Handbook of Set Theory

Foreman, Kanamori, and Magidor (eds.) August 5, 2006

2

Contents

I Forcing over models of determinacy 5 by Paul B. Larson 1 Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 ℙ_{max} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Sequences of models and countable closure . . . . . . . . . . . . 18 4 Generalized iterability . . . . . . . . . . . . . . . . . . . . . . . . 22 5 The basic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 ψ_{AC} and the Axiom of Choice . . . . . . . . . . . . . . . . . . . 35 7 Maximality and minimality . . . . . . . . . . . . . . . . . . . . . 36 8 Larger models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 9 Ω-logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.1 Variations for NS_{ω_{1}} . . . . . . . . . . . . . . . . . . . . . 53 10.2 Conditional variations for Σ_{2} sentences . . . . . . . . . . 57 4 CONTENTS

I. Forcing over models of determinacy

Paul B. Larson

The Axiom of Determinacy (AD) is the statement that all integer games of perfect information of length ω are determined. This statement contradicts the Axiom of Choice, and presents a radically different view of the universe of sets. Nonetheless, determinacy was a subject of intense study by the mid 1960’s, with an eye towards the possibility that some inner model of set theory satisfies AD (see, for example, the introductory remarks in [30]). Since strategies for these games can be coded by real numbers, the natural inner model to consider is L(ℝ), the smallest model of Zermelo-Fraenkel set theory containing the reals and the ordinals. This approach was validated by the following theorem of Woodin (see [13, 19]), building on work of Martin and Steel [23] and Foreman, Magidor and Shelah [6].

0.1 Theorem. If there exists a measurable cardinal which is greater than infinitely many Woodin cardinals, then the Axiom of Determinacy holds in L(ℝ).

A companion to Theorem 0.1, also due to Woodin (see [19]) and building on the work of Foreman, Magidor and Shelah [6], shows that the existence of a proper class of Woodin cardinals implies that the theory of L(ℝ) cannot be changed by set forcing. By Theorem 0.1, the Axiom of Determinacy is part of this fixed theory for L(ℝ).

0.2 Theorem. If δ is a limit of Woodin cardinals and there exists a measurable cardinal greater than δ, then no forcing construction in V_{δ} can change the theory of L(ℝ).

Theorem 0.2 has the following corollary. If P is a definable forcing construction in L(ℝ) which is homogeneous (i.e., the theory of the extension can be computed in the ground model), then the theory of the P -extension of L(ℝ) also cannot be changed by forcing (i.e., the P -extensions of L(ℝ) in all forcing extensions of V satisfy the same theory). This suggests that the absoluteness properties of L(ℝ) can be lifted to models of the Axiom of Choice, as Choice can be forced over L(ℝ).

In [33], Steel and Van Wesep made a major step in this direction, forcing over a model of a stronger form of determinacy than AD to produce a model of ZFC satisfying two consequences of AD, that δ_{∼}_{2}^{1} (the supremum of the lengths of the Δ_{2}^{1}-definable prewellordings of the reals) is ω_{2} and the nonstationary ideal on ω_{1} (NS_{ω_{1}}) is saturated. Woodin [36] later improved the hypothesis to AD^{L(ℝ)}.

In the early 1990’s, Woodin proved the following theorem, showing for the first time that large cardinals imply the existence of a partial order forcing the existence of a projective set of reals giving a counterexample to the Continuum Hypothesis. The question of whether ZFC is consistent with a projective witness to 𝔠 ≥ ω_{3} remains open.

0.3 Theorem. If NS_{ω_{1}} is saturated and there exists a measurable cardinal

then δ_{∼}_{2}^{1} = ω_{2}.

One important point in this proof is the fact that if NS_{ω_{1}} is saturated then every member of H(ω_{2}) (those sets whose transitive closure has cardinality less than ℵ_{2}) appears in an iterate (in the sense of the next section) of a countable model of a suitable fragment of ZFC. Since these countable models are elements of L(ℝ), their iterations induce a natural partial order in L(ℝ). With certain technical refinements, this partial order, called ℙ_{max}, produces an extension of L(ℝ) whose H(ω_{2}) is the direct limit of the structures H(ω_{2}) of models satisfying every forceable theory (and more). In particular, the structure H(ω_{2}) in the ℙ_{max} extension of L(ℝ) (assuming that AD holds in L(ℝ)) satisfies every Π_{2} sentence ϕ (in the language with predicates for NS_{ω_{1}} and each set of reals in L(ℝ)) for H(ω_{2}) such that for some integer n the theory ZFC + “there exist n Woodin cardinals” implies that ϕ is forceable. Furthermore, the partial order ℙ_{max} can be easily varied to produce other consistency results and canonical models.

The partial order ℙ_{max} and some of its variations (and many other related issues) are presented in [37]. The aim of this chapter is to prepare the reader for that book. First, we attempt to give a complete account of the basic analysis of the ℙ_{max} extension of L(ℝ), relative to published results. Then we briefly survey some of the issues surrounding ℙ_{max}, in particular ℙ_{max} variations and forcing over larger models of determinacy. We also briefly introduce Woodin’s Ω-logic, in order to properly state the maximality properties of the ℙ_{max} extension. For the most part, though, our focus is primarily on the ℙ_{max} extension of L(ℝ), and secondarily on ℙ_{max}-style forcing constructions as a means of producing consistency results. For other topics, such as the Ω-conjecture and the relationship between Ω-logic and the Continuum Hypothesis, we refer the reader to [40, 38, 39, 41, 2].

The material in this chapter is due to Woodin, except where noted otherwise. The author would like to thank Howard Becker and John Steel for advising him on parts of the material in Sections 4 and 9 respectively. He would also like to thank Andrés Caicedo, Neus Castells and Teruyuki Yorioka for making numerous helpful suggestions. The author was supported in part by the Fields Institute, FAPESP (Grant # 02/11551-3, University of São Paulo) and the Centre de Recerca Matemàtica of the Institut d’Estudis Catalans.

1. Iterations

The fundamental construction in the ℙ_{max} analysis is the iterated generic elementary embedding. These embeddings can have many forms, but we will concentrate on the following case. Suppose that I is a normal, uniform, proper ideal on ω_{1} (so I is a proper subset of P(ω_{1}) containing all the countable subsets, and such that whenever A is an I-positive set (i.e., in P(ω_{1})\I) and f: A → ω_{1} is a regressive function, f is constant on an I-positive set; notationally, we are going to act as though “proper” and “uniform” are contained in the definition of normal ideal, and similarly for “measure” and “ultrafilter”). Then forcing with the Boolean algebra P(ω_{1})/I creates a V - normal ultrafilter U on ω_{1}^{V}. By convention, we identify the wellfounded part of the ultrapower Ult(V, U) with its transitive collapse, and we note that this wellfounded part always contains ω_{2}^{V}. The corresponding elementary embedding j: V → Ult(V, U) has critical point ω_{1}^{V}, and since I is normal, for each A ∈ P(ω_{1})^{V}, A ∈ U if and only if ω_{1}^{V} ∈ j(A). Under certain circumstances, the corresponding ultrapower of V is wellfounded; if every condition in P(ω_{1})/I forces this, then I is precipitous.

For the most part, we will be concerned only with models of ZFC, but since occasionally we will want to deal with structures whose existence can be proved in ZFC, we define the fragment ZFC^{∘} to be the theory ZFC − Powerset − Replacement + “P(P(ω_{1})) exists” plus the following scheme, which is a strengthening of ω_{1}-Replacement: every (possibly proper class) tree of height ω_{1} definable from set parameters has a maximal branch (i.e., a branch with no proper extensions; in the cases we are concerned with, this just means a branch of length ω_{1}). By the Axiom of Choice here we mean that every set is the bijective image of an ordinal. We will use ZFC^{∘} in place of the theory ZFC^{∗} from [37], which asserts closure under the Gödel operations (see page 178 of [9]) plus a scheme similar to the one above. One advantage of using ZFC^{∗} is that H(ω_{2}) satisfies it (and thus so do its elementary submodels). On the other hand, it raises some technical points that we would rather avoid here. Some of these points appear in Woodin’s proof of Theorem 0.3. Our concentration is on ℙ_{max}, but we hope nonetheless that the reader will have no difficulty in reading the proofs of that theorem in [4, 37] after reading the material in this section.

With either theory, the point is that one needs to be able to prove the version of Łoś’s theorem asserting that ultrafilters on ω_{1} generate elementary embeddings, which amounts to showing the following fact. The fact follows immediately from the scheme above.

1.1 Fact. (ZFC^{∘}) Let n be an integer. Suppose that ϕ is a formula with n + 1 many free variables and f_{0}, …, f_{n−1} are functions with domain ω_{1}. Then there is a function g with domain ω_{1} such that for all α < ω_{1},

∃xϕ(x, f_{0}(α), …, f_{n−1}(α)) ⇒ ϕ(g(α), f_{0}(α), …, f_{n−1}(α)).

If M is a model of ZFC and κ is a cardinal of M of cofinality greater than ω_{1}^{M} (in M), then H(κ)^{M} satisfies ZFC^{∘} if it has |P(P(ω_{1}))|^{M} as a member.

Suppose that M is a model of ZFC^{∘} , I ∈ M is a normal ideal on ω_{1}^{M} and P(P(ω_{1}))^{M} is countable. Then there exist M-generic filters for the partial order (P(ω_{1})/I)^{M}. Furthermore, if j: M → N is an ultrapower embedding of this form, then P(P(ω_{1}))^{N} is countable, and there exist N-generic filters for (P(ω_{1})/j(I))^{N}. We can continue choosing generics in this way for up to ω_{1} many stages, defining a commuting family of elementary embeddings and using this family to take direct limits at limit stages.

We use the following formal definition.

1.2 Definition. Let M be a model of ZFC^{∘} and let I be an ideal on ω_{1}^{M} which is normal in M. Let γ be an ordinal less than or equal to ω_{1}. An iteration of (M, I) of length γ consists of models M_{α} (α ≤ γ), sets G_{α} (α < γ) and a commuting family of elementary embeddings j_{αβ}: M_{α} → M_{β} (α ≤ β ≤ γ) such that

• M_{0} = M,

• each G_{α} is an M_{α}-generic filter for (P(ω_{1})/j_{0α}(I))^{M_{α}},

• each j_{αα} is the identity mapping,

• each j_{α(α+1)} is the ultrapower embedding induced by G_{α},

• for each limit ordinal β ≤ γ, M_{β} is the direct limit of the system {M_{α}, j_{αδ} : α ≤ δ < β}, and for each α < β, j_{αβ} is the induced embedding.

If ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩ is an iteration of a pair (M, I) and each ω_{1}^{M_{α}} is wellfounded, then {ω_{1}^{M_{α}} : α < ω_{1}} is a club subset of ω_{1}. Note also that if ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩ is an iteration of a pair (M, I), then j[Ord^{M_{0}}] is cofinal in Ord^{M_{γ}}.

The models M_{α} in Definition 1.2 are called iterates of (M, I). If M is a model of ZFC^{∘} then an iteration of (M, NS_{ω_{1}}^{M}) is called simply an iteration of M and an iterate of (M, NS_{ω_{1}}^{M}) is called simply an iterate of M. When the individual parts of an iteration are not important, we sometimes call the elementary embedding j_{0γ} corresponding to an iteration an iteration itself. For instance, if we mention an iteration j: (M, I) → (M^{∗}, I^{∗}), we mean that j is the embedding j_{0γ} corresponding to some iteration

⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩

of (M, I), and that M^{∗} is the final model of this iteration and I^{∗} = j(I).

If M and I are as in Definition 1.2, then the pair (M, I) is iterable if every iterate of (M, I) is wellfounded. In this chapter, we are in general interested only in iterable pairs (M, I). Note that when checking for iterability it suffices to consider the countable length iterations, as any iteration of length ω_{1} whose final model is illfounded contains an illfounded model at some earlier stage. The following two lemmas show that if

• M is a transitive model of ZFC^{∘}+ Powerset containing ω_{1}^{V},

• |P(P(ω_{1})/I)|^{M} is countable,

• κ is a cardinal of M greater than |P(P(ω_{1})/I)|^{M} with cofinality greater than ω_{1}^{M} in M,

• I ∈ M is a normal precipitous ideal on ω_{1}^{M},

then the pair (H(κ)^{M}, I) is iterable. In particular, Lemma 1.5 below shows that every such pair (M, I) is iterable, and then Lemma 1.4 shows that (H(κ)^{M}, I) is iterable, as every iterate of (H(κ)^{M}, I) embeds into an iterate of (M, I). This argument is our primary means of finding iterable models.

1.3 Remark. Note that the statement that a given pair (M, I) is iterable is Π_{2}^{1} in any real x recursively coding the pair. One way to express this (not necessarily the most direct), is: for every countable model N of ZFC^{∘} with x as a member and every object J ∈ N such that N ⊧ “J is an iteration of (M, I)” and every function f from ω to the “ordinals” of the last model of J, either N is illfounded (i.e., there exists an infinite descending sequence of “ordinals” of N) or f(n + 1)“∉”f(n) for some integer n, where “∉” is the negation of the ∈-relation of the last model of J. Therefore, whether or not (M, I) is iterable is absolute between models of ZFC^{∘} containing the countable ordinals. Furthermore, assuming that x^{#} exists and letting γ denote ω_{1}^{L[x^{#}]}, any transitive model N of ZFC^{∘} containing L_{γ}[x^{#}] is correct about the iterability of (M, I), as L[x^{#}] is correct about it, and N thinks that L_{ω_{1}^{N}} [x^{#}] is correct about it. Similarly, if γ and δ are countable ordinals coded by reals y and z, then the existence of an iteration of (M, I) of length γ which is illfounded is a Σ_{1}^{1} fact about x and y, and the existence of an iteration of (M, I) of length γ such that the ordinals of the last model of the iteration have height at least (or, exactly) δ is a Σ_{1}^{1} fact about x, y and z.

The first lemma is easily proved by induction. The last part of the lemma uses the assumption that N is closed under ω_{1}^{M}-sequences in M (this is the main way in which the lemma differs from the corresponding lemma in [37] (Lemma 3.8)). In our applications, N will often be H(κ)^{M} for some cardinal κ of of M such that M ⊧ cf(κ) > ω_{1}^{M}, in which case H(κ)^{M} is indeed closed under ω_{1}^{M}-sequences in M.

1.4 Lemma. Suppose that M is a model of ZFC^{∘} and I ∈ M is a normal ideal on ω_{1}^{M}. Let N be a transitive model of ZFC^{∘} in M containing P(P(ω_{1})/I)^{M} and closed under ω_{1}^{M}-sequences in M. Let γ ≤ ω_{1} be an ordinal and let

⟨N_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩

be an iteration of (N, I). Then there exists a unique iteration

⟨M_{α}, G_{β}^{∗}, j_{αδ}^{∗} : β < α ≤ δ ≤ γ⟩

of (M, I) such that for all β < α ≤ γ, G_{β} = G_{β}^{∗} and

P(P(ω_{1})/I)^{N_{α}} = P(P(ω_{1})/I)^{M_{α}}.

Furthermore, N_{α} = j_{0α}^{∗}(N) for all α ≤ γ.

Given ordinals α, β, the partial order Col(α, β) is the set of partial functions from α to β whose domain has cardinality less than that of α, ordered by inclusion. In particular, Col(ω, β) makes β countable. Given ordinals α and β, Col(α, <β) is the partial order consisting of all finite partial functions p: β × α → β such that for all (δ, γ) ∈ dom(p), p(δ, γ) ∈ δ, ordered by inclusion (we will not use this definition until the next section).

The proof of Lemma 1.5 is a modification of standard arguments.

1.5 Lemma. Suppose that M is a transitive model of ZFC^{∘}+ Powerset and that I ∈ M is a normal precipitous ideal on ω_{1}^{M}. Suppose that j: (M, I) → (M^{∗}, I^{∗}) is an iteration of (M, I) whose length is in (ω_{1}^{V} + 1) ∩ M. Then M^{∗} is wellfounded.

Proof. If j and M^{∗} are as in the statement of the lemma, then M^{∗} is the union of all sets of the form j(H(κ)^{M}), where κ is a regular cardinal in M, and for each such κ > |P(P(ω_{1}))|^{M}, j↾H(κ)^{M} is an iteration of (H(κ)^{M}, I). If the lemma fails, then, we may let (γ̄, κ̄, η̄) be the lexicographically least triple (γ, κ, η) such that

• κ is a regular cardinal in M greater than |P(P(ω_{1})/I)|^{M},

• η < κ,

• there is an iteration ⟨N_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩ of (H(κ)^{M}, I) such that j_{0γ}(η) is not wellfounded.

Since I is precipitous in M, γ̄ is a limit ordinal, and clearly η̄ is a limit ordinal as well. Fix an iteration ⟨N_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ̄⟩ of (H(κ̄)^{M}, I) such that j_{0γ̄}(η̄) is not wellfounded, and let ⟨M_{α}, G_{β}, j′_{αδ} : β < α ≤ δ ≤ γ̄⟩ be the corresponding iteration of M as in Lemma 1.4. By the minimality of γ̄ we have that M_{α} is wellfounded for all α < γ̄. Since Nγ̄ is the direct limit of the iteration leading up to it, we may fix γ^{∗} < γ̄ and η^{∗} < j_{0γ^{∗}}(η̄) such that j_{γ^{∗}γ̄}(η^{∗}) is not wellfounded. Note that by Lemma 1.4, j′_{γ^{∗},γ̄}(η^{∗}) = j_{γ^{∗},γ̄}(η^{∗}) and j′_{γ^{∗},γ̄}(η̄) = j′_{γ^{∗},γ̄}(η̄).

[figure: M_{0}; M_{γ^{∗}}; Mγ̄; η̄; η^{∗}]

The key point is that if N is a model of ZFC^{∘}, J is a normal ideal on ω_{1}^{N} in N, γ is an ordinal and η is an ordinal in N, then the statement positing an iteration of (N, J) of length γ whose last model is illfounded below the image of η is a Σ_{1}^{1} sentence in a real parameter recursively coding N, η and γ, and so this statement is absolute between wellfounded models of ZFC^{∘} containing such a real. In particular, if

• N is a transitive model of ZFC^{∘} + Powerset,

• J is a normal ideal on ω_{1}^{N} in N,

• κ is a regular cardinal in N greater than |P(P(ω_{1}))|^{N},

• η < κ and γ are ordinals in N and β ∈ N is an ordinal greater than or equal to max{(2^{κ})^{N}, γ},

then if G is N-generic for Col(ω, β), then N[G] satisfies the correct answer for the assertion that there exists an iteration of (H(κ)^{N}, J) of length γ whose last model is illfounded below the image of η. Let ϕ(γ, κ, η, J) be the formula asserting that

• J is a normal ideal on ω_{1},

• κ is a regular cardinal greater than |P(P(ω_{1}))|,

• η < κ,

• letting β = max{2^{κ}, γ}, every condition (equivalently, some condition) in Col(ω, β) forces that there exists an iteration of (H(κ), J) of length γ whose last model is illfounded below the image of η.

