} be a condition in M_{β_{i}+1} extending s_{i} by one set which contains a_{β_{i+1}}, and, applying the induction hypothesis and elementarity we let s_{i+1} be a condition in M_{β_{i+1}+1} as desired, extending s′_{i}.

If a_{β_{i+1}} is not in M_{β_{i}+1}, we need to work harder to extend s_{i} while staying inside Σ(M_{α} ∩ H(θ)). In this case, let a(i, 0) = a_{β_{i+1}} and let γ(i, 0) be the largest δ in (β_{i}, β_{i+1}) such that a(i, 0) is not contained in M_{δ}. Let a(i, 1) be a finite subset of M_{γ(i,0)} ∩ X such that

[a(i, 1), M_{γ(i,0)} ∩ X] ⊆ Σ(M_{α} ∩ H(θ)).

Continue in this way, letting γ(i, j + 1) be the largest δ in [β_{i}, γ(i, j)) such that δ = β_{i} or a(i, j+1) is not contained in M_{δ}, and, if γ(i, j+1) > β_{i}, letting a(i, j + 2) be a finite subset of M_{γ(i,j+1)} ∩ X such that

[a(i, j + 2), M_{γ(i,j+1)} ∩ X] ⊆ Σ(M_{α} ∩ H(θ)).

As the γ(i, j)’s are decreasing, this sequence must stop at a point where a(i, j) ⊆ M_{β_{i}+1} and γ(i, j) = β_{i}. Let k be this j. As ⟨a(i, j) : j ≤ k⟩ is in M_{β_{i+1}+1}, we can argue in M_{β_{i+1}+1}, as follows.

Let t(i, k) be a condition in M_{β_{i}+1} extending s_{i} such that every member of t(i, k)\s_{i} contains a(i, k). Applying the induction hypothesis and elementarity, let s(i, k) be a condition in M_{γ(i,k−1)+1} extending t(i, k) which is (M_{δ}, ℙ_{Σ})-generic for every δ ≤ γ(i, k−1), and whose last member is M_{γ(i,k−1)} ∩X. For each positive j < k, let t(i, j) be a condition in M_{γ(i,j)+1} extending s(i, j+1) such that every member of t(i, j)\s(i, j+1) contains a(i, j), and let s(i, j) be a condition in M_{γ(i,j−1)+1} extending t(i, j) which is (M_{δ}, ℙ_{Σ})-generic for every δ ≤ γ(i, j −1), and whose last element is M_{γ(i,j−1)} ∩ X. Finally, let t(i, 0) be a condition in M_{γ(i,0)+1} extending s(i, 1) such that every member of t(i, 0) \ s(i, 1) contains a(i, 0), and let s_{i+1} be a condition in M_{β_{i+1}+1} extending t(i, 0) which is (M_{δ}, ℙ_{Σ})-generic for all δ ≤ β_{i+1} and whose last member is M_{β_{i+1}} ∩ X.

Then every member of s_{i+1}\s_{i} is in Σ(M_{α}∩H(θ)), as desired. Continuing in this way, the union of the s_{i}’s will be the desired condition. □

Now we return to our discussion of ψ_{1}. Let C⃗ be a ladder system and let κ_{i} (i < 3) be an increasing sequence of cardinals greater than ω_{2}. For a fixed A : ω_{1} → 2, we will define a totally proper forcing ℚ_{A,κ⃗,C⃗} which collapses κ_{2} to have cardinality ω_{1} and adds a function f : ω_{1} → 2^{ω} such that f(δ) is a code for A ↾ δ for each δ < ω_{1}, together with a witness N to the statement that (κ_{0}, κ_{1}, κ_{2}) codes f with respect to C⃗. In order to improve readability, we will suppress terms from subscripts which are either clear from the context or which do not influence the truth of a given statement.

The forcing ℚ_{A,κ⃗,C⃗} is the collection of all q such that:

(1) q is a function from some countable successor ordinal δ + 1 into [κ_{2}]^{ℵ_{0}};

(2) q is continuous and strictly ⊆-increasing;

(3) if ν ≤ δ is a limit ordinal, then there is a ν_{0} < ν and an r ∈ 2^{ω} such that r codes A ↾ ν and for all ν_{0} < ξ < ν,

Δ(s̄κ⃗(N_{ξ}, N_{ν}), r) ≥ n(N_{ξ}, N_{ν}).

This forcing can be viewed as a two step iteration in which we first add, by countable approximations, a function f : ω_{1} → 2^{ω} with the property that f(δ) codes A ↾ δ for each δ. Then we force to add a reflecting sequence (using the partial order described above) for the set mapping Σ_{f}, where Σ_{f}(N) is the set of all M in [κ_{2}]^{ℵ_{0}} such that M ⊆ N, M ∩ κ is bounded in N ∩ κ for κ in {ω_{1}, κ_{0}, κ_{1}, κ_{2}} and

Δ(s̄κ⃗(M, N), f(N ∩ ω_{1})) ≥ n(M, N).

It is not difficult to verify that this is an open set mapping and it will follow from arguments below that it is in fact an open stationary set mapping. Hence ℚ_{A,κ⃗,C⃗} can be regarded as a two step iteration of a σ-closed forcing followed by a forcing of the form ℙ_{Σ}.

Our goal in this section is to prove the following two lemmas. It then follows from Theorem 3.10 and standard book-keeping and chain condition arguments (see, e.g., [13, VIII], [20]) that if there is a inaccessible cardinal which is a limit of measurable cardinals, then there is a proper forcing extension with the same reals which satisfies ψ_{1}.

Lemma 4.2. If κ_{i} (i < 3) is an increasing sequence of measurable cardinals, then ℚ_{A,κ⃗,C⃗} is (<ω_{1})-semiproper.

Lemma 4.3. If ℙ is a totally proper forcing and κ⃗, C⃗, and Ȧ are ℙnames for objects as described above, then ℙ ∗ ℚ̇Ȧ is 1-semi-complete. In particular ℚ̇Ȧ is totally proper.

Remark 4.4. The reader may be puzzled as to why we constructed ℚ_{A,κ⃗,C⃗} by first forcing to produce the function f, since there are certainly functions f in V such that f(δ) codes A ↾ δ. The problem arises in proving Lemma 4.3 — the argument below does not go through unless we force the function f as we are building the corresponding reflecting sequence.

Remark 4.5. It is interesting to note that it is much easier to obtain the consistency of ψ_{1}[C⃗] with CH for some ladder system C⃗. Suppose that C⃗ is a ladder system on ω_{1}, ℙ is a totally proper forcing, and Ȧ is a ℙ-name for an element of 2^{ω_{1}}. If M is a suitable model for ℙ ∗ ℚ̇_{Ȧ,C⃗}, p is totally (M, ℙ)-generic, and q̇ is forced by p to be an element of M[G]∩ℚ̇_{Ȧ,C⃗}, then there is a ṙ such that (p, ṙ) is a totally (M, ℙ∗ℚ̇_{Ȧ,C⃗})- generic extension of (p, q̇) (those familiar with preservation theorems for not adding reals with proper forcing should notice that this almost never happens). This allows one to easily prove that if C⃗ is a fixed ladder system, then we can iterate forcings of the form ℚ̇_{Ȧ,C⃗} without adding reals (and without the complex iteration machinery which we are about to employ). This shows that if we allow a fixed ladder system as a parameter, we can force ψ_{1}[C⃗] ∧ CH over any model of ZFC (recall that 2^{ℵ_{0}} = 2^{ℵ_{1}} follows from the existence of a ladder system C⃗ such that both ψ_{1}[C⃗] and ψ_{2}[C⃗] hold). The difficulty arises when we want to quantify out the parameter C⃗ in order to obtain a Π_{2}-sentence. The final section of [19] contains an example of a pair ψ′_{1}[C⃗] and ψ′_{2} of Π_{2} sentences having these same properties except that ∀C⃗ψ′_{1}[C⃗] implies 2^{ℵ_{0}} = 2^{ℵ_{1}}.