Then, in M, (γ̄, κ̄, η̄) is the lexicographically least triple (γ, κ, η) such that ϕ(γ, κ, η, I) holds. Furthermore, since j′_{0γ^{∗}} is elementary, in M_{γ^{∗}},

(j′_{0γ^{∗}}(γ̄), j′_{0γ^{∗}}(κ̄), j′_{0γ^{∗}}(η̄))

is the least triple (γ, κ, η) such that ϕ(γ, κ, η, j′_{0γ^{∗}}(I)) holds. However, the tail of the iteration ⟨N_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ̄⟩ starting with N_{γ^{∗}} is an iteration of

(H(j_{0γ^{∗}}(κ̄))^{M_{γ^{∗}}} , j_{0γ^{∗}}(I))

(note that j′_{0γ^{∗}}(H(κ̄)^{M}) = j_{0,γ^{∗}}(H(κ̄)^{M})) of length less than or equal to γ̄ which in turn is less than or equal to j′_{0γ^{∗}}(γ̄). Furthermore, η^{∗} < j′_{0γ}(η̄) = j_{0γ}(η̄), and j_{γ^{∗}γ̄}(η^{∗}) is not wellfounded, which, by the correctness property mentioned above (using the fact that M_{γ^{∗}} is wellfounded) contradicts the minimality of j′_{0γ^{∗}}(η̄). ⊣

1.6 Example. Let M be any countable transitive model of ZFC in which there exists a measurable cardinal κ and a normal measure μ ∈ M on κ such that all countable iterates of M by μ are wellfounded. Iterating M by μ ω_{1} times, we obtain a model N of ZFC containing ω_{1} such that (V_{κ})^{M} = (V_{κ})^{N}. Now suppose that I is a normal precipitous ideal on ω_{1}^{M} in M. By Lemmas 1.4 and 1.5, ((V_{κ})^{M}, I) iterable.

Before moving on, we prove an important fact about iterations of iterable models which will show up later (in Lemmas 3.3, 6.2 and 7.8). This fact is a key step in Woodin’s proof of Theorem 0.3.

1.7 Lemma. Suppose that M is a countable transitive model of ZFC^{∘} and I ∈ M is a normal ideal on ω_{1}^{M} such that the pair (M, I) is iterable. Let x be a real coding the pair (M, I) under some recursive coding. Let

I = ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩

be an iteration of (M, I). Then every countable ordinal γ such that L_{γ}[x] satisfies ZFC is on the critical sequence of I.

Proof. Fix a countable ordinal γ such that L_{γ}[x] ⊧ ZFC. We want to see that for every η < γ there is a δ < γ such that the ordinals of the final model of every iteration of (M, I) of length η are contained in δ. To see this, fix η and let g ⊂ Col(ω, η) be L_{γ}[x]-generic. Then L_{γ}[x][g] ⊧ ZF C, and in L_{γ}[x][g], the set of ordertypes of the ordinals of iterates of (M, I) by iterations of length η is a Σ_{1}^{1} set in a real coding (M, I) and g. By the boundedness lemma for Σ_{∼}_{1}^{1} sets of wellorderings (see [24]), then, there is a countable (in L_{γ}[x][g]) ordinal δ such that all of these ordertypes are less than δ. Furthermore, the nonexistence of an iteration of (M, I) of length η such that δ can be embedded in an order-preserving way into the ordinals of the final model is absolute between L_{γ}[x][g] and V, by Σ_{∼}_{1}^{1}-absoluteness. ⊣

Lemma 1.7 has the following useful corollary. The case where γ is countable follows immediately from Lemma 1.7. The case where γ = ω_{1} follows by applying the countable case to a forcing extension where ω_{1} is collapsed.

1.8 Corollary. Suppose that M is a countable transitive model of ZFC^{∘}, I is normal ideal on ω_{1}^{M} in M, (M, I) is iterable and x is a real coding (M, I). Suppose that γ is an x-indiscernible less than or equal to ω_{1}, and let j: (M, I) → (M^{∗}, I^{∗}) be an iteration of (M, I) of length γ. Then the ordinals of M^{∗} have height less than the least x-indiscernible above γ.

We note one more useful fact about sharps. The fact can be proved directly using the remarkable properties of sharps, or by noting that the two functions implicit in the fact necessarily represent the same ordinal in any generic ultrapower.

1.9 Fact. Let x be a real and let γ be the least x-indiscernible above ω_{1}. Let π: ω_{1} → γ be a bijection. Then the set of α < ω_{1} such that o.t.(π[α]) is the least x-indiscernible above α contains a club.^{1}

1.10 Remark. Often in this chapter will we use recursive codings of elements of H(ω_{1}) by reals (by which we mean elements of ω^{ω}). The following coding is sufficient in all cases: fixing a recursive bijection π: ω × ω → ω, let x be a real coding the set of sets coded by those y ⊆ ω for which there exists an i < ω such that π(0, i) ∈ x and y = {j < ω | π(j + 1, i) ∈ x}. Note that under this coding, the relations “∈” and “=” are both Σ_{1}^{1}, since permutations of ω can give rise to different codes for the same set.

1.11 Remark. If there exists a precipitous ideal on ω_{1}, then A^{#} exists for every A ⊆ ω_{1}. To see this, note first of all that the existence of a precipitous ideal implies that for each real x there is an nontrivial elementary embedding from L[x] to L[x] in a forcing extension, which means that x^{#} exists already in the ground model. Furthermore, if I is a precipitous ideal on ω_{1} and j: V → M is the generic embedding derived from a V -generic filter G ⊂ P(ω_{1})/I, then P(ω_{1})^{V} ⊆ H(ω_{1})^{M}. Therefore, for every A ∈ P(ω_{1})^{V}, A ∈ M and M ⊧ “A^{#} exists.” Since M and V [G] have the same ordinals, V [G] and V then must also satisfy “A^{#} exists.”

Similarly, if (M, I) is an iterable pair, then M is correct about the sharps of the reals of M, since M is elementarily embeddeded into a transitive model containing ω_{1}, and thus M is correct about the sharps of the members of P(ω_{1})^{M}. In particular, if (M, I) is an iterable pair and A is in P(ω_{1})^{M}, then P(ω_{1})^{L[A]} ⊆ M, so M correctly computes ω_{1}^{L[A]}.

1Given a function f and subset X of the domain of X, we let f[X] denote the set {f(x) | x ∈ X}.

2. ℙ_{max}

We are now ready to define the partial order ℙ_{max}. We will make one modification of the definition given in [37] and require the conditions to satisfy ZFC^{∘} instead of the theory ZFC^{∗} defined in [37]. Our ℙ_{max} is a dense suborder of the original; furthermore, the basic analysis of the two partial orders is the same, though the proofs of Lemma 7.10 and Theorem 7.11 are less elegant than they might otherwise be.

Recall that MA_{ℵ_{1}} is the version of Martin’s Axiom for ℵ_{1}-many dense sets, i.e., the statement that whenever P is a c.c.c. partial order and D_{α} (α < ω_{1}) are dense subsets of P there is a filter G ⊂ P intersecting each D_{α}.

2.1 Definition. The partial order ℙ_{max} consists of all pairs ⟨(M, I), a⟩ such that

1. M is a countable transitive model of ZFC^{∘}+ MA_{ℵ_{1}},

2. I ∈ M and in M, I is a normal ideal on ω_{1},

3. (M, I) is iterable,

4. a ∈ P(ω_{1})^{M},

5. there exists an x ∈ P(ω)^{M} such that ω_{1}^{M} = ω_{1}^{L[a,x]}.

The order on ℙ_{max} is as follows: ⟨(M, I), a⟩ < ⟨(N, J), b⟩ if N ∈ H(ω_{1})^{M} and there exists an iteration j: (N, J) → (N^{∗}, J^{∗}) such that

• j(b) = a,

• j, N^{∗} ∈ M,

• I ∩ N^{∗} = J^{∗}.

We say that a pair (M, I) is a (ℙ_{max}) pre-condition if there exists an a such that ⟨(M, I), a⟩ is in ℙ_{max}.

2.2 Remark. If ⟨(M, I), a⟩ is a ℙ_{max} condition, then M is closed under sharps for reals (see Remark 1.11), and so a cannot be in L[x] for any real x in M. Therefore, a is unbounded in ω_{1}^{M}, and this in turn implies that the iteration witnessing that a given ℙ_{max} condition ⟨(M, I), a⟩ is stronger than another condition ⟨(N, J), b⟩ must have length ω_{1}^{M}.

2.3 Remark. To see that the order on ℙ_{max} is transitive, let j_{0} be an iteration witnessing that ⟨(M_{1}, I_{1}), a_{1}⟩ < ⟨(M_{0}, I_{0}), a_{0}⟩ and let j_{1} be an iteration witnessing that ⟨(M_{2}, I_{2}), a_{2}⟩ < ⟨(M_{1}, I_{1}), a_{1}⟩. Then j_{0} is an element if M_{1}, and it is not hard to check that j_{1}(j_{0}) witnesses that ⟨(M_{2}, I_{2}), a_{2}⟩ < ⟨(M_{0}, I_{0}), a_{0}⟩.

2. ℙ_{max} 15

2.4 Remark. As we shall see in Lemma 2.7, the requirement that the models satisfy MA_{ℵ_{1}}, along with condition (5) above, ensures that there is a unique iteration witnessing the order on each pair of comparable conditions. One can vary ℙ_{max} by removing condition (5) and the requirement that MA_{ℵ_{1}} holds, and replace a with a set of iterations of smaller models into M, as in the definition of the order, satisfying this uniqueness condition. Alternately, one can require that the models satisfy the statement ψ_{AC} (see Definition 6.1 and Remark 6.4), which implies that the image of any stationary, co-stationary subset of ω_{1} under an iteration determines the entire iteration.

2.5 Remark. Instead of using ideals on ω_{1}, we could use the stationary tower ℚ_{<δ} (see [19]) to produce the iterations giving the order on conditions. This gives us another degree of freedom in choosing our models, since in this case a small forcing extension of a condition is also a condition, roughly speaking. The resulting extension is essentially identical.

2.6 Remark. Given a real x, x^{†} (“x dagger”) is a real such that in L[x^{†}] there exists a transitive model M of ZFC containing ω_{1}^{V} ∪ {x} in which some ordinal countable in L[x^{†}] is a measurable cardinal (see [13]; this fact about x^{†} does not characterize it, but it is its only property that we require in this chapter). By [10], if there exists a measurable cardinal, then there is a partial order forcing that NS_{ω_{1}} is precipitous. By [20, 12], c.c.c. forcings preserve precipitousness of NS_{ω_{1}}. Essentially the same arguments show that if κ is a measurable cardinal and P is a c.c.c. forcing in the Col(ω, <κ)-extension, then there is a normal precipitous ideal on ω_{1} (which is κ) in the Col(ω, <κ) ∗ P -extension. By Lemmas 1.4 and 1.5, then, the statement that x^{†} exists for each real x implies that every real exists in the model M of some ℙ_{max} condition, and, by Lemma 2.8 below, densely many. However, the full strength of ℙ_{max} will require the consistency strength of significantly larger cardinals.

We will now prove two facts about iterations which are central to the ℙ_{max} analysis.

2.7 Lemma. Let ⟨(M, I), a⟩ be a condition in ℙ_{max} and let A be a subset of ω_{1}. Then there is at most one iteration of (M, I) for which A is the image of a. Furthermore, this iteration is in L[⟨(M, I), a⟩, A], if it exists.

Proof. The consequence of MA_{ℵ_{1}} that we need is known as almost disjoint coding [11]. This says that if Z = {z_{α} : α < ω_{1}} is a collection of infinite subsets of ω whose pairwise intersections are finite (i.e., Z is an almost disjoint family), then for every B ⊆ ω_{1} there exists a y ⊆ ω such that for all α < ω_{1}, α ∈ B if and only if y ∩ z_{α} is infinite.

Fix a real x in M such that ω_{1}^{M} = ω_{1}^{L[a,x]}, and let

Z = ⟨z_{α} : α < ω_{1}^{M}⟩

be the almost disjoint family defined recursively from the constructibility order in L[a, x] on P(ω)^{L[a,x]} by letting z_{α} be the (constructibly, in L[a, x]) least infinite subset of ω almost disjoint from each z_{β} (β < α).

Suppose that

I = ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩

and

I′ = ⟨M′_{α}, G′_{β}, j′_{αδ} : β < α ≤ δ ≤ γ′⟩

are two iterations of (M, I) such that j_{0γ}(a) = A = j′_{0γ′}(a). Then j_{0γ}(Z) = j′_{0γ′}(Z) (again, this uses Remark 1.11 to see that the constructibility order on reals in L[A, x] is computed correctly in M_{γ} and M′_{γ′}).

Since j_{0γ}(ω_{1}^{M}) = sup(j_{0γ}(a)) = sup(j′_{0γ′}(a)) = j′_{0γ′}(ω_{1}^{M}), if either iteration is an initial segment of the other, then the two iterations are the same. Supposing that this is not the case, let γ̄ be the length of the shortest initial segment of I which is not an initial segment of I′ and let γ^{∗} be the length of the shortest initial segment of I′ which is not an initial segment of I. Then γ̄ and γ^{∗} are both successor ordinals, and since the iterations up to their predecessors must be the same, they are both equal to η + 1 for some ordinal η, with M_{η} = M′_{η}.

Note that since j_{0γ}(Z) = j′_{0γ′}(Z), and the critical points of j_{(η+1)γ} and j′_{(η+1)γ′} are both greater than ω_{1}^{M_{η}}, j_{0(η+1)}(Z)_{ω_{1}^{Mη}} = j′_{0(η+1)}(Z)_{ω_{1}^{Mη}}. Let

⟨z_{α} : α < ω_{1}^{M_{η}}⟩ list the members of j_{0η}(Z) (and note that this is consistent with the definition of Z above). We now derive a contradiction by showing that G_{η} and G′_{η} are the same.

For every B ∈ P(ω_{1})^{M_{η}}, B ∈ G_{η} if and only if ω_{1}^{M_{η}} ∈ j_{η(η+1)}(B) and B ∈ G′_{η} if and only if ω_{1}^{M_{η}} ∈ j′_{η(η+1)}(B) . Fixing such a B, let y ∈ P(ω)^{M_{η}}

be such that for all α ∈ ω_{1}^{M_{η}}, y ∩ z_{α} is infinite if and only if α ∈ B. Then ω_{1}^{M_{η}} ∈ j_{η(η+1)}(B) if and only if j_{0(η+1)}(Z)_{ω_{1}^{Mη}} ∩ y is infinite, which holds if and only if j′_{0(η+1)}(Z)_{ω_{1}^{Mη}} ∩ y is infinite, which holds if and only if

ω_{1}^{M_{η}} ∈ j′_{η(η+1)}(B). Thus G_{η} and G′_{η} are the same.

For the last part of the lemma, note that the argument just given gives a definition for each G_{α} in terms of A, x and the iteration up to α. ⊣

One consequence of Lemma 2.7 is that, if G ⊂ ℙ_{max} is an L(ℝ)-generic filter, and A = ⋃{a | ⟨(M, I), a⟩ ∈ G}, then L(ℝ)[G] = L(ℝ)[A]. Therefore, the ℙ_{max} extension of L(ℝ) satisfies the sentence “V = L(P(ω_{1}))” (see the discussion at the beginning of Section 5).

2.8 Lemma. (ZFC^{∘}) If (M, I) is a pre-condition in ℙ_{max} and J is a normal ideal on ω_{1} then there exists an iteration j: (M, I) → (M^{∗}, I^{∗}) such that j(ω_{1}^{M}) = ω_{1} and I^{∗} = J ∩ M^{∗}.

Proof. First let us note that if ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩ is any iteration of (M, I), then j_{0ω_{1}}(I) ⊆ J ∩ M_{ω_{1}}. To see this, note that if E ∈ j_{0ω_{1}}(I), then E ∈ M_{ω_{1}} and E = j_{αω_{1}}(E′) for some α < ω_{1} and E′ ∈ j_{0α}(I). Then for all β ∈ [α, ω_{1}), j_{αβ}(E′) ∉ G_{β}, so ω_{1}^{M_{β}} ∉ E. Therefore, E is nonstationary, so E ∈ J by the normality of J.

Now, noting that J is a normal ideal, let {A_{iα} : i < ω, α < ω_{1}} be a collection of pairwise disjoint members of P(ω_{1}) \ J. We build an iteration ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩ by recursively choosing the G_{β}’s. As we do this, for each α < ω_{1} we let the set {B_{i}^{α} : i < ω} enumerate P(ω_{1})^{M_{α}}\j_{0α}(I). Given

⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩,

then, for some γ ≤ ω_{1}, if ω_{1}^{M_{γ}} ∈ A_{iα} for some i < ω and α ≤ γ, then (noting that there can be at most one such pair (i, α)) we let G_{γ} be any M_{γ}-generic filter for (P(ω_{1})/j_{0γ}(I))^{M_{γ}} with j_{αγ}(B_{i}^{α}) as a member. If ω_{1}^{M_{γ}} is not in A_{iα} for any i < ω and α ≤ γ, then we let G_{γ} be any M_{γ}-generic filter.

To see that this construction works, fix E ∈ P(ω_{1})^{M_{ω_{1}}} \ j_{0ω_{1}}(I). We need to see that E is not in J. We may fix i < ω and α < ω_{1} such that E = j_{αω_{1}}(B_{i}^{α}). Then F = (A_{iα} ∩ {ω_{1}^{M_{β}} : β ∈ [α, ω_{1})}) ⊆ E. Since F is the intersection of a club and set not in J, F is not in J, so E is not in J. ⊣

The construction in the proof of Lemma 2.8 appears repeatedly in the analysis of ℙ_{max}. In order to make our presentation of ℙ_{max} more modular (i.e., to avoid having to write out the proof of Lemma 2.8 repeatedly), we give the following strengthening of the lemma in terms of games. We note that the games defined here (and before Lemmas 3.5 and 5.2 and at the end of Section 10.2) are not part of Woodin’s original presentation of ℙ_{max}.

Suppose that (M, I) is a pre-condition in ℙ_{max}, let J be a normal ideal on ω_{1} and let B be a subset of ω_{1}. Let G((M, I), J, B) be the following game of length ω_{1} where Players I and II collaborate to build an iteration

⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩

of (M, I) of length ω_{1}. In each round α, if α ∈ B, then Player I chooses a set A_{α} in P(ω_{1})^{M_{α}} \ j_{0α}(I) and then Player II chooses an M_{α}-generic filter G_{α} contained in (P(ω_{1})/j_{0α}(I))^{M_{α}} with A_{α} ∈ G_{α}. If α ∉ B, then Player II chooses any M_{α}-generic filter G_{α} ⊆ (P(ω_{1})/j_{0α}(I))^{M_{α}}. After all ω_{1} many rounds have been played, Player I wins if j_{0ω_{1}}(I) = J ∩ M_{ω_{1}}.

The proof of Lemma 2.9 is almost identical to the proof of Lemma 2.8.

2.9 Lemma. (ZFC^{∘}) Suppose that (M, I) is a pre-condition in ℙ_{max}, let J be a normal ideal on ω_{1}, and let B be a subset of ω_{1}. Then Player I has a winning strategy in G((M, I), J, B) if and only if B ∉ J.

Using Remark 2.6 and Lemmas 2.7 and 2.8, we can show that ℙ_{max} is homogeneous and countably closed. By homogeneity we mean the following property: for each pair of conditions p_{0}, p_{1} in ℙ_{max} there exist conditions q_{0}, q_{1} such that q_{0} ≤ p_{0}, q_{1} ≤ p_{1} and the suborders of ℙ_{max} below the conditions q_{0} and q_{1} are isomorphic. The importance of this property is that it implies that the theory of the ℙ_{max} extension can be computed in the ground model.

2.10 Lemma. If x^{†} exists for each real x, then ℙ_{max} is homogeneous.

Proof. Let p_{0} = ⟨(M_{0}, I_{0}), a_{0}⟩ and p_{1} = ⟨(M_{1}, I_{1}), a_{1}⟩ be conditions in ℙ_{max}. By Remark 2.6, we can fix a pre-condition (N, J) with p_{0}, p_{1} ∈ H(ω_{1})^{N}. Applying Lemma 2.8 in N, we may fix iterations j_{0} : (M_{0}, I_{0}) → (M_{0}^{∗}, I_{0}^{∗}) and j_{1} : (M_{1}, I_{1}) → (M_{1}^{∗}, I_{1}^{∗}) in N such that I_{0}^{∗} = J ∩ M_{0}^{∗} and I_{1}^{∗} = J ∩ M_{1}^{∗}. Letting a_{0}^{∗} = j_{0}(a_{0}) and a_{1}^{∗} = j_{1}(a_{1}), then,

q_{0} = ⟨(N, J), a_{0}^{∗}⟩

and

q_{1} = ⟨(N, J), a_{1}^{∗}⟩

are conditions in ℙ_{max} and j_{0} and j_{1} witness that q_{0} ≤ p_{0} and q_{1} ≤ p_{1} respectively.

Now, if q′_{0} = ⟨(N′, J′), a′⟩ is a condition below q_{0}, then there is an iteration j′: (N, J) → (N^{∗}, J^{∗}) witnessing this. Then a′ = j′(a_{0}^{∗}), and q′_{1} = ⟨(N′, J′), j′(a_{1}^{∗})⟩ is a condition below q_{1}. Let π be the map with domain the suborder of ℙ_{max} below q_{0} which sends each ⟨(N′, J′), a′⟩ to the corresponding ⟨(N′, J′), j′(a_{1}^{∗})⟩ as above. By Lemma 2.7, this map is an isomorphism between the suborders below q_{0} and q_{1} respectively. ⊣

In order to show that ℙ_{max} is countably closed, we must define a new class of iterations.

3. Sequences of models and countable closure

For each i < ω, let p_{i} = ⟨(M_{i}, I_{i}), a_{i}⟩ be a ℙ_{max} condition, and for each such i let j_{i(i+1)} : (M_{i}, I_{i}) → (M_{i}^{∗}, I_{i}^{∗}) be an iteration witnessing that p_{i+1} < p_{i}. Let {j_{ik} : i ≤ k < ω} be the commuting family of embeddings generated by the j_{i(i+1)}’s. Let a = ⋃{a_{i} : i < ω}. By Lemma 2.7, for each i < ω there is a unique iteration j_{iω} : (M_{i}, I_{i}) → (N_{i}, J_{i}) sending a_{i} to a. Since each (M_{i}, I_{i}) is iterable, each N_{i} is wellfounded, and the structure (⟨(N_{i}, J_{i}) : i < ω⟩, a) satisfies the following definition.

3.1 Definition. A limit sequence is a pair (⟨(N_{i}, J_{i}) : i < ω⟩, a) such that the following hold for all i < ω:

1. N_{i} is a countable transitive model of ZFC^{∘}+ MA_{ℵ_{1}},

2. J_{i} ∈ N_{i} and in N_{i}, J_{i} is a normal ideal on ω_{1}, 3. ω_{1}^{N_{i}} = ω_{1}^{N_{0}},

4. for all k < i, N_{k} ∈ H(ω_{2})^{N_{i}},

5. for all k < i, J_{k} = J_{i} ∩ N_{k},

6. a ∈ P(ω_{1})^{N_{0}},

7. there exists an x ∈ P(ω)^{N_{0}} such that ω_{1}^{N_{0}} = ω_{1}^{L[a,x]}.

A structure ⟨(N_{i}, J_{i}) : i < ω⟩ is a pre-limit sequence if there exists a set a such that (⟨(N_{i}, J_{i}) : i < ω⟩, a) is a limit sequence.

If we write a sequence as ⟨N_{k} : k < ω⟩ (as we will in Section 10.1), the ideals are presumed to be the nonstationary ideal on ω_{1}.

If p_{i} (i < ω) is a descending sequence of ℙ_{max} conditions, then the limit sequence corresponding to p_{i} (i < ω) is the structure

(⟨(N_{i}, J_{i}) : i < ω⟩, a)

defined above. Note that in this case each ⟨(N_{i}, J_{i}), a⟩ is a condition in ℙ_{max}.

If ⟨(N_{i}, J_{i}) : i < ω⟩ is a pre-limit sequence, then a filter

G ⊂ ⋃{P(ω_{1})^{N_{i}} \ J_{i} : i < ω}

is a ⋃{N_{i} : i < ω}-normal ultrafilter for the sequence if for every regressive function f on ω_{1}^{N_{0}} in any N_{i}, f is constant on some member of G. Given such G and ⟨(N_{i}, J_{i}) : i < ω⟩, we form the ultrapower of the sequence by letting N_{i}^{∗} be the ultrapower of N_{i} formed from G and all functions f: ω_{1}^{N_{0}} → N_{i} existing in any N_{k} (this ensures that the image of each N_{i} in the ultrapower of each N_{k} (k > i) is the same as the ultrapower of N_{i}). As usual, we identify the transitive parts of each N_{i}^{∗} with their transitive collapses. If (for each i < ω) we let j_{i}^{∗} be the induced embedding of N_{i} into N_{i}^{∗} then for each i < k < ω, j_{i}^{∗} = j_{k}^{∗}↾N_{i}, so we can let j^{∗} = ∪{j_{i}^{∗} : i < ω} be the embedding corresponding to the ultrapower of the pre-limit sequence.

3.2 Definition. Let ⟨(N_{i}, J_{i}) : i < ω⟩ be a pre-limit sequence, and let γ be an ordinal less than or equal to ω_{1}. An iteration of ⟨(N_{i}, J_{i}) : i < ω⟩ of length γ consists of pre-limit sequences ⟨(N_{i}^{α}, J_{i}^{α}) : i < ω⟩ (α ≤ γ), normal ultrafilters G_{α} (α < γ) and a commuting family of embeddings j_{αβ} (α ≤ β ≤ γ) such that

• ⟨(N_{i}^{0}, J_{i}^{0}) : i < ω⟩ = ⟨(N_{i}, J_{i}) : i < ω⟩

• for all α < γ, G_{α} ⊆ ⋃{P(ω_{1})^{N_{i}^{α}} \ J_{i}^{α} : i < ω} is a normal ultrafilter for the sequence ⟨(N_{i}^{α}, J_{i}^{α}) : i < ω⟩, and j_{α(α+1)} is the corresponding embedding, • for each limit ordinal β ≤ γ, ⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩ is the direct limit of

the system {⟨(N_{i}^{α}, J_{i}^{α}) : i < ω⟩, j_{αδ} : α ≤ δ < β} and for each α < β j_{αβ} is the induced embedding.

As with iterations of single models, we sometimes describe an iteration of a pre-limit sequence by fixing only the initial sequence, the final sequence and the embedding between them. An iterate of a pre-limit sequence p̄ is a pre-limit sequence appearing in an iteration of p̄. If every iterate of a pre-limit sequence is wellfounded, then the sequence is iterable.

By Lemma 1.7 and Corollary 1.8, pre-limit sequences derived from descending chains {⟨(M_{i}, I_{i}), a_{i}⟩ : i < ω} in ℙ_{max} satisfy the hypotheses of the following lemma, letting x_{i} be any real in M_{i+1} coding M_{i}. Yet another way to vary ℙ_{max} is to replace the model M in the definition of ℙ_{max} conditions with sequences satisfying this hypothesis. This approach is used for the order ℚ_{max}^{∗} defined in section 10.1.

3.3 Lemma. Suppose that p̄ = ⟨(N_{i}, J_{i}) : i < ω⟩ is a pre-limit sequence, and suppose that for each i < ω there is a real x_{i} ∈ N_{i+1} such that x_{i}^{#} ∈ N_{i+1} and

• the least x_{i}-indiscernible above ω_{1}^{N_{0}} is greater than the ordinal height of N_{i},

• every club subset of ω_{1}^{N_{0}} in N_{i} contains a tail of the x_{i}-indiscernibles below ω_{1}^{N_{0}}.

Then p̄ is iterable.

Proof. First we will show that any iterate of p̄ is wellfounded if its version of ω_{1} is wellfounded. Then we will show that the ω_{1} of each iterate of p̄ is wellfounded.

For the first part, if ⟨(N_{i}^{∗}, J_{i}^{∗}) : i < ω⟩ is an iterate of p̄, then by elementarity the ordinals of each N_{i}^{∗} embed into the least x_{i}-indiscernible above ω_{1}^{N_{0}^{∗}}. So, if ω_{1}^{N_{0}^{∗}} is actually an ordinal (i.e., is wellfounded), then N_{i+1}^{∗} constructs this next x_{i}-indiscernible correctly, and so N_{i}^{∗} is wellfounded.

We prove the second part by induction on the length of the iteration, noting that the limit case follows immediately, and the successor case follows from the case of an iteration of length 1. What we want to see is that if G is a normal ultrafilter for p̄ and j is the induced embedding, then j(ω_{1}^{N_{0}}) = ⋃{N_{i} ∩ Ord : i < ω}. Notice that for each x_{i}, if f_{i}: ω_{1}^{N_{0}} → ω_{1}^{N_{0}} is defined by letting f_{i}(α) be the least x_{i}-indiscernible above α, then j(f_{i})(ω_{1}^{N_{0}}) is the least indiscernible of x_{i} above ω_{1}^{N_{0}}. Thus

j(ω_{1}^{N_{0}}) ≥ sup{j(f_{i})(ω_{1}^{N_{0}}) : i < ω} = ⋃{N_{i} ∩ Ord : i < ω}.

For the other direction, let h: ω_{1}^{N_{0}} → ω_{1}^{N_{0}} be a function in some N_{i}. Then the closure points of h contain a tail of the x_{i}-indiscernibles, which means that f_{i} > h on a tail of the ordinals below ω_{1}^{N_{0}}, so [f_{i}]_{G} > [h]_{G}. Thus j(ω_{1}^{N_{0}}) = ⋃{N_{i} ∩ Ord : i < ω}. ⊣

The following lemma has essentially the same proof as Lemma 2.8, and shows (given that x^{†} exists for each real x) that ℙ_{max} is countably closed. The point is that if ⟨p_{i} : i < ω⟩ is a descending sequence of ℙ_{max} conditions, letting p̄ = (⟨(N_{i}, J_{i}) : i < ω⟩, a) be the limit sequence corresponding to ⟨p_{i} : i < ω⟩, if (M, I) is a ℙ_{max} pre-condition with {p_{i} : i < ω}, p̄ ∈ H(ω_{1})^{M}, then by letting j^{∗} be an iteration of p̄ resulting from applying Lemma 3.4 inside of M, the embedding j^{∗}(j_{iω}) (where j_{iω} is as defined in the first paragraph of this section) witnesses that ⟨(M, I), j^{∗}(a)⟩ is below p_{i} in ℙ_{max}, for each i < ω.

3.4 Lemma. (ZFC^{∘}) Suppose that ⟨(N_{i}, J_{i}) : i < ω⟩ is an iterable pre-limit sequence, and let I be a normal ideal on ω_{1}. Then there is an iteration

j^{∗}: ⟨(N_{i}, J_{i}) : i < ω⟩ → ⟨(N_{i}^{∗}, J_{i}^{∗}) : i < ω⟩

such that j^{∗}(ω_{1}^{N_{0}}) = ω_{1} and J_{i}^{∗} = I ∩ N_{i}^{∗} for each i < ω.

Suppose that ⟨(N_{i}, J_{i}) : i < ω⟩ is an iterable pre-limit sequence, let I be a normal ideal on ω_{1}, and let B be a subset of ω_{1}. Let

G_{ω}(⟨(N_{i}, J_{i}) : i < ω⟩, I, B)

be the following game of length ω_{1} where Players I and II collaborate to build an iteration of ⟨(N_{i}, J_{i}) : i < ω⟩ consisting of pre-limit sequences ⟨(N_{i}^{α}, J_{i}^{α}) : i < ω⟩ (α ≤ ω_{1}), normal ultrafilters G_{α} (α < ω_{1}) and a family of embeddings j_{αβ} (α ≤ β ≤ ω_{1}), as follows. In each round α, let

X_{α} = ⋃{P(ω_{1})^{N_{i}^{α}} \ J_{i}^{α} : i < ω}.

If α ∈ B, then Player I chooses a set A ∈ X_{α}, and then Player II chooses a ⋃{N_{i}^{α} : i < ω}-normal filter G_{α} contained in X_{α} with A ∈ G_{α}. If α is not in B, then Player II chooses any ⋃{N_{i}^{α} : i < ω}-normal filter G_{α} contained in X_{α}. After all ω_{1} many rounds have been played, Player I wins if J_{i}^{ω_{1}} = I ∩ N_{i}^{ω_{1}} for each i < ω.

Lemma 3.4 can be rephrased in terms of games as follows.

3.5 Lemma. (ZFC^{∘}) Suppose that ⟨(N_{i}, J_{i}) : i < ω⟩ is an iterable pre-limit sequence, let I be a normal ideal on ω_{1} and let B be a subset of ω_{1}. Then Player I has a winning strategy in G_{ω}(⟨(N_{i}, J_{i}) : i < ω⟩, I, B) if and only if B ∉ I.

At this point, we have gone as far with the ℙ_{max} analysis as daggers can take us.

4. Generalized iterability

The following definition gives a generalized iterability property with respect to a given set of reals. In the ℙ_{max} analysis, these sets of reals often code ℙ_{max}-names for sets of reals.

4.1 Definition. Let A be a set of reals. If M is a transitive model of ZFC^{∘} and I is an ideal on ω_{1}^{M} which is normal and precipitous in M, then the pair (M, I) is A-iterable if

• (M, I) is iterable,

• A ∩ M ∈ M,

• j(A ∩ M) = A ∩ M^{∗} whenever j: (M, I) → (M^{∗}, I^{∗}) is an iteration of (M, I).

4.2 Remark. The definition of A-iterability in [37] is more general than this one, in ways which we won’t require.

In order to achieve the full effects of forcing with ℙ_{max} over a given model (for now we will deal with L(ℝ)) we need to see (and in fact it is enough to see) that for each A ⊆ ℝ in the model there exists a ℙ_{max} pre-condition (M, I) such that

• (M, I) is A-iterable,

• ⟨H(ω_{1})^{M}, A ∩ M⟩ ≺ ⟨H(ω_{1}), A⟩.

As it turns out, the existence of such a condition for each A ⊆ ℝ in L(ℝ) is equivalent to the statement that the Axiom of Determinacy holds in L(ℝ) (see pages 285-290 of [37]).

There are two basic approaches to studying the ℙ_{max} extension. One can think of V as being a model of some form of determinacy, and use determinacy to analyze the ℙ_{max} forcing construction and its corresponding extension. Alternately, one can assume that Choice holds and certain large cardinals exist and use these large cardinals to analyze the ℙ_{max} extension of some inner model of ZF satisfying determinacy. Accordingly, the existence of A-iterable conditions (for a given set A) can be derived from determinacy or from large cardinals. We give here an example of each method, quoting some standard facts which we will briefly discuss.

The proof from large cardinals uses weakly homogeneous trees. Very briefly, a countably complete tower is a sequence of measures ⟨σ_{i} : i < ω⟩ such that each σ_{i} is a measure on Z^{i} for some fixed underlying set Z and for every sequence {A_{i} ∈ i < ω} of sets such that each A_{i} ∈ σ_{i} there exists a function g ∈ Z^{ω} such that g↾i ∈ A_{i} for all i < ω. Given a tree T ⊂ (ω × Z)^{<ω}, for some set Z, the projection of T is the set

p[T] = {y ∈ ω^{ω} | ∃z ∈ Z^{ω} ∀i < ω(x↾i, z↾i) ∈ T }.

Given a set Z and a cardinal δ, a tree T ⊆ ω ×Z is δ-weakly homogeneous if there exists a countable family Σ of δ-complete measures on Z^{<ω} such that for each x ∈ ω^{ω}, x ∈ p[T] if and only if there exists a sequence of measures {σ_{i} : i < ω} ⊆ Σ such that

• for all i < ω, {z ∈ Z^{i} | (x↾i, z) ∈ T } ∈ σ_{i},

• ⟨σ_{i} : i < ω⟩ forms a countably complete tower.

A set of reals A is δ-weakly homogeneously Suslin if there exists a δ-weakly homogeneous tree T whose projection is A, and weakly homogeneously Suslin if it is δ-weakly homogeneously Suslin for some uncountable ordinal δ. The following fact is standard.

4.3 Theorem. Let θ be a regular cardinal, suppose that T ∈ H(θ) is a weakly homogeneous tree on ω × Z, for some set Z. Let δ ≥ 2^{ω} be an ordinal such that there exists a countable collection Σ of δ^{+}-complete measures witnessing the weak homogeneity of T. Then for every elementary submodel Y of H(θ) of cardinality less than δ with T, Σ ∈ Y there is an elementary submodel X of H(θ) containing Y such that X ∩ δ = Y ∩ δ, and such that, letting S be the image of T under the transitive collapse of X, p[S] = p[T].