In [3], the proof of Lemma 5 actually yields the following lemma (stated in our notation).

Lemma 4.6. Suppose that κ_{i} (i < 3) is an increasing sequence of regular cardinals above ω_{1} and C⃗^{i} (i < l) (for some l ∈ ω) is a sequence of ladder systems on ω_{1}. If M is a countable elementary submodel of H(θ) for θ sufficiently large and E ⊆ [κ_{2}]^{ℵ_{0}} is a club in M, then there is an n such that for any σ in 2^{<ω} there is an N in E ∩ M such that

o(x^{i} \ n, y^{i} \ n, z^{i} \ n) = σ
n^{i}(N, M) ≤ n

for all i < l, where x^{i}, y^{i}, y^{i}, and n^{i} are computed from M and N as in the computation of sC⃗^{i} ; κ⃗(N, M).

We will now prove Lemmas 4.2 and 4.3.

Proof of 4.3. Let ℙ be proper and force that:

• κ̇_{i} (i < 3) is an increasing sequence of regular cardinals above ω_{1},

• ⟨Ċ_{ξ} : ξ ∈ Lim(ω_{1})⟩ is a ladder system on ω_{1}, and

• Ȧ is a function from ω_{1} to 2. Let ℚ̇ denote ℚ̇_{Ȧ,κ⃗,C⃗}, N_{0} ∈ N_{1} be suitable models for ℙ ∗ ℚ̇, G be an (N_{0}, ℙ)-generic filter, and q be an element of ℚ^{N_{0}[G]}. Observe that there is a condition in G deciding κ̇_{i} to be some κ_{i} for each i < 3. Let κ̄_{i} denote the image of κ_{i} under the transitive collapse of N_{0}. Furthermore, if δ = N_{0} ∩ ω_{1}, then there is an A : δ → 2 such that for every α < δ, there is a condition in G forcing Ȧ ↾ α̌ = Ǎ ↾ α̌. Fix an r in 2^{ω} such that r codes A ↾ δ.

Notice that, by CH, if p is (N_{1}, ℙ)–semi-generic and a lower bound for G, then p forces that the value of Ċ_{δ} is some element of N_{1}, where δ = N_{0} ∩ω_{1} (although it need not decide which is this value). A similar statement is true concerning Ċκ̄_{i} for each i < 3. Let C_{δ}^{j} (j < ω) and Cj ; κ̄_{i} (j < ω) enumerate all cofinal subsets of δ and κ̄_{i}, respectively, of ordertype ω which are elements of N_{1}. Let D_{i} (i < ω) enumerate all dense open subsets of ℚ which are in N_{0}[G].

We will now build a sequence q_{i} (i < ω) of conditions in ℚ^{N_{0}[G]} such that:

• q_{i+1} ≤ q_{i} and q_{i+1} is in N_{0}[G] ∩ D_{i};
• if ξ is in dom(q_{i+1}) \ dom(q_{i}), then M = q_{i+1}(ξ) satisfies
Δ(s̄^{j}(M, N_{0} ∩ κ_{2}), r) ≥ n^{j}(M, N_{0} ∩ κ_{2}) for all j ≤ i,
where s^{j} and n^{j} are computed using C_{δ}^{j} and Cj ; κ̄_{i} (i < 3).

If this can be done, then any condition p̄ which is an (N_{1}, ℙ)–semigeneric lower bound for G will force that there is some i_{0} < ω such that Ċ_{δ} = Č_{δ}^{i_{0}}, and therefore that q_{i} (i < ω) will have a lower bound (namely the union of this sequence).

Suppose that we have constructed q_{i} and we wish to construct q_{i+1}. Following [18, 3.1] (or Lemma 4.1), it is sufficient to demonstrate that there is an countable elementary submodel M of H((2^{κ_{2}})^{+}) such that D_{i} and q_{i} are in M and

Δ(s̄^{j}(M ∩ κ_{2}, N_{0} ∩ κ_{2}), r) ≥ n^{j}(M, N_{0} ∩ κ_{2})

holds for all j ≤ i. Let E be the collection of all sets of the form M ∩ κ_{2} such that M is a countable elementary submodel of H((2^{κ_{2}})^{+}) such that q_{i} and D_{i} are in M. Let n be given as in Lemma 4.6 and let σ = r ↾ (n + 1). Find an M in E such that

o(x^{j} \ n, y^{j} \ n, z^{j} \ n) = σ̄,
n^{j}(M, N_{0} ∩ κ_{2}) ≤ n

for all j ≤ i. Then s̄^{j}(M, N_{0} ∩ κ_{2}) contains r ↾ n as an initial part and therefore

Δ(s̄^{j}(M, N_{0} ∩ κ_{2}), r) ≥ n^{j}(M, N_{0} ∩ κ_{2}).

This finishes the proof. □

Now we are ready to turn to the proof of Lemma 4.2. Since ℚ_{A,κ⃗,C⃗} decomposes as an iteration of a σ-closed partial order followed by a forcing of the form ℙ_{Σ}, it is sufficient to verify the (< ω_{1})-semiproperness of the second factor. In fact we will show that if κ⃗ consists of measurable cardinals, f : ω_{1} → 2^{ω} is any function, and Σ_{f,κ⃗} is the open set mapping associated to f as above, then ℙ_{Σ_{f,κ⃗}} is (<ω_{1})-semiproper.

For the rest of this section, let κ⃗ = ⟨κ_{0}, κ_{1}, κ_{2}⟩ be a fixed increasing sequence of three measurable cardinals, and fix a normal ultrafilter U_{i} on each κ_{i}. Let f be any fixed function from ω_{1} to 2^{ω} and let C⃗ be a fixed ladder system on ω_{1}. We will denote ℙ_{Σ_{f,κ⃗}} by ℙ.

Let θ be sufficiently large for ℙ and let ◁ be a well ordering of H(θ). Given subsets M, I of H(θ) with I ⊆ κ_{2} ∈ M, we use cl(M, I) to denote the set of values g(η_{0}, …, η_{n−1}), where g is a function in M with domain κ_{2}^{<ω} and {η_{0}, …, η_{n−1}} is a finite subset of I.

Still fixing θ, ◁, κ⃗ and U⃗, given an i ≤ 2 and an elementary submodel M of (H(θ), ∈, ◁) of cardinality less than κ_{i}, we will say that (M_{ξ})_{ξ<κ_{i}} is the iteration of M relative to U_{i} in case (M_{ξ})_{ξ<κ_{i}} is the unique ⊆– continuous sequence such that M_{0} = M and such that, for all ξ < κ_{i}, M_{ξ+1} = cl(M_{ξ}, {η_{ξ}}), where η_{ξ}^{i} = min(⋂(U_{i} ∩ M_{ξ})). We will also call (η_{ξ}^{i})_{ξ<κ} the critical sequence of M relative to U_{i}. We will use the following well-known facts repeatedly in the proof of Lemma 4.10.

Fact 4.7. For θ, ◁ and κ⃗ as above, if M is an elementary submodel of (H(θ), ∈, ◁) and I ⊆ κ_{2} ∈ M, then cl(M, I) is an elementary submodel of (H(θ), ∈, ◁).

Fact 4.8. Let θ, ◁, κ⃗ and U⃗ be as above. Fix i ≤ 2 and let M be an elementary submodel of H(θ) such that U_{i}, κ_{2} ∈ M. If η ∈ ⋂(M ∩ U_{i}), then cl(M, {η}) ∩ κ_{i} is an end-extension of M ∩ κ_{i}.

Fact 4.9. Let θ, ◁, κ⃗ and U⃗ be as above. Fix i ≤ 2 and let M be an elementary submodel of H(θ) such that U_{i}, κ_{2} ∈ M. Let I be a subset of κ_{i} and let μ ∈ M be a regular cardinal greater than κ_{i}. Then

sup(cl(M, I) ∩ μ) = sup(M ∩ μ).