Proof. Fixing θ, T, Σ and δ as in the statement of the theorem, the theorem follows from the following fact. Suppose that Y is an elementary submodel of H(θ) with T, Σ ∈ Y and |Y | < δ, and fix x ∈ p[T]. Fix a countably complete tower {σ_{i} : i < ω} ⊆ Σ such that for all i < ω, {a ∈ Z^{i} : (x↾i, a) ∈ T } ∈ σ_{i}, and for each i < ω, let A_{i} = ⋂(σ_{i} ∩Y). Then since {σ_{i} : i < ω} is countably complete, there exists a z ∈ Z^{ω} such that for all i < ω, z↾i ∈ A_{i}. Then the pair (x, z) forms a path through T, and, letting

Y [z] = {f(z↾i) | i < ω ∧ f: Z^{i} → H(θ) ∧ f ∈ Y },

Y [z] is an elementary submodel of H(θ) containing Y and {z↾i : i < ω}, and, since each σ_{i} is δ^{+}-complete, Y ∩ δ = Y [z] ∩ δ. Repeated application of this fact for each real in the projection of T proves the theorem. ⊣

Proofs of the following facts about weakly homogeneous trees and weakly homogeneously Suslin sets of reals appear in [19]. Some of these facts follow directly from the definitions, and none are due to the author. Theorem 4.6 derives ultimately from [21].

4.4 Fact. For every cardinal δ, the collection of δ-weakly homogeneously Suslin sets of reals is closed under countable unions and continuous images.

4.5 Theorem. (Woodin) If δ is a limit of Woodin cardinals and there exists a measurable cardinal above δ then every set of reals in L(ℝ) is <δ-weakly homogeneously Suslin (i.e., γ-weakly homogeneously Suslin for all γ < δ).

4.6 Theorem. If δ is a cardinal and T is a δ-weakly homogeneous tree, then there is a tree S such that p[T] = ω^{ω} \ p[S] in all forcing extensions by partial orders of cardinality less than δ (including the trivial one).

4.7 Theorem. (Woodin) If δ is a Woodin cardinal and A is a δ^{+}-weakly homogeneously Suslin set of reals, then the complement of A is <δ-weakly homogeneously Suslin.

Note also that if S and T are trees whose projections are disjoint, then they remain disjoint in all forcing extensions, as there is a ranking function on the tree of attempts to build a real in both projections. This fact plus Theorem 4.6 gives the following corollary.

4.8 Corollary. If δ is a cardinal and T_{0} and T_{1} are δ-weakly homogeneous trees with the same projection, then T_{0} and T_{1} still have the same projection in all forcing extensions by forcings of cardinality less than δ.

Given a set of reals A, a set of reals B is projective in A if it can be defined by a projective formula (i.e., all unbounded quantifiers ranging over reals) with A as a parameter. Fact 4.4 and Theorem 4.7 together imply that if δ is a limit of Woodin cardinals then the set of <δ-weakly homogeneously Suslin sets of reals is projectively closed.

The following theorem is a generalized existence result which is useful in analyzing variations of ℙ_{max}.

4.9 Theorem. Let γ be a strongly inaccessible cardinal, let A be a set of reals, and suppose that θ is a strong limit cardinal of cofinality greater than ω_{1} such that every set of reals projective in A is γ^{+}-weakly homogeneously Suslin as witnessed by a tree and a set of measures in H(θ). Let X be a countable elementary submodel of H(θ) with γ, A ∈ X, and let M be the transitive collapse of X ∩ H(γ). Let N be any forcing extension of M in which there exists a normal precipitous ideal I on ω_{1}^{N}. Let j: (N, I) → (N^{∗}, I^{∗}) be any iteration of (N, I). Then

• N^{∗} is wellfounded,

• N ∩ A ∈ N,

• j(N ∩ A) = N^{∗} ∩ A,

• ⟨H(ω_{1})^{N^{∗}}, A ∩ N^{∗}, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Proof. Let {A_{i} : i < ω} be a listing in X of the sets of reals projective in A (with A_{0} = A), and let {T_{i} : i < ω} and {Σ_{i} : i < ω} be sets in X such that each T_{i} is a γ^{+}-weakly homogeneous tree (as witnessed by Σ_{i}) projecting to A_{i}. By the proof of Theorem 4.3, there is an elementary submodel Y of H(θ) containing X such that X ∩ H(γ) = Y ∩ H(γ) and such that, letting M^{+} be the transitive collapse of Y, and letting, for each i < ω, S_{i} be the image of T_{i} under this collapse, p[S_{i}] = A_{i}. Note that since there are sets projective in A which are not the projections of countable trees, ω_{1} ⊆ M^{+}. Now, let N be a forcing extension of M with I a normal precipitous ideal on ω_{1}^{N} in N, and let N^{+} be the corresponding extension of M^{+}. Let j: (N, I) → (N^{∗}, I^{∗}) be an iteration of (N, I). By Lemmas 1.4 and 1.5, j extends to an iteration of (N^{+}, I) (which we will also call j), and N^{∗} is wellfounded. Furthermore, for each i < ω there is a j < ω such that S_{i} and S_{j} project to complements. Then

• p[S_{i}] ⊆ p[j(S_{i})],

• p[S_{j}] ⊆ p[j(S_{j})],

• p[j(S_{i})] ∩ p[j(S_{j})] = ∅,

which means that p[S_{i}] = p[j(S_{i})], so j(N ∩ A_{i}) = N^{∗} ∩ A_{i}.

To verify the last part of the theorem, noting that the theory of H(ω_{1}) is recursive in the theory of ℝ (see Remark 1.10), we need to see that for each formula ϕ (with a predicate for A) with all unbounded quantifiers ranging over the reals, if A_{i} is the set of reals satisfying ϕ, then in N^{+}, S_{i} projects to the set of reals satisfying ϕ (i.e., that this relationship is preserved in the forcing extension from M^{+} to N^{+}). This is easily shown by induction on the complexity of formulas, using Theorem 4.6 and Corollary 4.8. Note first of all that this holds for all such ϕ whose quantifiers are all bounded, as ⟨H(ω_{1})^{N^{+}}, A ∩ N^{+}, ∈⟩ is elementary in ⟨H(ω_{1}), A, ∈⟩ for such formulas simply by virtue of being a substructure. The verification for ∧ and ∃ follows from Corollary 4.8. Letting γ^{∗} be the image of γ under the collapse of Y, fix integers i and j and formulas ϕ_{i} and ϕ_{j} such that S_{i} and S_{j} project to the sets of reals satisfying ϕ_{i} and ϕ_{j} respectively in both M^{+} and N^{+}. Then, working in M^{+}, we can directly construct γ^{∗}-weakly homogeneous trees T and T′ projecting to the sets of reals satisfying ∃xϕ_{i}(x) and ϕ_{i} ∧ ϕ_{j} respectively in both M^{+} and N^{+}. Then, letting k and k′ be integers such that S_{k} and S_{k′} project in M^{+} to the sets of reals satisfying ∃xϕ_{i}(x) and ϕ_{i}∧ϕ_{j} respectively, Corollary 4.8 gives us that p[T] = p[S_{k}] and p[T] = p[S_{k′}] in N^{+}. The verification for ¬ follows similarly from Corollary 4.8 and Theorem 4.6. ⊣

Alternately, we can derive the existence of A-iterable conditions from determinacy. The proof from determinacy requires the following fact: if AD holds and Z is a set of ordinals, then there is an inner model of ZFC containing the ordinals with Z as a member in which some countable ordinal is a measurable cardinal. Note that a tree of finite sequences of ordinals can easily be coded as a set of ordinals (see [25], for instance).

The following unpublished theorem of Woodin is more than sufficient, but in the spirit of completeness we will not use it, since its proof is well beyond the scope of this chapter. Given a model M and a set X, HOD_{X}^{M} is the class of hereditarily ordinal definable sets (using X as a parameter), as computed in M. It is a standard fact that this model satisfies ZFC.

4.10 Theorem. ( AD) Suppose that Z is a set of ordinals. Then there exists a real x such that for all reals z with x ∈ L[Z, z], ω_{2}^{L[Z,z]} is a Woodin cardinal in HOD_{{Z}}^{L[Z,z]}.

The following theorem is sufficient for our purposes.

4.11 Theorem. ( AD) For every subset Z of L, there is an inner model N of ZFC containing {Z} and the ordinals such that some countable ordinal is measurable in N.

Proof. For each increasing function f: ω → ω_{1}, let s(f) be the supremum of the range of f, and let F (f) be the filter on s(f) consisting of all subsets of s(f) which contain all but finitely many members of the range of f. For each such f, let N(f) be the inner model L[Z, F (f)]. We will find an f such that the restriction of F (f) to N(f) is a countably complete ultrafilter in N(f), i.e., such that

(+) for every function g from s(f) to ω in N(f), g is constant on a set in F (f).

Note the following facts.

1. If f_{0} and f_{1} are functions from ω to ω_{1} whose ranges are the same modulo a finite set, then F (f_{0}) = F (f_{1}) and so not only are the models N(f_{0}) and N(f_{1}) the same, but their canonical wellorderings are the same also.

2. Using the canonical wellordering of each N(f), there is a function G choosing for each increasing f: ω → ω_{1} a function G(f): s(f) → ω failing condition (+) above, if one exists.

The key consequence of AD is the partition property ω_{1} → (ω_{1})_{ω^{ω}}^{ω} (see [8] or pages 391-396 of [13]), which says that for every function from the set of increasing ω-sequences from ω_{1} to ω^{ω} (the set of functions from ω to ω) there is an uncountable E ⊆ ω_{1} such that the function is constant on the set of increasing ω-sequences from E.

Now, for each increasing f: ω → ω_{1}, let P (f) be the constant function 0 if F (f) satisfies conditions (+) in N(f). If f fails condition (+) in N(f), then let P (f)(0) be 1 and let P (f)(n+1) = G(f)(f(n)) for all n ∈ ω. Let E be an uncountable subset of ω_{1} such that P (f) is the same for all increasing f: ω → E. We show that the constant value is the constant function 0. If the constant value corresponds to a failure of (+), then there is an i: ω → ω such that for all increasing f: ω → E, for all n ∈ ω, G(f)(f(n)) = i(n).

Then i must be constant, since if n ∈ ω is such that i(n) ≠ i(0), then if f is an increasing function from ω to E and g: ω → E is defined by letting g(m) = f(m + n), then G(f) ≠ G(g), contradicting the fact that F (f) = F (g). But if i is constant, then for every increasing f: ω → E, G(f) is constant on a set in F (f), contradicting the failure of (+). ⊣

4.12 Theorem. Suppose that the Axiom of Determinacy holds in L(ℝ), and let A be a set of reals in L(ℝ). Then there exists a condition ⟨(M, I), a⟩ in ℙ_{max} such that

• A ∩ M ∈ M,

• ⟨H(ω_{1})^{M}, A ∩ M⟩ ≺ ⟨H(ω_{1}), A⟩,

• (M, I) is A-iterable.

• if M^{+} is any forcing extension of M and J is a normal precipitous ideal on ω_{1}^{M^{+}} in M^{+} then A ∩ M^{+} ∈ M^{+} and (M^{+}, J) is A-iterable, and if j: (M^{+}, J) → (M^{∗}, J^{∗}) is any iteration of (M^{+}, J), then

⟨H(ω_{1})^{M^{∗}}, A ∩ M^{∗}⟩ ≺ ⟨H(ω_{1}), A⟩.

Proof. Work in L(ℝ). If there is an A ⊆ ℝ which is a counterexample to the statement of the theorem, then we may assume that there exists such an A which is Δ ∼_{1}^{2}. This follows from the Solovay Basis Theorem (see [8]), which says (in ZF) that every nonempty Σ ∼_{1}^{2} collection of sets of reals has a member which is Δ ∼_{1}^{2}. We give a quick sketch of the proof. Note first of all that for any ordinal α the transitive collapse any elementary submodel of L_{α}(ℝ) containing ℝ is a set of the form L_{β}(ℝ) for some ordinal β ≤ α. Now, if α is any ordinal, there exist (in L(ℝ)) an elementary submodel X of L_{α}(ℝ) containing ℝ and a surjection π: ℝ → X, so if α is the least ordinal such that a member of a given Σ ∼_{1}^{2} set exists in L_{α+1}(ℝ) then there is a surjection (in L(ℝ)) from ℝ onto L_{α+1}(ℝ), and a formula ϕ and a real x such that some member of the set is defined over L_{α}(ℝ) by ϕ from x. By the minimality of α, that member has Σ ∼_{1}^{2} and Π ∼_{1}^{2} definitions using x and incorporating ϕ.

Towards a contradiction, fix a Δ ∼_{1}^{2} counterexample A. By [22], the pointclass Σ_{1}^{2} has the scale property in L(ℝ), which means that every subset of ℝ×ℝ which is Δ_{∼}_{1}^{2} in L(ℝ) is the projection of a tree in L(ℝ) on the product of ω and some ordinal, and can be uniformized by a function which is Δ_{∼}_{1}^{2} in L(ℝ). (We refer the reader to [8, 13, 24] for a discussion of scales and their corresponding trees. Briefly, if B ⊆ ℝ × ℝ is the projection of a tree T on ω × ω × γ (for some ordinal γ) then for each real x such that there exists a y with (x, y) in B, we can recursively define functions f(x): ω → ω and g(x): ω → γ as follows: if (m, α) is the lexicographically least pair in ω × γ such that there exist a real y and a function a: ω → γ such that • y extends f(x)↾n and y(n) = m,

• a extends g(x)↾n and a(n) = α,

• (x, y, a) is a path through T,

then f(x)(n) = m and g(x)(n) = α. Then f uniformizes B, and if T is the tree corresponding to a Σ_{∼}_{1}^{2} scale on B, then f is Δ ∼_{1}^{2}.) Now, Δ ∼_{1}^{2} is closed under complements, projections and countable unions, so there exists a Δ ∼_{1}^{2} set B ⊆ ℝ × ℝ such that whenever F: ℝ → ℝ is a function uniformizing B and N is a transitive model N of ZF closed under F,

⟨H(ω_{1})^{N}, A ∩ N, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Fix such B and F, both Δ ∼_{1}^{2}. Let S, S^{∗}, T, T^{∗} be trees (on ω × γ, for some ordinal γ) in L(ℝ) projecting to A, the complement of A, F and the complement of F respectively. Note that any transitive model of ZF with T as a member is closed under F.

Now by Theorem 4.11, we may fix a transitive model N of ZFC and a countable ordinal γ such that N contains the ordinals, S, S^{∗}, T and T^{∗} are elements of N and γ is a measurable cardinal in N. Since N ⊆ L(ℝ) and L(ℝ) satisfies AD, ω_{1}^{V} is a limit of strongly inaccessible cardinals in N. Let δ be any strongly inaccessible cardinal in N between γ and ω_{1}^{V}. Recall (Remark 2.6) that if we choose an N-generic filter G for the forcing consisting of Col(ω, <γ) followed by the standard c.c.c. iteration to make Martin’s Axiom hold, as defined in N, then if we let I be the normal ideal generated by an ideal in N dual to a fixed normal measure on γ in N, I is a precipitous ideal in N[G] and (N_{δ}[G], I) is iterable, by Lemmas 1.4 and 1.5. It suffices now to fix a forcing extension M^{+} of N_{δ}[G] in which there exists a normal precipitous ideal J on ω_{1}^{M^{+}} and to show that the second part of the conclusion of the theorem holds for M^{+} and J. Let N^{+} be the corresponding forcing extension of N[G]. Since S ∈ N^{+}, A∩M^{+} ∈ M^{+}. Fix an iteration j: (M^{+}, J) → (M^{∗}, J^{∗}). By Lemma 1.5 there is an iteration j^{∗}: (N^{+}, J) → (N^{∗}, J^{∗}) such that j^{∗}↾M^{+} = j. Now, p[S] ⊆ p[j^{∗}(S)] and p[S^{∗}] ⊆ p[j^{∗}(S^{∗})], and further, by absoluteness p[j^{∗}(S)] ∩ p[j^{∗}(S^{∗})] = ∅, so p[S] = p[j^{∗}(S)]. Similarly, p[T] = p[j^{∗}(T)]. Then N^{∗} is closed under F, so we have that

⟨H(ω_{1})^{M^{∗}}, A ∩ M^{∗}, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Furthermore, j(A ∩ M^{+}) = p[j^{∗}(S)] ∩ M^{∗}, so j(A ∩ M^{+}) = A ∩ M^{∗}. This shows that A is not in fact a counterexample to the statement of the theorem. ⊣

Suppose that A is a set of reals and x is a real coding a condition p in ℙ_{max} by some recursive coding, and let B be the set of reals coding members of A × {x}. Then if (M, I) is a B-iterable pair such that

⟨H(ω_{1})^{M}, B ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), B, ∈⟩,

then (M, I) is A-iterable and p ∈ H(ω_{1})^{M}. Therefore, the existence, for each A ⊆ ℝ in L(ℝ), of an A-iterable pair (M, I) such that

⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩

implies that for each A ⊆ ℝ in L(ℝ) the set of ⟨(M, I), a⟩ in ℙ_{max} such that (M, I) is A-iterable and ⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩ is dense.

5. The basic analysis

With the existence of A-iterable conditions (for all sets of reals A in L(ℝ)) in hand, we can now prove the most important fact about the ℙ_{max} extension, that every subset of ω_{1} in the extension is the image of a member of the generic filter under the iteration of that member induced by the generic filter.

Formally, if G ⊂ ℙ_{max} is a set of pairwise compatible conditions, then since the elementary embedding witnessing the order on a pair of ℙ_{max} conditions has critical point the ω_{1} of the smaller model, for each pair ⟨(M, I), a⟩, ⟨(N, J), b⟩ in G, a ∩ γ = b ∩ γ, where γ = min{ω_{1}^{M}, ω_{1}^{N}}. For any such G, we let

A_{G} = ⋃{a | ∃(M, I) ⟨(M, I), a⟩ ∈ G}.

By Lemma 2.7, for any such G, for any member ⟨(M, I), a⟩ of G there is a unique iteration of (M, I) sending a to A_{G}. Using this fact, we define P(ω_{1})_{G} to be the collection of all E such that there exists a condition ⟨(M, I), a⟩ ∈ G and a set e ∈ P(ω_{1})^{M} such that j(e) = E, where j is the unique iteration of (M, I) sending a to A_{G}. Likewise, we define I_{G} to be the collection of all E such that there exists a condition ⟨(M, I), a⟩ ∈ G and a set e ∈ I such that j(e) = E, where j is the unique iteration of (M, I) sending a to A_{G}.

We state the following theorem from the point of view of the ground model (so in particular, the universe V in the statement of the theorem does not satisfy AC). We have seen that large cardinals and determinacy each apply that the hypothesis of the theorem is satisfied in L(ℝ), but as we shall see, it can hold in other models as well.

5.1 Theorem. (ZF) Assume that for every A ⊆ ℝ there exists a ℙ_{max} condition ⟨(M, I), a⟩ such that (M, I) is A-iterable and

⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Suppose that G ⊂ ℙ_{max} is a V -generic filter. Then in V [G] the following hold.

(a) P(ω_{1}) = P(ω_{1})_{G}.

(b) NS_{ω_{1}} = I_{G}.

(c) δ_{∼}_{2}^{1} = ω_{2}.

(d) NS_{ω_{1}} is saturated.

Before proving Theorem 5.1, we prove another iteration lemma in terms of games in order to separate out some commonly needed details.