Lemma 4.2 follows from combining Lemma 4.1 with Lemma 4.10, since whenever N and N^{∗} are suitable models for a partial order ℙ with N ⊆ N^{∗} and N ∩ ω_{1} = N^{∗} ∩ ω_{1}, any q ∈ ℙ which is (N^{∗}, ℙ)- generic is (N, ℙ)-semi-generic.

Lemma 4.10. Let α < ω_{1} be a limit ordinal and let ⟨N_{ξ} : ξ ≤ α⟩ be a suitable tower in H(θ) for ℙ such that each N_{ξ} is a countable elementary submodel of (H(θ), ∈, ◁). Then there is a suitable tower ⟨N_{ξ}^{∗} : ξ ≤ α⟩ in H(θ) such that for each ξ ≤ α:

• N_{ξ}^{∗} is a countable elementary submodel of (H(θ), ∈, ◁) of the form cl(N_{ξ}, I) for some I ⊆ κ_{2}, with N_{ξ}^{∗} ∩ ω_{1} = N_{ξ} ∩ ω_{1}; • if ξ ≤ α is a limit ordinal, then there is a ξ_{0} < ξ such that

Δ(s̄(N_{ν}^{∗} ∩ κ_{2}, N_{ξ}^{∗} ∩ κ_{2}), f(N_{ξ}^{∗} ∩ ω_{1})) ≥ n(N_{ν}^{∗} ∩ κ_{2}, N_{ξ}^{∗} ∩ κ_{2})

whenever ξ_{0} < ν < ξ, where s denotes sC⃗ ; κ_{0},κ_{1}.

Proof. By induction on α. We start by proving the lemma for α = ω, in which case we will prove the lemma with one additional conclusion, discussed below. Let (N_{j}^{0})_{j<ω} and (η_{i}^{0})_{i<ω} be the respective initial segments of length ω of the iteration of N_{ω} and the critical sequence of N_{ω}, both relative to U_{0}. Let N^{0} = ⋃_{j<ω} N_{j}^{0}. Let (N_{j}^{1})_{j<ω} and (η_{i}^{1})_{i<ω} be the respective initial segments of length ω of the iteration of N^{0} and the critical sequence of N^{0}, both relative this time to U_{1}. Let N^{1} = ⋃_{j<ω} N_{j}^{1}. Finally, let (N_{j}^{2})_{j<ω} and (η_{i}^{2})_{i<ω} be the respective initial segments of length ω of the iteration of N^{1} and the critical sequence of N^{1}, both relative to U_{2}.

Each model N_{j}^{∗} will be of the form cl(N_{j}, ⋃_{r<3}{η_{i}^{r} : i ∈ I_{j}^{r}}) for suitable finite subsets I_{j}^{r} of ω (for r < 3). It will follow in particular from Fact 4.8 that N_{j} ∩ ω_{1} = N_{j}^{∗} ∩ ω_{1}, so that n(N_{j}, N_{ω}) = n(N_{j}^{∗}, N_{ω}^{∗}). Furthermore, we will choose the sets I_{j}^{r} so that j ⊆ I_{j}^{r} ⊆ I_{j+1}^{r} for all r < 3 and all j < ω. This will ensure that each N_{j}^{∗} is a member of N_{j+1}^{∗} and also that we already know at the beginning of the construction exactly which set N_{ω}^{∗} = ⋃_{j<ω} N_{j}^{∗} is going to be. Specifically, N_{ω}^{∗} will be

cl(N_{ω}, ⋃_{r<3}{η_{i}^{r} : i < ω}) = ⋃_{j<ω} cl(N_{j}, ⋃_{r<3}{η_{i}^{r} : i < ω}) = ⋃_{j<ω} N_{j}^{2}.

Let δ = N_{ω} ∩ ω_{1}. Let π be the collapsing function of N_{ω}^{∗}, and let C^{0} = π^{−1}“C_{π(κ_{0})}, C^{1} = π^{−1}“C_{π(κ_{1})} and C^{2} = π^{−1}“C_{π(κ_{2})}.

For each j < ω and r < 3, I_{j}^{r} will be of the form

j ∪ ( ⋃_{j′<j} I_{j′}^{r}) ∪ {i_{k}^{r} : k < n}

for a suitable increasing sequence (i_{k}^{r})_{k<n} of integers above ⋃_{j′<j} I_{j′}^{r} to be defined as follows. Let j < ω be given and suppose that I_{j′}^{r} have been chosen for all r < 3 and j′ < j.

Set

M_{0} = cl(N_{j}, ⋃_{r<3}{η_{i}^{r} : i ∈ j ∪ ( ⋃_{j′<j} I_{j′}^{r})}).

Let n = n(N_{j}, N_{ω}). If n = 0 we can let N_{j}^{∗} = M_{0}. Otherwise, let ⟨p_{0}, …, p_{n−1}⟩ be f(δ) ↾ n. By the choice of (η_{i}^{r})_{i<ω} (for r < 3) together with Fact 4.8, each of {η_{i}^{0}}_{i<ω}, {η_{i}^{1}}_{i<ω} and {η_{i}^{2}}_{i<ω} is cofinal in κ_{0}∩N_{ω}^{∗}, κ_{1}∩N_{ω}^{∗} and κ_{2}∩N_{ω}^{∗}, respectively. Choose integers i_{k}^{0}, i_{k}^{1} and i_{k}^{2} (0 ≤ k ≤ n − 1) and models M_{t} (1 ≤ t ≤ 2n), satisfying the following conditions.

• j ∪ (⋃_{j′<j} I_{j′}^{0}) < i_{0}^{0} < … < i_{n−1}^{0};

• j ∪ (⋃_{j′<j} I_{j′}^{1}) < i_{0}^{1} < … < i_{n−1}^{1};

• j ∪ (⋃_{j′<j} I_{j′}^{2}) < i_{0}^{2} < … < i_{n−1}^{2};

• for all k ∈ {0, …, n − 1},

– sup(M_{2k} ∩ κ_{0}) < η_{i_{k}^{0}}^{0} and C^{0} ∩ η_{i_{k}^{0}}^{0} has size strictly bigger than both

|C^{1} ∩ sup(M_{2k} ∩ κ_{1})|

and

|C^{2} ∩ sup(M_{2k} ∩ κ_{2})|;

– M_{2k+1} = cl_{j}(M_{2k} ∪ {η_{i_{k}^{0}}^{0}});

– if p_{n−1−k} = 0, then

|C^{0} ∩ sup(M_{2k+1} ∩ κ_{0})| < |C^{1} ∩ η_{i_{k}^{1}}^{1}| < |C^{2} ∩ η_{i_{k}^{2}}^{2}|;

– if p_{n−1−k} = 1, then

|C^{0} ∩ sup(M_{2k+1} ∩ κ_{0})| < |C^{1} ∩ η_{i_{k}^{2}}^{2}| < |C^{2} ∩ η_{i_{k}^{1}}^{1}|;

– M_{2k+2} = cl(M_{2k+1}, {η_{i_{k}^{1}}^{1}, η_{i_{k}^{2}}^{2}}).

Note the following consequences of these choices (and Facts 4.8 and 4.9 and the fact that each η_{i}^{r} is regular), for all k ∈ {0, …, n − 1} .

• M_{2k+1} ∩ [sup(M_{2k} ∩ κ_{0}), η_{i_{k}^{0}}^{0}) = ∅;

• M_{2k+2} ∩ κ_{0} = M_{2k+1} ∩ κ_{0};

• for all μ ∈ {η_{i_{k′}^{1}}^{1} : k′ < k} ∪ {κ_{1}},

sup(M_{2k} ∩ μ) = sup(M_{2k+1} ∩ μ) < η_{i_{k}^{1}}^{1};

• M_{2k+2} ∩ [sup(M_{2k+1} ∩ κ_{1}), η_{i_{k}^{1}}^{1}) = ∅;

• for all μ ∈ {η_{i_{k′}^{2}}^{2} : k′ < k} ∪ {κ_{2}},

sup(M_{2k} ∩ μ) = sup(M_{2k+1} ∩ μ) < η_{i_{k}^{2}}^{2};

• M_{2k+2} ∩ [sup(M_{2k+1} ∩ κ_{2}), η_{i_{k}^{2}}^{2}) = ∅.