Suppose that p is a ℙ_{max} condition, let J be a normal ideal on ω_{1} and let B be a subset of ω_{1}. Let G_{ω_{1}}(p, J, B) be the game where Players I and II collaborate to build a descending ω_{1}-sequence of ℙ_{max} conditions p_{α} = ⟨(M_{α}, I_{α}), a_{α}⟩ below p, where in round α < ω_{1}, I chooses p_{α} if α ∉ B, and II chooses p_{α} if α ∈ B. At the end of the game, II wins if, letting A = ⋃{a_{α} : α < ω_{1}} and letting j_{α}: (M_{α}, I_{α}) → (M_{α}^{∗}, I_{α}^{∗}) (for each α < ω_{1}) be the iteration of (M_{α}, I_{α}) sending a_{α} to A, j_{α}(I_{α}) = J ∩ M_{α}^{∗} holds for each α < ω_{1}.

5.2 Lemma. (ZFC^{∘}) Suppose that x^{†} exists for every real x. Let p be a condition in ℙ_{max}, let J be a normal ideal on ω_{1} and let B be a subset of ω_{1}. Then II has a winning strategy in G_{ω_{1}}(p, J, B) if and only if B ∉ J.

Proof. The interesting direction is showing that II has a winning strategy if B ∉ J, and for this direction it suffices to consider the case where B consists entirely of limit ordinals (we have no use for the other direction and leave its proof to the reader). The strategy for II uses the usual trick. Partition B into J-positive sets {B_{i}^{α} : α < ω_{1}, i < ω}, and as the p_{α} are chosen, let {E_{i}^{α} : i < ω} enumerate P(ω_{1})^{M_{α}} \ I_{α} for each α.

Fix a ladder sytem {h_{α} : α ∈ B} (so each h_{α} is an increasing function from ω to α with cofinal range). Having constructed our sequence of p_{α}’s up to some limit stage β in B, let

(⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩, a_{β}^{∗})

be the limit sequence corresponding to the descending sequence

⟨p_{h_{β}(i)} : i < ω⟩,

and for each i < ω let j′_{iβ} be the unique iteration of (M_{h_{β}(i)}, I_{h_{β}(i)}) sending a_{h_{β}(i)} to a_{β}^{∗}. Since the dagger of each real exists, we may fix a ℙ_{max} precondition (M_{β}, I_{β}) with

(⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩, a_{β}^{∗}) ∈ H(ω_{1})^{M_{β}}.

As in Lemma 3.4 (more precisely, using Lemma 3.5), we let j′_{β} be an iteration of ⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩ in M_{β} such that

j′_{β}(J_{i}^{β}) = I_{β} ∩ j′_{β}(N_{i}^{β})

for each i < ω, with the extra stipulation that if

ω_{1}^{N_{0}^{β}} ∈ B_{k}^{γ}

for some γ < β and k < ω, then, letting i′ be the least element i of ω such that h_{β}(i) ≥ γ,

j′_{i′β}(j_{γh_{β}(i′)}(E_{k}^{γ}))

is in the normal filter corresponding to the first step of this iteration of ⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩ (note that j′_{i′β}(j_{γh_{β}(i′)}(E_{k}^{γ})) is J_{i′}^{β}-positive by the agreement of ideals imposed by the ℙ_{max} order). Then, letting a_{β} = j′_{β}(a_{β}^{∗}), we have that

ω_{1}^{N_{0}^{β}} ∈ j_{γβ}(E_{k}^{γ}).

Since for each i < ω and α < ω_{1} the set of β ∈ B_{i}^{α} such that ω_{1}^{N_{0}^{β}} = β is J-positive, by playing in this manner Player II ensures that the image of each E_{i}^{α} is J-positive. ⊣

We separate out the following standard argument as well.

5.3 Lemma. Suppose that x^{†} exists for every real x, and let G ⊂ ℙ_{max} be an L(ℝ)-generic filter. Let p_{0} = ⟨(M, I), a⟩ be a ℙ_{max} condition in G, and suppose that P ∈ M is a set of ℙ_{max} conditions such that p ≥ p_{0} for every p ∈ P. Let j be the unique iteration of (M, I) sending a to A_{G}. Then every member of j(P) is in G.

Proof. Let ⟨M_{α}, G_{β}, j_{αδ}^{∗} : β < α ≤ δ ≤ ω_{1}⟩ be the iteration corresponding to j, and fix q = ⟨(N_{0}, J_{0}), b_{0}⟩ in j(P). Fix α_{0} < ω_{1} such that q ∈ j_{0α_{0}}^{∗}(P), and let j_{q} (in M_{α_{0}}) be the iteration of (N_{0}, J_{0}) sending b_{0} to j_{0α_{0}}^{∗}(a). By the genericity of G there is a condition p_{1} = ⟨(N_{1}, J_{1}), b_{1}⟩ in G such that p_{1} ≤ p_{0} and α_{0} < ω_{1}^{N_{1}}. Then ⟨M_{α}, G_{β}, j_{αδ}^{∗} : β < α ≤ δ ≤ ω_{1}^{N_{1}}⟩ is in M_{ω_{1}^{N_{1}}} and is the unique iteration of (M, I) sending a to b_{1}. Since

j_{q}(J_{0}) = j_{0α_{0}}^{∗}(I) ∩ j_{q}(N_{0})

and

j_{0ω_{1}^{N_{1}}}^{∗} (I) = J_{1} ∩ M_{ω_{1}^{N_{1}}} ,

j_{α_{0}ω_{1}^{N_{1}}}^{∗} (j_{q}) witnesses that q ≥ p_{1}. ⊣ Proof of Theorem 5.1. (a) Let τ be a ℙ_{max}-name in L(ℝ) for a subset of ω_{1}, and let A be the set of reals coding (under some fixed recursive coding) the set of triples (p, α, i) such that p ∈ ℙ_{max}, α < ω_{1}, i ∈ 2 and, if i = 1 then p ⊩ α̌ ∈ τ, otherwise p ⊩ α̌ ∉ τ. Let p = ⟨(N, J), d⟩ be any condition in ℙ_{max} and let (M, I) be an A-iterable pre-condition such that

• p ∈ H(ω_{1})^{M},

• ⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Applying Lemma 5.2 in M, there exists a descending sequence of ℙ_{max} conditions p_{α} = ⟨(N_{α}, J_{α}), d_{α}⟩ (α < ω_{1}^{M}) such that

(1) p_{0} = p,

(2) each p_{α+1} decides the sentence “α̌ ∈ τ,”

(3) letting D = ⋃{d_{α} : α < ω_{1}^{M}}, for each α < ω_{1}, j_{α}(J_{α}) = I ∩ j_{α}(N_{α}), where j_{α} is the unique iteration of (N_{α}, J_{α}) sending d_{α} to D.

Note that conditions (1) and (2) are easily satisfied, using the fact that ⟨H(ω_{1})^{M}, A∩M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩, and we may apply Lemma 5.2 (letting B be the set of countable limit ordinals) inside M to meet Condition (3) since in M the dagger of each real exists.

Now, letting D be as in Condition (3) above, ⟨(M, I), D⟩ is a ℙ_{max} condition below each p_{α}. Let e be the subset of ω_{1}^{M} in M such that for each α < ω_{1}, α ∈ e ⇔ p_{α+1} ⊩ α̌ ∈ τ.

Suppose that p′ = ⟨(M, I), D⟩ ∈ G, and let

⟨M_{α}, G_{β}, j_{αδ}^{∗} : β < α ≤ δ ≤ ω_{1}⟩

be the unique iteration of (M, I) sending D to A_{G}. We want to see that j_{0ω_{1}}^{∗}(e) = τ_{G}. Let ⟨q_{α} : α < ω_{1}⟩ = j_{0ω_{1}}^{∗}(⟨p_{α} : α < ω_{1}^{M}⟩). By the elementarity of j_{0α^{∗}}^{∗} and the A-iterability of (M, I), for each γ < ω_{1}, q_{γ+1} ⊩ γ̌ ∈ τ if γ ∈ j_{0α^{∗}}(e) and q_{γ+1} ⊩ γ̌ ∉ τ if γ ∉ j_{0α^{∗}}(e). By Lemma 5.3, each q_{γ} is in G, so j^{∗}(e) = τ_{G}.

(b) The fact that I_{G} = NS_{ω_{1}} follows almost immediately. If E ∈ I_{G}, then there is a condition ⟨(M, I), a⟩ in G, an e ∈ I and an iteration j of (M, I) sending e to E. Then E is disjoint from the critical sequence of this iteration and therefore nonstationary. On the other hand, if E is a nonstationary subset of ω_{1} in V [G], then there is a club C disjoint from E and a condition ⟨(M, I), a⟩ in G, sets e, c ∈ P(ω_{1})^{M} and an iteration j of (M, I) sending e and c to E and C respectively. Then c must be a club subset of ω_{1}^{M} in M, so e ∈ I, which means that E is in I_{G}.

(c) That δ_{∼}_{2}^{1} = ω_{2} also follows almost immediately, using Corollary 1.8 and the standard fact that if x^{#} exists for every real x, then δ_{∼}_{2}^{1} is equivalent to u_{2}, the second uniform indiscernible (the least ordinal above ω_{1} which is an indiscernible of every real) (see [34, 37]). So, showing that δ_{∼}_{2}^{1} = ω_{2} then amounts to showing that for every γ < ω_{2} there is a real x such that the least x-indiscernible above ω_{1} is greater than γ. Working in V [G], fix γ ∈ [ω_{1}, ω_{2}) and a wellordering π of ω_{1} of length γ. By the first part of this theorem, we may fix a condition ⟨(M, I), a⟩ ∈ G and an e ∈ P(ω_{1} × ω_{1})^{M} such that j(e) = π, where j is the iteration of (M, I) sending a to A_{G}. Then γ is in j(M), and so is less than the least indiscernible above ω_{1} of any real coding (M, I), by Corollary 1.8.

(d) To show that NS_{ω_{1}} is saturated in V [G], we show that for any set D ⊆ P(ω_{1})\NS_{ω_{1}} which is dense under the subset order, there is a subset D′ of D of cardinality ℵ_{1} whose diagonal union contains a club. So, following the proof of the first part of this theorem, let τ be a name for such a set D. Let A be the set of reals coding (by a fixed recursive coding) the set of pairs (p, e) such that p = ⟨(M, I), a⟩ is a condition in ℙ_{max}, e ∈ P(ω_{1})^{M} \ I and p forces that j(ě) ∈ τ, where j is the unique iteration of (M, I) sending a to A_{G}.

Let p = ⟨(N, J), b⟩ be any condition in ℙ_{max} and let (M, I) be an Aiterable pre-condition such that

• p ∈ H(ω_{1})^{M},

• ⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Fix a partition {B_{i}^{α} : α < ω_{1}, i < ω} in M of pairwise disjoint I-positive sets whose diagonal union is I-large. Fix also a function g: ω_{1}^{M} × ω → ω_{1}^{M} in M such that g(α, i) ≥ α for all (α, i) ∈ dom(g).

Working in M, we are going to build a descending sequence of ℙ_{max} conditions p_{α} = ⟨(N_{α}, J_{α}), b_{α}⟩ (with the order on conditions witnessed by a commuting family of embeddings j_{αβ}), enumerations {e_{i}^{α} : i < ω} in M of each set P(ω_{1})^{N_{α}} \ J_{α} and sets d_{α} (α ≤ β ≤ ω_{1}^{M}) such that

(4) p_{0} = p,

(5) each d_{α} ∈ P(ω_{1})^{N_{α+1}} \ J_{α+1} and, if α = g(β, i) for some β ≤ α and i < ω, then d_{α} ⊆ j_{β(α+1)}(e_{i}^{β}) and (p_{α+1}, d_{α}) is coded by a real in A,

(6) for each (β, i) ∈ dom(g), B_{i}^{α} \ j_{(g(β,i)+1)ω_{1}^{M}} (d_{g(β,i)}) is nonstationary.

Conditions (5) and (6) together imply that our sequence will satisfy Condition (3) from part (a) of this proof. Furthermore, Conditions (4) and (5) here are easily achieved, by the assumptions on τ. In particular, for each α < ω_{1}^{M}, by the assumptions on τ there exists a pair (p^{∗}, d^{∗}) such that p^{∗} ≤ p_{α} and Condition (5) holds with p^{∗} in the role of p_{α+1} and d^{∗} in the role of d_{α}, and we let (p_{α+1}, d_{α}) be any such pair. Condition (6) implies that the diagonal union of the sets j_{(g(β,i)+1)ω_{1}^{M}} (d_{g(β,i)}) will be I-large.

Condition (6) is achieved in almost exactly the same way as Condition (3) in the first part of the proof (but not exactly the same way; unfortunately we cannot quote Lemma 5.2). Fix a ladder system {h_{α} : α < ω_{1} limit} in M. Having constructed our sequence of p_{α}’s up to some limit stage β, let (⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩, b_{β}^{∗}) be the limit sequence corresponding to the descending sequence ⟨p_{h_{β}(i)} : i < ω⟩, and again for each i < ω let j′_{iβ} be the unique iteration of (N_{h_{β}(i)}, J_{h_{β}(i)}) sending b_{h_{β}(i)} to b_{β}^{∗}. Fix a pre-condition (N_{β}, J_{β}) in M with

(⟨N_{i}^{β} : i < ω⟩, b_{β}^{∗}) ∈ H(ω_{1})^{N_{β}}.

As in Lemma 3.4, we let j′_{β} be an iteration of ⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩ in N_{β} such that

j′_{β}(J_{i}^{β}) = J_{β} ∩ j′_{β}(N_{i}^{β})

for each i < ω, with the extra stipulation that if

ω_{1}^{N_{0}^{β}} ∈ B_{k}^{γ}

for some γ < β and k < ω with g(γ, k) < β, then, letting i′ be the least i ∈ ω such that h_{β}(i) ≥ g(γ, k),

j′_{i′β}(j_{(g(γ,k)+1)h_{β}(i′)}(d_{g(γ,k)}))

is in the filter corresponding to the first step of this iteration of the sequence ⟨(N_{i}^{β}, J_{i}^{β}) : i < ω⟩, ensuring (once we let b_{β} = j′_{β}(b_{β}^{∗})) that

ω_{1}^{N_{0}^{β}} ∈ j_{(g(γ,k)+1)β}(d_{g(γ,k)}).

Then since {ω_{1}^{N_{0}^{β}} : β < ω_{1} limit} is a club subset of ω_{1}^{M}, Condition (6) is satisfied.

Now, letting B = ⋃{b_{α} : α < ω_{1}^{M}}, ⟨(M, I), B⟩ is a ℙ_{max} condition below each p_{α}. For each α < ω_{1} and i < ω, let d′_{αi} = j_{(g(α,i)+1)ω_{1}^{M}} (d_{g(α,i)}). Then the diagonal union of

A = {d′_{αi} : α < ω_{1}^{M}, i < ω}

contains an I-large subset of ω_{1}^{M} in M.

Suppose that ⟨(M, I), B⟩ ∈ G, and let ⟨M_{α}, G_{β}, j_{αδ}^{∗} : β < α ≤ δ ≤ ω_{1}⟩ be the unique iteration of (M, I) sending B to A_{G}. Then the diagonal union of j_{0ω_{1}}^{∗}(A) contains the critical sequence of j_{0ω_{1}}^{∗}, which is a club. We want to see that j_{0ω_{1}}^{∗}(A) ⊆ τ_{G}.

Let ⟨q_{α} : α < ω_{1}⟩ = j_{0ω_{1}}^{∗}(⟨p_{α} : α < ω_{1}^{M}⟩). By Lemma 5.3, each q_{α} is in G. Since (M, I) is A-iterable, each member of j^{∗}(A) is forced to be in τ_{G} by some q_{α}, so j^{∗}(A) ⊆ τ_{G}. ⊣

5.4 Remark. It is shown in [16] that, under the hypothesis of Theorem 5.1, Todorcevic’s Open Coloring Axiom [35] holds in the ℙ_{max} extension. The proof in that paper can be greatly simplified by using Lemmas 5.2 and 5.3 to separate out the standard details.

6. ψ_{AC} and the Axiom of Choice 35

6. ψ_{AC} and the Axiom of Choice

We have not yet shown that the ℙ_{max} extension of L(ℝ) satisfies the Axiom of Choice. We shall do this by showing (assuming that AD holds in L(ℝ)) that the following axiom holds there.

We let o.t.(X) denote the ordertype of a linear order X.

6.1 Definition. ψ_{AC} is the statement that for every pair A, B of stationary, co-stationary subsets of ω_{1}, there exists a bijection π between ω_{1} and some ordinal γ such that the set {α < ω_{1} | α ∈ A ⇔ o.t.(π[α]) ∈ B} contains a club.

Using a partition {A_{α} : α < ω_{1}} of ω_{1} into stationary sets, ψ_{AC} allows us to define an injection from 2^{ω_{1}} into ω_{2}. Since the ℙ_{max} extension of L(ℝ) satisfies the sentence “V = L(P(ω_{1})),” this is enough to see that AC holds there. Let B^{∗} be any stationary, co-stationary subset of ω_{1}. For each X ⊆ ω_{1}, let A_{X} = ⋃{A_{α} : α ∈ X}, and let γ_{X} be the ordinal given by ψ_{AC}, where A_{X} is in the role of A, and B^{∗} is in the role of B. Let X_{0} and X_{1} be distinct subsets of ω_{1}, and let E be the (stationary) symmetric difference of A_{X_{0}} and A_{X_{1}}. Supposing towards a contradiction that γ_{X_{0}} = γ_{X_{1}}, let π_{0} and π_{1} be bijections and C_{0} and C_{1} club subsets of ω_{1} witnessing ψ_{AC} for the pairs A_{X_{0}}, B^{∗} and A_{X_{1}}, B^{∗} respectively. Then there is a club subset D of ω_{1} such that o.t.(π_{0}[α]) = o.t.(π_{1}[α]) for all α ∈ D. Then E ∩ C_{0} ∩ C_{1} ∩ D is nonempty, which gives a contradiction, since α ∈ A_{X_{0}} ⇔ o.t.(π_{0}[α]) ∈ B ⇔ o.t.(π_{1}[α]) ∈ B ⇔ α ∈ A_{X_{1}} for all α ∈ C_{0} ∩ C_{1} ∩ D. Therefore, ψ_{AC} implies that 2^{ω_{1}} = ω_{2}. In fact, it also implies that 2^{ω} = 2^{ω_{1}}, but we will not take the time to show this (it follows from a result of Shelah proved in Section 3.2 of [37]); we already know from Theorem 5.1 that the Continuum Hypothesis fails in the ℙ_{max} extension (assuming AD in L(ℝ)).

That ψ_{AC} holds in the ℙ_{max} extension follows from part (a) of Theorem 5.1 and the following lemma.

6.2 Lemma. (ZFC^{∘}) Suppose that (M, I) is a pre-condition in ℙ_{max}, and let A, B ∈ M be I-positive subsets of ω_{1}^{M} whose complements in ω_{1}^{M} are also I-positive. Let J be a normal ideal on ω_{1}. Then there exist an iteration j: (M, I) → (M^{∗}, I^{∗}) of (M, I) of length ω_{1}, an ordinal γ < ω_{2}, and a bijection π: ω_{1} → γ such that I^{∗} = J ∩ M^{∗} and

{α < ω_{1} | α ∈ j(A) ⇔ o.t.(π[α]) ∈ j(B)}

contains a club.

Proof. Let x be a real coding (M, I). Using Fact 1.9, it suffices to construct an iteration ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩ such that for every α which is a limit of countable x-indiscernibles, j_{0α}(A) ∈ G_{α} if and only if j_{0α^{∗}}(B) ∈ G_{α^{∗}}, where α^{∗} is the least x-indiscernible above α. Note that by the proof of Lemma 1.7, ω_{1}^{M_{γ}} = γ for each x-indiscernible γ, so in particular, each such γ is on the critical sequence.

We construct our iteration using the the winning strategy for Player I in G(ω_{1} \ E) from Lemma 2.9, where E is the set of countable ordinals of the form α^{∗} as above, where α is a limit of x-indiscernibles. This ensures that j_{0ω_{1}}(I) = J ∩M_{ω_{1}}. To complete the construction, we recursively choose each G_{α^{∗}} (α^{∗} ∈ E) in such a way that j_{0α^{∗}}(B) ∈ G_{α^{∗}} if and only if j_{0α}(A) ∈ G_{α}. Fact 1.9 implies that any iteration satisfying these conditions satisfies the conclusion of the lemma. ⊣

Stated in the fashion of Theorem 5.1, we have shown the following.

6.3 Theorem. (ZF) Assume that for every A ⊆ ℝ there exists a ℙ_{max} condition ⟨(M, I), a⟩ such that (M, I) is A-iterable and

⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Suppose that G ⊂ ℙ_{max} is a V -generic filter. Then ψ_{AC} holds in V [G].

6.4 Remark. Suppose that A, B are stationary, co-stationary subsets of ω_{1}, π: ω_{1} → γ is a bijection (for some γ < ω_{2}) and the set

{α < ω | α ∈ A ⇔ o.t.(π[α]) ∈ B}

contains a club subset of ω_{1}. Then for any normal ideal I on ω_{1}, A is the Boolean value in the partial order P(ω_{1})/I that γ ∈ j(B), where j is the induced embedding. It follows that if (M, I) is any iterable pair with M a countable transitive model of ZFC^{∘}+ ψ_{AC} and B is any stationary, costationary subset of ω_{1}^{M} in M, then the image of B under any iteration of (M, I) determines the entire iteration. This in turn implies that one can replace MA_{ℵ_{1}} with ψ_{AC} in the definition of ℙ_{max} without significantly changing the corresponding analysis; in some cases the analysis is easier with ψ_{AC}.

7. Maximality and minimality

In this section we will show that if certain large cardinals exist in V then the ℙ_{max} extension of the inner model L(ℝ) is maximal, in that all forceable Π_{2} sentences for H(ω_{2}) hold there, and that it is minimal, in that every subset of ω_{1} added by the generic filter for ℙ_{max} generates the entire extension. We will also show that a certain form of this maximality characterizes the ℙ_{max} extension.

The following theorem is an immediate consequence of part (a) of Theorem 5.1, and it implies in particular that MA_{ℵ_{1}} holds in the ℙ_{max} extension. Corollary 7.7 below is the maximal version of this fact.

7.1 Theorem. Assume that for every A ⊆ ℝ there exists a ℙ_{max} condition ⟨(M, I), a⟩ such that (M, I) is A-iterable and

⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩.

Let ψ be a binary formula without unbounded quantifiers in a language with one additional unary predicate. Suppose that it is a theorem of ZFC + “NS_{ω_{1}} is precipitous” that for every X ∈ H(ω_{2}) there is a partial order preserving stationary subsets of ω_{1} and the precipitousness of NS_{ω_{1}} and forcing the formula ∃Y ψ(X, Y) to hold in the structure ⟨H(ω_{2}), NS_{ω_{1}}, ∈⟩ of the forcing extension. Then ∀X∃Y ψ(X, Y) holds in the ℙ_{max} extension of V.

To show that the ℙ_{max} extension is Π_{2}-maximal, we will use the following theorem of Woodin (see [7, 5, 28, 37]; Foreman, Magidor and Shelah originally proved the theorem from the existence of supercompact cardinal [6]). An ideal I on ω_{1} is presaturated if for any A ∈ P(ω_{1}) \ I and any sequence ⟨A_{i} : i < ω⟩ of maximal antichains in P(ω_{1}) \ I there exists a B ∈ P(A) \ I such that there are at most ℵ_{1} many X ∈ ⋃{A_{i} : i < ω} such that X ∩ B ∉ I. It is straightforward to check that normal presaturated ideals on ω_{1} are precipitous.

7.2 Theorem. If δ is a Woodin cardinal, then every condition in the partial order Col(ω_{1}, <δ) forces that NS_{ω_{1}} is presaturated.

Using this we can prove the following theorem.

7.3 Theorem. Suppose that δ is a limit of Woodin cardinals, and let κ > δ be a measurable cardinal. Let A be a set of reals in L(ℝ). Let p = ⟨(M, I), a⟩ be a condition in ℙ_{max}, and let ψ be a binary formula without unbounded quantifiers in the language with two additional unary predicates, such that ∀X∃Y ψ(X, Y) holds in ⟨H(ω_{2}), NS_{ω_{1}}, A, ∈⟩. Let x be an element of H(ω_{2})^{M}. Then there exists a ℙ_{max} condition q = ⟨(N, J), b⟩ such that

• q ≤ p,

• (N, J) is A-iterable,

• if j: (M, I) → (M^{∗}, I^{∗}) is the unique iteration of (M, I) sending a to

b, then

⟨H(ω_{2})^{N}, J, A ∩ N, ∈⟩ ⊧ ∃y ψ(j(x), y).

Proof. By Theorem 4.5, since there exist infinitely many Woodin cardinals below a measurable cardinal, all sets of reals in L(ℝ) are <δ-weakly homogeneously Suslin. Let γ be the least Woodin cardinal, let γ′ be the least strongly inaccessible cardinal greater than γ, and let θ be a regular cardinal greater than κ. Let X be a countable elementary submodel of H(θ) with γ, γ′, A, p ∈ X. Applying Lemma 2.8, there is an iteration j: (M, I) → (M^{∗}, I^{∗}) of (M, I) in X of length ω_{1} such that I^{∗} = NS_{ω_{1}} ∩M^{∗}. Then there is a y ∈ H(ω_{2}) ∩ X such that

⟨H(ω_{2}), NS_{ω_{1}}, A, ∈⟩ ⊧ ψ(j(x), y).

Let M be the transitive collapse of X ∩ H(γ′) and, letting γ^{∗} be the image of γ under this collapse, let g be M-generic for Col(ω_{1}, <γ^{∗})^{M}. By Theorem 7.2, NS_{ω_{1}}^{M[g]} is precipitous in M[g]. Let h be M[g]-generic for the standard forcing to make MA_{ℵ_{1}} hold. Then by Theorem 4.9, (M[g][h], NS_{ω_{1}}^{M[g][h]}) is A-iterable. Letting b = j(a) completes the proof of the theorem. ⊣

7.4 Definition. Given a cardinal κ, a set of reals A is κ-universally Baire if there exist trees S and T (contained in ω × Z for some set Z) such that p[S] = A and S and T project to complements in all extensions by forcing constructions of cardinality less than κ.

Theorem 4.6 shows that for any cardinal κ, κ-weakly homogeneously Suslin sets of reals are κ-universally Baire.

If κ is a cardinal, A is a κ-universally Baire set of reals and V [G] is an extension of V by a forcing construction of cardinality less than κ, then we let A(G) be the union of all sets of the form (p[S])^{V [G]}, where S is a tree in V whose projection in V is contained in A. (The notation A_{G} is often used here, but we are already using that for something else.) Note that for any pair of trees S and T in V witnessing that A is κ-universally Baire (i.e., such that p[S] = A and S and T project to complements in all extensions by forcing constructions of cardinality less than κ), (p[S])^{V [G]} = A(G).

The following theorem (due to Woodin) is proved in [19]. The set ℝ^{#} was introduced in [31]. All we need to know here is that ℝ^{#} is a set of reals coding the theory of L(ℝ) in the language with constants for each real and ω many ordinal indiscernibles, and that if ℝ^{#} exists then each set of reals in L(ℝ) is definable in L(ℝ) from a real and a finite set of these indiscernibles.

7.5 Theorem. Suppose that δ is a limit of Woodin cardinals below a measurable cardinal. Then ℝ^{#} is <δ-weakly homogeneous, and if M is any forcing extension of V by a forcing construction in V_{δ} then (ℝ^{#})^{V} = (ℝ^{#})^{M} ∩V.

This gives the following strengthening of Theorem 0.2.

7.6 Theorem. Suppose that δ is a limit of Woodin cardinals below a measurable cardinal, and let A be a set of reals in L(ℝ). Then if G is a V -generic filter contained in a partial order in V_{δ}, then in V [G] there is an elementary embedding from L(A, ℝ^{V}) to L(A(G), ℝ^{V [G]}) sending A to A(G).

By Theorems 5.1 and 7.6, Theorem 7.3 has the following corollary.

7.7 Corollary. Suppose that δ is a limit of Woodin cardinals such that there exists a measurable cardinal above δ, and let A be a set of reals in L(ℝ). Suppose that ϕ is a Π_{2} sentence in the language with two additional unary predicates and P is a partial order in V_{δ} which forces that ϕ holds in the structure ⟨H(ω_{2}), NS_{ω_{1}}, A(G), ∈⟩ (where A(G) is defined relative to P). Then ϕ holds in the structure ⟨H(ω_{2}), NS_{ω_{1}}, A, ∈⟩ when computed in the ℙ_{max} extension of L(ℝ) (of V).

Theorem 7.12 below is a sort of converse to Corollary 7.7. First we will show that every new subset of ω_{1} in the ℙ_{max} extension generates the entire generic filter (Theorem 7.11).

Note that the conclusion of the following lemma corresponds to Condition 5 in the definition of ℙ_{max}.

7.8 Lemma. Suppose that the dagger of each real exists. Let ⟨(M′, I′), a′⟩ be a ℙ_{max} condition and let e be an element of P(ω_{1})^{M′}. Then there exist a ℙ_{max} pre-condition (N, J) with (M′, I′) ∈ H(ω_{1})^{N} and an iteration j: (M′, I′) → (M^{∗}, I^{∗}) in N such that

• j(ω_{1}^{M′}) = ω_{1}^{N},

• I^{∗} = J^{∗} ∩ M^{∗},

and either

1. for some x ∈ P(ω)^{N}, j(e) ∈ L[x], or

2. for some x ∈ P(ω)^{N}, ω_{1}^{N} = ω_{1}^{L[j(e),x]}.

Proof. Fix a limit sequence (⟨(M_{i}, I_{i}) : i < ω⟩, a) corresponding to any descending ω-sequence in ℙ_{max} starting with ⟨(M′, I′), a′⟩, and let (N, J) be a ℙ_{max} pre-condition with {(M′, I′), (⟨(M_{i}, I_{i}) : i < ω⟩, a)} ∈ H(ω_{1})^{N}. Let j′ be the iteration of (M′, I′) sending a′ to a. Now one of two things must hold. Either there exist i < ω, γ < ω_{2}^{M_{i}} and a bijection f: ω_{1}^{M_{0}} → γ in M_{i} such that {α < ω_{1}^{M_{0}} : o.t.(f[α]) ∈ j′(e)} and {α < ω_{1}^{M_{0}} : o.t.(f[α]) ∉ j′(e)} are both I_{i}-positive subsets of ω_{1}^{M_{0}} in M_{i}, or there are no such i, γ, f.

If there is no such triple, then the image of j′(e) is the same under every iteration of ⟨(M_{i}, I_{i}) : i < ω⟩ of length ω_{1}^{N}. Let x be a real in N coding ⟨(M_{i}, I_{i}) : i < ω⟩. There exist iterations of ⟨(M_{i}, I_{i}) : i < ω⟩ of length ω_{1}^{N} in forcing extensions of L[x] by the partial order Col(ω, <ω_{1}^{N}), and since this partial order is homogeneous, this fixed image of j′(e) exists in L[x]. Letting j be any suitable (for example, using a strategy for Player I in G(ω_{1}) as in Theorem 3.5) such iteration in N of length ω_{1}^{N}, then, j(j′) is an iteration of (M′, I′) satisfying the first conclusion of the lemma.

If there is such a triple, note that there is a real y in M_{i+1} such that γ is definable in M_{i+1} (absolutely, in fact) from ω_{1}^{M_{0}} and y (for instance, we could let y be the sharp of any real whose least indiscernible above ω_{1}^{M_{0}} is greater than γ). In particular, we may fix a ternary formula ϕ such that γ is the unique ordinal such that ϕ(γ, y, ω_{1}^{M_{0}}) holds in L[y]. Let

A = {α < ω_{1}^{M_{0}} : o.t.(f[α]) ∈ j′(e)}.

Then A is the Boolean value of the statement that γ is in the image of j′(e). Let x be a real in N coding (⟨(M_{i}, I_{i}) : i < ω⟩, a). Then just as in the proof of Lemma 1.7, the indiscernibles of x are on the critical sequence of any iteration of ⟨(M_{i}, I_{i}) : i < ω⟩. Fix a set B ⊆ ω_{1}^{N} in N such that ω_{1}^{N} = ω_{1}^{L[B]}. Working in N, build an iteration j (with partial iterations j_{αβ} (α ≤ β ≤ ω_{1}^{N}) and normal filters G_{α} (α < ω_{1}^{N})) of ⟨(M_{i}, I_{i}) : i < ω⟩, using a winning strategy for Player I in the game G(ω_{1}^{N} \ E) from Theorem 3.5, where E is the set of countable x-indiscernibles which are not limits of x-indiscernibles (note that x^{#} ∈ N, as (N, J) is iterable, so N contains the sharps for all its reals). When j_{0α}(ω_{1}^{M_{0}}) is in E, we put j_{0α}(A) in the normal filter G_{α} if and only if η ∈ B, where j_{0α}(ω_{1}^{M_{0}}) is the ηth successor xindiscernible. Having completed the construction of our iteration, we have that B is constructible from j(j′(e)), y and x^{#}: B is the set of η < ω_{1}^{N} such that, letting ι_{η} be the η-th successor x-indiscernible, the unique ordinal γ^{∗} satisfying ϕ(γ^{∗}, y, ι_{η}) in L[y] is in j(j′(e)). Then j(j′) is an iteration of (M′, I′) satisfying the second conclusion of the lemma. ⊣

For the rest of this section we fix the following notation: if B is a subset of ω_{1}, we let F_{B} be the set of conditions ⟨(M, I), b⟩ in ℙ_{max} such that there exists an iteration j: (M, I) → (M^{∗}, I^{∗}) such that j(b) = B and I^{∗} = NS_{ω_{1}} ∩ M^{∗}.

Woodin defines the following axiom.

7.9 Definition. Axiom (∗) is the statement that AD holds in L(ℝ) and L(P(ω_{1})) is a ℙ_{max} extension of L(ℝ).

The proofs of Theorems 7.11 and 7.12 use the following lemma.

7.10 Lemma. Assume that axiom (∗) holds, and let B be a subset of ω_{1} such that there exists a real z such that ω_{1} = ω_{1}^{L[z,B]}. Then the set F_{B} is a filter.

Proof. Fix an L(ℝ)-generic filter G ⊂ ℙ_{max} such that L(P(ω_{1})) = L(ℝ)[G]. Fix a real x such that ω_{1} = ω_{1}^{L[x,A_{G}]}. As in the proof of Lemma 2.7, let {a_{α} : α < ω_{1}} be the almost disjoint family of subsets of ω constructed in L[x, A_{G}] by recursively taking a_{α} to be the first real in the L[x, A_{G}] constructibility order almost disjoint from each a_{β} (β < α). Now let y ⊆ ω be such that for all α < ω, a_{α} ∩ y is infinite if and only if α ∈ B. Let

p_{0} = ⟨(M_{0}, I_{0}), b_{0}⟩ and p_{1} = ⟨(M_{1}, I_{1}), b_{1}⟩

be members of F_{B}, as witnessed by iterations j_{0} and j_{1} respectively, and let C_{0} and C_{1} be the respective critical sequences of j_{0} and j_{1}. Let ⟨(N, J), a⟩ be a member of G with x, y, z, p_{0}, p_{1} ∈ H(ω_{1})^{N} and sets c_{0} and c_{1} in P(ω_{1})^{N} such that, for j the unique iteration of (N, J) sending a to A_{G}, j(c_{0}) = C_{0} and j(c_{1}) = C_{1}. Then c_{0} = C_{0} ∩ ω_{1}^{N}, c_{1} = C_{1} ∩ ω_{1}^{N}, and c_{0} and c_{1} are both club subsets of ω_{1}^{N}. Since ω_{1}^{L[z,B]} = ω_{1}, ω_{1}^{L[z,b_{0}]} = ω_{1}^{M_{0}} and ω_{1}^{L[z,b_{1}]} = ω_{1}^{M_{1}}. Since x ∈ N, {a_{α} : α < ω_{1}^{N}} is in N (it satisfies the same definition in N relative to x and a that {a_{α} : α < ω_{1}} satisfies relative to x and A_{G}). Since y ∈ N, B ∩ ω_{1}^{N} ∈ N, and the unique iterations

j_{0}^{∗}: (M_{0}, I_{0}) → (M_{0}^{∗}, I_{0}^{∗})

and

j_{1}^{∗}: (M_{1}, I_{1}) → (M_{1}^{∗}, I_{1}^{∗})

sending b_{0} to B∩ω_{1}^{N} and b_{1} to B∩ω_{1}^{N} respectively are in N, and furthermore j(B ∩ ω_{1}^{N}) = B. Once we see that I_{0}^{∗} = J ∩ M_{0}^{∗} and I_{1}^{∗} = J ∩ M_{1}^{∗} we will be done. The two proofs are the same. For (M_{0}, I_{0}), if E ∈ J ∩ M_{0}^{∗}, then since E ∈ J, j(E) ∈ NS_{ω_{1}}^{L(ℝ)[G]}. Now, j(j^{∗}) is an iteration of (M_{0}, I_{0}) sending b_{0} to B, and so it is equal to j_{0}. Then j(E) is the image of E under the tail of the iteration j_{0} starting with (M_{0}^{∗}, I_{0}^{∗}). So j(E) ∈ j_{0}(M_{0}), and since j_{0}(I_{0}) = NS_{ω_{1}}^{L(ℝ)[G]} ∩ j_{0}(M_{0}), j(E) ∈ j_{0}(I_{0}), and so E ∈ I_{0}^{∗}. ⊣

Note that if G ⊂ ℙ_{max} is an L(ℝ)-generic filter, then F_{(A_{G})} is a filter containing G, and so by the genericity of G, F_{(A_{G})} = G.