Now it is not hard to check that the string ⟨p_{n−1}, …, p_{0}⟩ is a terminal segment of s(M_{2n}∩κ_{2}, N^{∗}∩κ_{2}), which means that we can let N_{j}^{∗} = M_{2n}.

We note one additional aspect of this construction: each η_{i}^{r} is in each member of N_{ω} ∩ U_{i}. From this and Fact 4.8 it follows that for each j ∈ ω and any countable elementary submodel P of (H(θ), ∈, ◁) such that U⃗ ∈ P ∈ N_{ω},

cl(P, ⋃_{r<3}{η_{i}^{r} : i ∈ I_{j}^{r}}) ∩ ω_{1} = P ∩ ω_{1}.

This completes the proof for α = ω.

Now we can prove the lemma for general α < ω_{1} by induction on α, α a limit ordinal. Let (N_{ν})_{ν≤α} be a tower as in the hypothesis of the lemma for α, and assume that the lemma is true for all β < α. Let (α_{j})_{j<ω} be any increasing sequence of limit ordinals with supremum α. Apply the case α = ω to the sequence (N_{α_{j}+1})_{j<ω} to obtain the models N_{α_{j}+1}^{∗} for j < ω. Let N_{α}^{∗} = ⋃_{j<ω} N_{α_{j}+1}^{∗}. Applying the additional conclusion of the case α = ω, we have that there is a ⊆-increasing sequence of finite sets ⟨E_{j} : j < ω⟩ such that for each j < ω,

• E_{j} ⊆ N_{α_{j}+1}^{∗} ∩ κ_{2};

• N_{α_{j}+1}^{∗} = cl(N_{α_{j}+1}, E_{j});

• for all countable elementary submodels P of (H(θ), ∈, ◁) in N_{α},

cl(P, E_{j}) ∩ ω_{1} = P ∩ ω_{1};

• for all Q such that E_{j} ⊆ Q ⊆ N_{α_{j}+1}^{∗},

Δ(s̄(Q, N_{α}^{∗}∩κ_{2}), f(N_{α}^{∗}∩ω_{1})) ≥ n(N_{α_{j}+1}^{∗}∩κ_{2}, N_{α}^{∗}∩κ_{2}) = n(Q∩κ_{2}, N_{α}^{∗}∩κ_{2}).

We can now build the rest of the sequence of N_{β}^{∗}’s by working separately on each interval [α_{j} + 2, α_{j+1}] inside of N_{α_{j+1}+1}^{∗} (this omits the construction for the first interval, which can be taken care of by setting α_{−1} = −2). Fixing such a j, for each β ∈ [α_{j} + 2, α_{j+1}], let N_{β}^{0} = cl(N_{β}, E_{j+1}). Now apply the induction hypothesis inside of N_{α_{j+1}+1}^{∗} to the sequence ⟨N_{β}^{0} : α_{j} + 2 ≤ β ≤ α_{j+1}⟩ to obtain the desired sequence ⟨N_{β}^{∗} : α_{j} + 2 ≤ β ≤ α_{j+1}⟩. □

5. Concluding remarks

The Π_{2} sentences which we employed to resolve Problem 1.1 are quite ad hoc in nature and it is natural to ask whether there are simpler examples. In particular, it is unclear whether there are Π_{2}-sentences which have already been studied in the literature which solve Problem 1.1. Also, while it is reasonable to expect that the consistency of ψ_{1} (with or without CH) requires an inaccessible cardinal, it is unclear whether consistency of ψ_{1} with CH requires, for instance, a measurable cardinal. This seems largely a technical question, but one whose solution (either positive or negative) would likely involve new ideas and give new insight how models of CH can (or cannot) be obtained by iterated forcing.

Until the present article, the study of preservation theorems for not adding reals largely centered on the degree to which (< ω_{1})-properness can be dispensed with in theorems like Theorem VIII.4.5 of [20] (which is the precursor to [8] and Theorems 3.3 and 3.10 above). This question has now been resolved as well in the third author’s forthcoming [16]: some hypothesis beyond a completeness assumption is necessary.