Now we show that any new subset of ω_{1} added by forcing with ℙ_{max} generates the entire extension.

7.11 Theorem. Suppose that (∗) holds. Then for every B ∈ P(ω_{1}) \ L(ℝ), F_{B} is an L(ℝ)-generic filter for ℙ_{max}, and L(P(ω_{1})) = L(ℝ)[F_{B}].

Proof. By Lemma 7.8, there is a real z such that L[z, B] correctly computes ω_{1}. By Lemma 7.10, F_{B} is a filter. Now, let ⟨(M, I), a⟩ be a condition in G such that z ∈ M and for some b ∈ P(ω_{1})^{M}, j(b) = B, for j the unique iteration of (M, I) sending a to A_{G}. As in Lemma 2.10, the mapping π sending each condition ⟨(N, J), c⟩ below ⟨(M, I), a⟩ to the condition ⟨(N, J), b^{∗}⟩, where b^{∗} is the image of b by the iteration of (M, I) sending a to c, is an isomorphism. The image of G under π, F_{B}, is then an L(ℝ)-generic filter in ℙ_{max}. Furthermore, π is in L(ℝ), so G is in L(ℝ)[F_{B}]. ⊣

7.12 Theorem. Suppose AD holds in L(ℝ), and that for every Π_{2} sentence ϕ in the language with two additional unary predicates, if A ∈ P(ℝ) ∩ L(ℝ) and

⟨H(ω_{2}), A, NS_{ω_{1}}, ∈⟩^{L(ℝ)^{ℙmax}} ⊧ ϕ

then

⟨H(ω_{2}), A, NS_{ω_{1}}, ∈⟩ ⊧ ϕ.

Then for every B ∈ P(ω_{1}) \ L(ℝ), F_{B} is an L(ℝ)-generic filter and

H(ω_{2}) = H(ω_{2})^{L(ℝ)[F_{B}]}.

Proof. Let P denote P(ω_{1})\⋃{L[x] : x ∈ ℝ} (under AD^{L(ℝ)} this is the same as P(ω_{1}) \ L(ℝ), but we want to make the relevant syntax more explicit). The sentence asserting that F_{B} is a filter for every B ∈ P is Π_{2} in H(ω_{2}) with parameters for NS_{ω_{1}} and the set of ℙ_{max} conditions, and by Lemma 7.10, this sentence holds in the ℙ_{max} extension of L(ℝ). If X ∈ L(ℝ) is a dense subset of ℙ_{max}, then the statement that F_{B}∩X is nonempty for every B ∈ P is Π_{2} in H(ω_{2}) with parameters for NS_{ω_{1}}, X and the set of ℙ_{max} conditions, and this sentence holds in the ℙ_{max} extension of L(ℝ) by Theorem 7.11. Thus, for every B ∈ P, F_{B} is L(ℝ)-generic. Finally, the following statement is Π_{2} in H(ω_{2}) with parameters for NS_{ω_{1}} and the set of ℙ_{max} conditions, and holds in the ℙ_{max} extension: for every E ⊆ ω_{1} and for every B ∈ P there is a ℙ_{max} condition ⟨(M, I), b⟩ and an iteration j: (M, I) → (M^{∗}, I^{∗}) such that E ∈ P(ω_{1})^{M^{∗}}, j(b) = B and I^{∗} = NS_{ω_{1}} ∩ M^{∗}. Fixing a set B ∈ P, then, since {x, B} ∈ L(ℝ)[F_{B}], H(ω_{2}) ⊆ H(ω_{2})^{L(ℝ)[F_{B}]}. ⊣

Theorem 7.11 gives us another way to characterize the ℙ_{max} extension of L(ℝ), this time without mention of ℙ_{max}. For the definition below, we fix the following notation. If g is a filter contained in Col(ω, <ω_{1}), then for each α < ω_{1} we let

S_{α}^{g} = {β | ∃p ∈ g p(0, β) = α}

and, for each τ ⊆ ω_{1} × Col(ω, <ω_{1}), we let

I_{g}(τ) = {α | ∃p ∈ g (α, p) ∈ τ}.

7.13 Definition. Axiom (_{∗}^{∗}) is the statement that x^{#} exists for every real number x and if X is a nonempty subset of P(ω_{1}) which is definable from real and ordinal parameters then there exists a real x and a set

τ ⊆ ω_{1} × Col(ω, <ω_{1})

such that τ ∈ L[x] and such that for all filters g ⊂ Col(ω, <ω_{1}), if g is L[x]-generic and if for each α < ω_{1}, S_{α}^{g} is stationary, then I_{g}(τ) ∈ X.

The converse of the following theorem also holds (see Sections 5.7 and 5.8 of [37]), though its proof is beyond the scope of this chapter.

7.14 Theorem. Axiom (∗) implies that Axiom (_{∗}^{∗}) holds in L(P(ω_{1})).

Proof. First note that AD implies that the sharp of every real exists. Now let G be an L(ℝ)-generic filter such that L(P(ω_{1})) = L(ℝ)[G], and fix a set X as in the statement of Axiom (_{∗}^{∗}). Let p = ⟨(M, I), a⟩ be a condition in G such that for some b ∈ P(ω_{1})^{M}, p forces that j(b) ∈ X, for j the unique iteration of (M, I) sending a to A_{G}. Let x be a real such that ⟨(M, I), a⟩ ∈ H(ω_{1})^{L[x]}. Now, if g ⊂ Col(ω, <ω_{1}) is L[x]-generic, then as in the proof of Lemma 2.8, in L[x][g] there is an iteration

{M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}}

of (M, I) such that for each β < ω_{1} and each set e ∈ P(ω_{1})^{M_{β}} \ j_{0β}(I) there is an α < ω_{1} such that S_{α}^{g} \ j_{βω_{1}}(e) is nonstationary. Let σ be an Col(ω, <ω_{1})-name in L[x] for the embedding j_{0ω_{1}} corresponding to such an iteration, and let τ be the set of pairs (α, p) ∈ ω_{1} × Col(ω, <ω_{1}) such that p ⊩ α̌ ∈ σ(b̌). Now suppose that g ⊂ Col(ω, <ω_{1}) is L[x]-generic and that each S_{α}^{g} is stationary. Then σ_{g} is an iteration of (M, I), and since there exists a real z such that

ω_{1}^{L[z,σ_{g}(a)]} = ω_{1},

σ_{g}(a) is not in L(ℝ). Then Theorem 7.11 implies that G_{σ_{g}(a)} (as in the statement of that theorem) is an L(ℝ)-generic filter for ℙ_{max}. Since each S_{α}^{g} is stationary, σ_{g} witnesses that ⟨(M, I), a⟩ is in G_{σ_{g}(a)}, which means that σ_{g}(b) is in X. Since σ_{g}(b) = I_{g}(τ), we are done. ⊣

Theorem 7.14 has the following immediate corollary. By a perfect subtree of 2^{<ω_{1}} we mean a tree of height ω_{1} such that every node is extended by a pair of incompatible nodes, and such that every countable increasing sequence has a node extending it.

7.15 Corollary. Suppose that AD holds in L(ℝ) and let G ⊂ ℙ_{max} be an L(ℝ)-generic filter. Let ϕ be a unary formula with parameters for elements of L(ℝ) and suppose that there exists a subset of ω_{1} in L(ℝ)[G] \ L(ℝ) satisfying ϕ. Then there is a perfect subtree T of 2^{<ω_{1}} such that every subset of ω_{1} corresponding to a path through T satisfies ϕ.

Given an ordinal β, Martin’s Maximum^{+β} (MM^{+β}, derived from [6]) is the statement that whenever P is a partial order such that forcing with P preserves stationary subsets of ω_{1}, ⟨D_{α} : α < ω_{1}⟩ is a sequence of dense subsets of P and ⟨τ_{α} : α < β⟩ is a sequence of P -names for stationary subsets of ω_{1}, there is a filter G ⊂ P such that G ∩ D_{α} is nonempty for each α < ω_{1} and {γ < ω_{1} | ∃p ∈ G p ⊩ γ̌ ∈ τ_{α}} is stationary for each α < β.

It is shown in [15] that MM^{+ω} does not imply (∗), if the existence of a supercompact limit of supercompact cardinals is consistent with ZFC. The question of whether MM^{+ω_{1}} implies (∗) remains open. We mention the following two test cases, consequences of (∗) which have not been shown from large cardinals to be provably forceable by a semi-proper partial order. We omit the proofs, as they appear in full in [37] (Theorem 7.16 appears in [37] as Theorem 5.74(5) and Theorem 7.19 appears as Theorem 6.124).

7.16 Theorem. Suppose that (_{∗}^{∗}) holds. Then for every A ⊆ ω_{1} which is not constructible from a real, there exist a real x and an L[x]-generic filter g ⊂ Col(ω, <ω_{1}) such that L[x][g] = L[x, A].

The statement of Theorem 7.19 requires the following definitions.

7.17 Definition. A tree T ⊆ {0, 1}^{<ω_{1}} is weakly special if for all countable X ≺ ⟨H(ω_{2}), T, ∈⟩, if b: ω_{1} ∩ X → {0, 1} is a cofinal branch of T_{X} not in M_{X}, then there is a bijection π: ω → ω_{1}^{M_{X}} definable in the structure ⟨M_{X}, T_{X}, b, ∈⟩, where ⟨M_{X}, T_{X}, ∈⟩ is the transitive collapse of X.

7.18 Definition. Φ_{♢}^{+} is the statement that for each A ⊆ ω_{1} there exists a B ⊆ ω_{1} such that, letting T_{B} = {0, 1}^{<ω_{1}} ∩ L[B],

• A ∈ L[B],

• T_{B} is weakly special,

• every branch of T_{B} is in L[B].

7.19 Theorem. Axiom (∗) implies Φ_{♢}^{+}.

One consequence of Theorem 7.19 is that there are no weak Kurepa trees (subtrees of {0, 1}^{<ω_{1}} of cardinality ℵ_{1} with ℵ_{2} many cofinal branches) in any ℙ_{max} extension.

8. Larger models

The forcing construction ℙ_{max} can be applied to larger models than L(ℝ), if they satisfy (ostensibly) stronger forms of determinacy.

8.1 Definition. A set of reals A is ∞-Borel if there exists a set of ordinals S, an ordinal α and a binary formula ϕ such that

A = {y ∈ ℝ | L_{α}[S, y] ⊧ ϕ(S, y)}.

The ordinal Θ is defined to be the least ordinal which is not a surjective image of ℝ. The notion of continuity in the definition below refers to the discrete topology on λ, not the interval topology. Dependent Choice (DC) is a weak form of the Axiom of Choice saying that every tree of height ω with no terminal nodes has a cofinal branch; DC_{ℝ} (Dependent Choice for reals) is the restriction of DC to trees on the reals.

8.2 Definition. (ZF + DC_{ℝ}) AD^{+} is the conjunction of the following two statements.

• Every set of reals is ∞-Borel.

8. Larger models 45

• If λ < Θ and π: λ^{ω} → ω^{ω} is a continuous function, then π^{−1}(A) is determined for every A ⊆ ω^{ω}.

It is an open question whether AD implies AD^{+}, though it is known that AD^{+} holds in all models of AD of the form L(A, ℝ), where A is a set of reals (some of the details of the argument showing this appear in [8]).

The following consequences of AD^{+} are enough to prove that ℙ_{max} conditions exist in suitable generality. 8.3 Theorem. ( ZF + DC_{ℝ}) If AD^{+} holds and V = L(P(ℝ)) then

• the pointclass Σ_{1}^{2} has the scale property,

• every Σ_{1}^{2} set of reals is the projection of a tree in HOD,

• every true Σ_{1}-sentence is witnessed by a Δ_{∼}_{1}^{2} set of reals.

Adapting the proof of Theorem 4.12, then, we have the following. 8.4 Theorem. Suppose that Γ ⊆ P(ℝ) is a pointclass closed under continuous preimages and that L(Γ, ℝ) ⊧ DC_{ℝ} +AD^{+}. Then for every set of reals A in L(Γ, ℝ) there is a ℙ_{max} precondition (M, I) such that

• A ∩ M ∈ M,

• ⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}), A, ∈⟩,

• (M, I) is A-iterable.

The corresponding parts of the proof of Theorem 5.1 then go through to give the following. 8.5 Theorem. Suppose that Γ ⊆ P(ℝ) is a pointclass closed under continuous preimages such that L(Γ, ℝ) ⊧ DC_{ℝ} + AD^{+}. Suppose that G ⊂ ℙ_{max} is L(Γ, ℝ)-generic. Then the following hold in L(Γ, ℝ)[G]:

• P(ω_{1}) = P(ω_{1})_{G},

• I_{G} is the nonstationary ideal,

• δ_{∼}_{2}^{1} = ω_{2},

• I_{G} is saturated.

If there is no surjection in L(Γ, ℝ) from ℝ × Ord onto Γ, then Γ is not wellordered in the ℙ_{max} extension of L(Γ, ℝ). Producing a model of Choice then requires the following step, which appears with proof in [37] as Theorem 9.36. The statement ω_{2}-DC says that <ω_{2}-closed trees of height ω_{2} with no terminal nodes have cofinal branches.

8.6 Theorem. Suppose that Γ ⊆ P(ℝ) is a pointclass closed under continuous preimages such that L(Γ, ℝ) ⊧ DC_{ℝ} + AD^{+} + “Θ is regular”. Suppose that G ⊂ ℙ_{max} is L(Γ, ℝ)-generic. Then L(Γ, ℝ)[G] ⊧ ω_{2}-DC.

The axiom AD_{ℝ} is the statement that all two player games of perfect information of length ω where the players play real numbers are determined. This statement easily implies DC_{ℝ} and in the context of DC is properly stronger than AD^{+}. Theorem 8.7 below lists some properties of the ℙ_{max} extension of a model of AD_{ℝ} plus “Θ is regular.” Many of the corresponding proofs proceed by finding a ℙ_{max} condition satisfying axiom (∗) and satisfying the conclusion of Theorem 8.4 for a suitable set A. We emphasize that part (1) of the conclusion of Theorem 8.7 says that in the ℙ_{max} extension of L(Γ, ℝ), L(P(ω_{1})) is a ℙ_{max} extension of L(ℝ), not (merely) L(Γ, ℝ).

Martin’s Maximum^{++}(𝔠) is the restriction of Martin’s Maximum^{++} to partial orders of cardinality the continuum (which it implies is ℵ_{2}). The statement ♢_{ω}(ω_{2}) says that there is a sequence

{A_{γ} : γ < ω_{2} ∧ cf(γ) = ω}

such that each A_{γ} is a subset of γ and such that for every B ⊆ ω_{2}, the set of α < ω_{2} of countable cofinality such that B ∩ α = A_{α} is stationary. Woodin shows in Section 5.2 of [37] that ♢_{ω}(ω_{2}) follows from Martin’s Maximum. Part (4) of the conclusion of Theorem 8.7 is due to Dan Seabold [26]. Chang’s Conjecture is the statement that for each function F: [ω_{2}]^{<ω} → ω_{2} there exists an X ⊆ ω_{2} of ordertype ω_{1} such that F [[X]^{<ω}] ⊆ X (i.e., that the set of subsets of ω_{2} of ordertype ω_{1} is stationary, in the sense of [19]). It is an open question whether Chang’s Conjecture holds in the ℙ_{max} extension of L(ℝ) whenever L(ℝ) satisfies AD. This question has been resolved (negatively) for ℚ_{max} (see Remark 10.7).

Parts (5), (6) and (7) of Theorem 8.7 show that ℙ_{max} can be used to produce consistency results at ω_{2} as well as at ω_{1}. We let NS_{ω_{2}}^{ω} denote the nonstationary ideal on ω_{2} concentrating on the ordinals of cofinality ω. The ideal NS_{ω_{2}}^{ω} is weakly presaturated if for every S ∈ P(ω_{2}) \ NS_{ω_{2}}^{ω} and every function f: S → ω_{2} there exist a ordinal γ < ω_{3} and a bijection π: ω_{2} → γ such that

{α ∈ S | f(α) < o.t.(π[α])} ∉ NS_{ω_{2}}^{ω}.

A normal ideal I on ω_{2} is semi-saturated if whenever U is a set generic V -normal ultrafilter on ω_{2} contained in P(ω_{2}) \ I, Ult(V, U) is wellfounded.

Sections 9.6 and 9.7 of [37] contain stronger consistency results than parts (4), (5) and (6) of Theorem 8.7, but merely stating these facts would take us too far afield.

8.7 Theorem. Suppose that Γ ⊆ P(ℝ) is a pointclass closed under continuous preimages such that L(Γ, ℝ) ⊧ AD_{ℝ} + “Θ is regular.” Suppose that G ⊂ ℙ_{max} is L(Γ, ℝ)-generic, and let

H ⊆ Col(ω_{3}, H(ω_{3}))^{L(Γ,ℝ)[G]}

be an L(Γ, ℝ)[G]-generic filter. Then the following hold in L(Γ, ℝ)[G][H]:

1. Axiom (∗),

2. Martin’s Maximum^{++}(𝔠),

3. ♢_{ω}(ω_{2}),

4. Chang’s Conjecture

5. NS_{ω_{2}}^{ω} is precipitous,

6. NS_{ω_{2}}^{ω} is weakly presaturated,

7. there is a normal semi-saturated ideal on ω_{2} containing NS_{ω_{2}}^{ω}.

9. Ω-logic

In this section we will briefly describe the relationship between ℙ_{max} and Woodin’s Ω-logic as presented in [37] (our presentation of Ω-logic, however, will follow the one in [42]). Let T be a set of sentences and let ϕ be a sentence, both in the language of set theory. Then T ⊧_{Ω} ϕ (ϕ is Ω_{T} -valid) if for every forcing construction P and every ordinal α, if V_{α}^{P} ⊧ T then V_{α}^{P} ⊧ ϕ. We will define below the conjectured proof-theoretic complement to this model-theoretic notion.

A set of reals A is universally Baire if it is κ-universally Baire for all cardinals κ (see Definition 7.4). Woodin has shown that if δ is a limit of Woodin cardinals, then a set of reals is δ-universally Baire if and only if it is <δ-weakly homogeneously Suslin (a proof is given in [19]). Given a universally Baire set of reals A, a transitive model N of ZFC is said to be A-closed if, whenever P is a partial order in N and G ⊂ P is V -generic (not just N-generic), then N[G] ∩ A(G) is in N[G]. Lemmas 9.2 and 9.3 give useful reformulations of A-closure, and are relatively easy to prove. The proof of Lemma 9.2 uses the following fact, which will show up again in the proof of Theorem 9.4 and in Section 10.1. For a proof of Theorem 9.1, see page 516 of [9] or the appendix of [19].