The present article underscores that the notion of completeness is not as robust as one might hope. The results in this paper show that there is an important distinction between 1-semi-complete iterations and ω-complete iterations. In [8], the apparent added flexibility of 2complete over 1-complete iterations was important to the argument. While this was largely dismissed as a technical detail at the time, it may now warrant further investigation.

References

[1] Abraham, Uri. Proper Forcing. Handbook of Set Theory. Foreman, Kanamori, and Magidor (eds.) Springer, 2010

[2] Uri Abraham and Stevo Todorčević. Partition properties of ω_{1} compatible with CH. Fund. Math., 152(2) (1997) 165–181.

[3] Andrés E. Caicedo and Boban Veličković. The bounded proper forcing axiom and well orderings of the reals. Mathematical Research Letters, 13(3) (2006) 393–408.

[4] Keith Devlin and Saharon Shelah. A weak version of ♢ which follows from 2^{ℵ_{0}} < 2^{ℵ_{1}}. Israel Journal of Math, 29(2–3) (1978) 239–247.

[5] Keith J. Devlin and Hȧvard Johnsbrȧten. The Souslin problem. Lecture Notes in Mathematics, Vol. 405. Springer-Verlag, Berlin, 1974.

[6] Hans-Dieter Donder and Ulrich Fuchs. Revised countable support iterations. unpublished note.

[7] Todd Eisworth and Peter Nyikos. First countable, countably compact spaces and the continuum hypothesis. Trans. Amer. Math. Soc., 357(11) (2005) 4269– 4299.

[8] Todd Eisworth and Judith Roitman. CH with no Ostaszewski spaces. Trans. Amer. Math. Soc., 351(7) (1999) 2675–2693.

[9] Todd Eisworth and Justin Tatch Moore. Iterated forcing and the Continuum Hypothesis. To appear in Appalachian Set Theory Workshop booklet, Dec. 2011.

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

[11] Moti Gitik. Nonsplitting subsets of 𝒫_{κ}(κ^{+}). Jour. Symb. Logic , 50(4) (1985) 881–894.

[12] Thomas Jech. Set theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded.

[13] Kenneth Kunen. An introduction to independence proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1983.

[14] Paul Larson and Saharon Shelah. Bounding by canonical functions, with CH. J. Math. Log., 3(2) (2003), 193–215.

[15] Tadatoshi Miyamoto. A limit stage construction for iterating semiproper preorders. In Proceedings of the 7th and 8th Asian Logic Conferences, pages 303– 327, Singapore, 2003. Singapore Univ. Press.

[16] Justin Tatch Moore. Forcing axioms and the Continuum Hypothesis, part II: transcending ω_{1}-sequences of reals. ArXiv preprint 1110.5037 dated October 23, 2011.

[17] Justin Tatch Moore. The Proper Forcing Axiom. Proceedings of the 2010 meeting of the International Congress of Mathematicians, Hyderabad, India.

[18] Justin Tatch Moore. Set mapping reflection. J. Math. Log., 5(1) (2005) 87–97.

[19] Justin Tatch Moore. ω_{1} and −ω_{1} may be the only minimal uncountable order types. Michigan Math. Journal, 55(2) (2007) 437–457.

[20] Saharon Shelah. Proper and improper forcing. Springer-Verlag, Berlin, second edition, 1998.

[21] Robert Solovay and Stanley Tennenbaum. Iterated Cohen extensions and Souslin’s problem. Ann. of Math., 94 (1971) 201–245.

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

Institute of Discrete Mahematics and Geometry, Technische Universität Wien, Vienna, Austria

  E-mail address: david.aspero@tuwien.ac.at

Department of Mathematics, Miami University, Oxford, OH 45056, USA

  E-mail address: larsonpb@muohio.edu

Department of Mathematics, Cornell University, Ithaca, NY 14853– 4201, USA

  E-mail address: justin@math.cornell.edu