9.1 Theorem. (McAloon) If ℙ is a partial order and forcing with ℙ makes ℙ countable, then ℙ is forcing-equivalent to Col(ω, |ℙ|).

Theorem 9.1 implies that that every partial order ℙ regularly embeds into Col(ω, |ℙ|), which is forcing-equivalent to ℙ × Col(ω, |ℙ|).

9.2 Lemma. Given a universally Baire set of reals A, a model M of ZFC is A-closed if and only if for all ordinals γ ∈ M, the set of pairs (τ, p) ∈ H(|γ|^{+})^{M} such that τ is a Col(ω, γ)-name in M for a real, p is a condition in Col(ω, γ) and p forces in V that the realization of τ is in A(G) is in M.

Lemma 9.3 shows that for countable models, it is not necessary to consider V -generic filters. The point is that, even without assuming the existence of large cardinals, if

• A is a universally Baire set of reals,

• M is a countable transitive model of ZFC,

• P is a partial order in M,

• p is a condition in P, and

• τ is a P -name in M for a real number

then p forces in V that τ_{G} is in A(G) if and only if there exists (in V) a collection {D_{i} : i < ω} of dense subsets of P such that τ_{g} ∈ A for every M-generic filter g ⊂ P containing p and intersecting each D_{i}.

9.3 Lemma. Let A be a universally Baire set of reals and let M be a countable transitive model of ZFC. Then M is A-closed if and only if for each partial order P in M there exists (in V) a collection {D_{i} : i < ω} of dense subsets of P such that M[G] ∩ A ∈ M[G] for every M-generic filter g ⊂ P intersecting each D_{i}.

Note that if A is the set of reals coding wellorderings of ω (under some fixed recursive coding), then (expanding to the class of ω-models of ZFC) A-closure is equivalent to wellfoundedness.

Let T be a theory containing ZFC and let ϕ be a sentence, both in the language of set theory. Then T ⊢_{Ω} ϕ (T implies ϕ in Ω-logic) if there exists a set of reals A such that

• L(A, ℝ) ⊧ DC_{ℝ} + AD^{+},

• every set of reals in L(A, ℝ) is universally Baire,

• for every countable A-closed model M and every ordinal α ∈ M, if V_{α}^{M} satisfies T then V_{α}^{M} satisfies ϕ.

A sentence ϕ is Ω_{ZFC}-consistent if ZFC ⊬_{Ω} ¬ϕ. The first two conditions above ensure that the set of reals A is sufficiently canonical, and hold of all universally Baire sets of reals in the presence of a proper class of Woodin cardinals. The third condition says that A serves as a sort of proof of ϕ, in the sense that ϕ holds in all models which are closed under a certain function corresponding to A.

The following theorem shows that statements which can be forced to hold (along with ZFC) in suitable initial segments of the universe are Ω_{ZFC}- consistent. Note that the proof shows the stronger fact that for every universally Baire set of reals A, all forceable statements hold in models N which are A-closed in the stronger sense that N[G] ∩ A ∈ N[G] for all N-generic filters G.

9.4 Theorem. Suppose that A is a universally Baire set of reals and that κ is a strongly inaccessible cardinal. Then any forcing extension (in V) of any transitive collapse of any elementary submodel of V_{κ} containing A is A-closed.

Proof. First note that A is universally Baire in V_{κ}. To see this, note that for any partial order P in V_{κ}, if S and T are trees witnessing the universal Baireness of A for P, then by taking a P -name τ in V_{κ} for all the reals of the P -extension and then taking an elementary submodel (of a sufficiently large H(θ)) of size less than κ containing {S, T } ∪ τ, the images of S and T under the transitive collapse of this elementary submodel are in V_{κ} and witness the universal Baireness of A for P.

Now let X be an elementary submodel of V_{κ} with A as an element, and let M be the transitive collapse of X. Let P be a partial order in X, let P̄ be the image of P under the transitive collapse of X, and let g ⊂ P̄ be an M-generic filter. Let τ be a P -name in X for a partial order, and let τ̄ be the image of τ under the transitive collapse of X. We want to see that whenever h ⊆ τ̄_{g} is a V -generic filter, then M[g][h] ∩ A(g, h) is in M[g][h]. Let γ ∈ X ∩ κ be a cardinal greater than |P ∗ τ| and let S and T be trees in X witnessing the universal Baireness of A for Col(ω, γ). Then S and T project to complements in any forcing extension of V by either P ∗ τ or τ̄_{g}.

Let σ be a τ̄_{g}-name in M[g] for a real. Let S̄ and T̄ be the images of S and T under the transitive collapse of X. Let h ⊆ τ̄_{g} be V -generic. Then σ_{h} is in exactly one of (p[S])^{V [h]} and (p[T])^{V [h]}, and by the elementarity of the collapsing map, σ_{h} is in exactly one of (p[S̄])^{M[g][h]} and (p[T̄])^{M[g][h]}. Since (p[S̄])^{M[g][h]} ⊆ (p[S])^{V [h]} and (p[T̄])^{M[g][h]} ⊆ (p[T])^{V [h]}, and since A(h) = (p[S])^{V [h]}, σ_{h} is in A(h) if and only it is in (p[S̄])^{M[g][h]}. Putting all of this together, we have that

M[g][h] ∩ A(h) = (p[S̄])^{M[g][h]},

which shows that M[g] is A-closed. ⊣

Woodin has shown that the axiom (∗) is Ω_{ZFC}-consistent.

9.5 Theorem. Suppose that there exists a proper class of Woodin cardinals and that there is an inaccessible cardinal which is a limit of Woodin cardinals. Then the theory

ZFC + (∗)

is Ω_{ZFC}-consistent.

The proof of Theorem 9.5 requires one to force (∗) over larger models than L(ℝ), in particular, models of the form L(S, ℝ), where for some strongly inaccessible limit of Woodin cardinals κ, S is a <κ-weakly homogeneous tree. A proof that such models can satisfy AD^{+} appears in [19]. Note that it is not known whether there are large cardinals whose existence implies that one can force over V to make (∗) hold. Woodin has conjectured that (ordertype) ω^{2} many Woodin cardinals are sufficient. Of course, if MM^{++} implies (∗) (we discussed this question in Section 7) then one supercompact cardinal is sufficient. Woodin’s Ω-conjecture asserts that if there exist proper class many Woodin cardinals then for every sentence ϕ, ∅ ⊧_{Ω} ϕ if and only if ∅ ⊢_{Ω} ϕ.

Recall that for a set x, x^{†} is a set of the same cardinality as x coding a theory extending ZFC + “there exists a measurable cardinal” with constants for each member of x and for two classes of indiscernibles (above and below the measurable cardinal). If there exist proper class many Woodin cardinals, then the set D of reals coding (under some fixed recursive coding) pairs (x, i), where x is a real number, i is an integer and i ∈ x^{†} is universally Baire. Any D-closed model M then has the property that for any set x, x^{†} exists in any forcing extension of M where x is countable, which since x^{†} is unique means that x^{†} exists in M already (an easy way to say this uses the fact that Col(ω, |x|) is homogeneous, though this is not necessarily the most direct way). Thus for every ordinal α ∈ M, there exist an inner model N of M containing V_{α}^{M} (definable in M), an ordinal κ > α which is a measurable cardinal in N and a set μ which is a κ-complete nonprincipal measure on κ in N such that all iterates of N by μ are wellfounded. As in Example 1.6, then, if M is a D-closed model and I is a normal precipitous ideal on ω_{1}^{M} in M, then every rank initial segment of M is a rank initial segment of a model N such that (N, I) is iterable, and so (M, I) is also iterable. Using this we have that every Π_{2} sentence for ⟨H(ω_{2}), NS_{ω_{1}}, ∈⟩ which is Ω_{ZFC}-consistent with the existence of a precipitous ideal on ω_{1} holds in the ℙ_{max} extension. Using the canonical inner models for Woodin cardinals we can do more, however. In the next few paragraphs we will sketch the proof of Theorem 9.6 below.

We are going to need a number of facts from inner model theory. Unfortunately, we do not have a reference for the exact facts that we need (though [32] is very close), which is why this is just a sketch. For each set x, let M(x) denote the minimal model (i.e., sound, sufficiently iterable premouse) of ZFC plus “there exists an ordinal λ which is a limit of Woodin cardinals such that V_{λ}^{#} exists.” This theory implies that ℝ^{#} is <λ-weakly homogeneously Suslin, and so there exist in M(x) trees S and T on ω × λ witnessing in M(x) that ℝ^{#} and its complement are λ-universally Baire. Furthermore, from the point of view of V, S and T project to a subset of ℝ^{#} and a subset of ℝ \ ℝ^{#}, respectively. The property of M(x) that we need is the following: if δ is a Woodin cardinal in M(x) below λ, γ is an ordinal below δ and y is a subset of ω, then there exist a partial order P (this partial order was discovered by Woodin and is typically called the extender algebra) of cardinality δ in M(x) and an elementary embedding j: M(x) → M′ with critical point greater than γ such that

• y is M′-generic for j(P),

• p[j(S)] ⊆ ℝ^{#},

• p[j(T)] ⊆ ℝ \ ℝ^{#}.

Furthermore, there is a canonical coding of M(x) by a set of integers when x is itself a set of integers, giving rise to a universally Baire set B consisting of the pairs (x, i) such that i is part of the code for M(x). A B-closed model of ZFC then contains M(x) for every set x ∈ M.

Now let ϕ be a Π_{2} sentence for H(ω_{2}) (of the form ∃X∀Y ψ(X, Y)) with predicates for NS_{ω_{1}} and a given set of reals A in L(ℝ). Let z be a real number coding a given ℙ_{max} condition and a real which codes A relative to ℝ^{#} (see the remarks before Theorem 7.5). Suppose that N is a countable Bclosed model of ZFC satisfying ϕ and containing z. Let x be a wellordering of H(ω_{2})^{N} in N. Then H(ω_{2})^{M(x)} = H(ω_{2})^{N}. Let γ be the least strongly inaccessible cardinal in M(x) above the least Woodin cardinal. Let S and T be trees in M(x) witnessing the λ-universal Baireness of ℝ^{#} and its complement, where λ is the least limit of Woodin cardinals in M(x). We want to see that whenever we make NS_{ω_{1}} precipitous by any forcing in V_{γ}^{M(x)} (getting a generic filter g) and then iterate V_{γ}^{M(x)[g]} by NS_{ω_{1}}, we iterate correctly for ℝ^{#}. Given this, if g is such a generic filter for a forcing preserving stationary subsets of ω_{1}^{M(x)} then V_{γ}^{M(x)[g]} is an A-iterable model such that ∃Y ψ(X, Y) holds in H(ω_{2})^{V_{γ}^{M(x)[g]}} for all X ∈ H(ω_{2})^{V_{γ}^{M(x)}}, and by a density argument then, ϕ holds in the ℙ_{max} extension.

Towards a contradiction, choose a bad generic filter g and bad iteration k. Let j: M(x) → M′ be the embedding in the previous paragraph (with critical point above γ) such that we can add g and k to M′ by forcing with the extender algebra for the image of the least Woodin cardinal in M(x) above γ. Then M′[g, k] has a bad iteration of V_{γ}^{M(x)[g]} in it, and by Lemma 1.4 this iteration extends to an iteration of M′[g] (which we will also call k), which means that

k(ℝ^{#} ∩ V_{γ}^{M(x)[g]}) = p[k(j(S))] ∩ k(V_{γ}^{M(x)[g]})

and

k((ℝ \ ℝ^{#}) ∩ V_{γ}^{M(x)[g]}) = p[k(j(T))] ∩ k(V_{γ}^{M(x)[g]}).

But j(S) and j(T) are j(λ)-universally Baire in M′, so they project to complements in M′[g, k]. Furthermore,

p[j(S)] ⊆ ℝ^{#}

and

p[j(T)] ⊆ ℝ \ ℝ^{#}.

Since p[j(S)] ⊆ p[k(j(S))] and p[j(T)] ⊆ p[k(j(T))], p[j(S)] = p[k(j(S))] and p[j(T)] = p[k(j(T))], contradicting that k is a bad iteration. This argument shows the following.

9.6 Theorem. If there exists a proper class of Woodin cardinals, then for every set of reals A in L(ℝ), every Ω_{ZFC}-consistent Π_{2} sentence for ⟨H(ω_{2}), NS_{ω_{1}}, A, ∈⟩ holds in the ℙ_{max} extension of L(ℝ).

Another strengthening of Theorem 0.2, using the absoluteness of ℝ^{#}, is the following.

9.7 Theorem. Suppose that there exists a proper class of Woodin cardinals. Then for every sentence ϕ, either ZFC ⊢_{Ω} L(ℝ) ⊧ ϕ or ZFC ⊢_{Ω} L(ℝ) |̸= ϕ.

Since ℙ_{max} is a homogeneous forcing extension of L(ℝ), this gives the following.

9.8 Theorem. Suppose that there exists a proper class of Woodin cardinals. Then for every sentence ϕ, either

ZFC + (∗) ⊢_{Ω} L(P(ω_{1})) ⊧ ϕ

or

ZFC + (∗) ⊢_{Ω} L(P(ω_{1}) |̸= ϕ.

Note that since ℝ^{#} is not in L(ℝ), the Continuum Hypothesis (plus the existence of ℝ^{#}) implies that L(P(ω_{1})) is not contained in a forcing extension of L(ℝ). Moreover, Woodin has shown (see Theorem 10.183 of [37]) that if ψ is any sentence for which Theorem 9.8 holds with ψ in the place of (∗), then ZFC + ψ implies in Ω-logic that the Continuum Hypothesis is false.

10. Variations

The ℙ_{max} method is fairly flexible, and the partial order ℙ_{max} can be varied in a number of ways. We present here two types of variations. The first is an example of the utility of ℙ_{max} for manipulating ideals on ω_{1}. The second illustrates a method for producing extensions which are Π_{2}-maximal for H(ω_{2}) relative to a fixed Σ_{2} sentence. Several other variations appear in [37, 43]. Still others appear in [3, 17].

10.1. Variations for NS_{ω_{1}}

An ideal I on ω_{1} is ℵ_{1}-dense if the Boolean algebra P(ω_{1})/I has a dense subset of cardinality ℵ_{1}. In unpublished work, Woodin showed that starting from a huge cardinal one can force the existence of a normal ℵ_{1}-dense ideal on ω_{1}. Shelah later showed [27] that, starting from a supercompact cardinal, one can force that the nonstationary ideal restricted to a fixed stationary subset of ω_{1} is ℵ_{1}-dense. The ℙ_{max} variation ℚ_{max}^{∗} discussed here, when applied to a model of the form L(ℝ) satisfying AD, produces a model of ZFC in which NS_{ω_{1}} is ℵ_{1}-dense; by unpublished work of Woodin, this shows that the Axiom of Determinacy and the ℵ_{1}-density of NS_{ω_{1}} are equiconsistent. To date, ℙ_{max} variations are the only known means for producing models in which NS_{ω_{1}} is ℵ_{1}-dense.

Using the result of Shelah mentioned above, the partial order ℚ_{max} below can be used to obtain the ℵ_{1}-density of NS_{ω_{1}} from a supercompact cardinal below ω Woodin cardinals below a measurable. This hypothesis is not optimal, but unlike with ℚ_{max}^{∗}, we can give all the details here (aside from one argument, we have already done so).

By Theorem 9.1, the ℵ_{1}-density of a σ-ideal on ω_{1} is witnessed by a function from ω_{1} to H(ω_{1}) of the following form.

10.1 Definition. Given a normal ℵ_{1}-dense ideal I on ω_{1}, Y_{Coll}(I) is the set of functions f: ω_{1} → H(ω_{1}) satisfying the following conditions (where for each p ∈ Col(ω, ω_{1}) we let S_{p}^{f} = {α < ω_{1} | p ∈ f(α)} ):

• for each α < ω_{1}, f(α) is a filter in Col(ω, 1 + α)

• for each p ∈ Col(ω, ω_{1}), S_{p}^{f} ∉ I,

• for each S ∈ P(ω_{1})/I, there exists a condition p ∈ Col(ω, ω_{1}) such that S_{p}^{f} \ S ∈ I.

10.2 Definition. The partial order ℚ_{max} consists of the set of pairs of the form ⟨(M, I), f⟩ satisfying the following conditions:

1. M is a countable transitive model of ZFC^{∘},

2. I is a normal ℵ_{1}-dense ideal on ω_{1}^{M} in M,

3. (M, I) is iterable,

4. f ∈ (Y_{Coll}(I))^{M}.

The order on ℚ_{max} is as follows: ⟨(N, J), g⟩ < ⟨(M, I), f⟩ if M ∈ H(ω_{1})^{N} and there exists an iteration j: (M, I) → (M^{∗}, I^{∗}) such that

• j(f) = g,

• j, M^{∗} ∈ N, • I^{∗} = M^{∗} ∩ J.

If ⟨(M, I), f⟩ is a ℚ_{max} condition, then by the normality of I in M, the image of f under any iteration of (M, I) determines the entire iteration.

The only new argument we need to give in the ℚ_{max} analysis is the following. The corresponding versions for iterating sequences of models and for building descending ω_{1}-sequences of conditions are essentially the same.

10.3 Lemma. Suppose that J is a normal ℵ_{1}-dense ideal on ω_{1}, and let g be a function in Y_{Coll}(J). Suppose that ⟨(M, I), f⟩ is a condition in ℚ_{max}. Then there is an iteration j: (M, I) → (M^{∗}, I^{∗}) of length ω_{1} such that

• {α < ω_{1} | j(f)(α) ≠ g(α)} ∈ J,

• I^{∗} = M^{∗} ∩ J.

Proof. Note that the second conclusion follows from the first. Let

⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩

be any iteration of (M, I) satisfying the condition that whenever β < ω_{1} is such that j_{0β}(ω_{1}^{M}) = β and g(β) is M_{β}-generic for Col(ω, β), then G_{β} is the corresponding filter in P(ω_{1})^{M_{β}}/j_{0β}(I), i.e., for each p ∈ Col(ω, β), S_{p}^{j_{0β}(f)} ∈ G_{β} if and only if p ∈ g(β). It is immediate that G_{β} is M_{β} generic, and that the choice of G_{β} makes j_{0(β+1)}(f)(β) = g(β). It remains to see that the set of β < ω_{1} such that g(β) is not M_{β}-generic for Col(ω, β) is in J. To see this, let A be subset of ω_{1} coding ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ ω_{1}⟩ under some fixed recursive coding. Then for club many η < ω_{1}, j_{0η}(ω_{1})^{M} = η and ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ η⟩ ∈ L[A ∩ η]. Every condition in P(ω_{1})/J forces that (letting k be the induced elementary embedding) k(g)(ω_{1}^{V}) is a V -generic (and thus L[A]-generic) filter in Col(ω, ω_{1}^{V}), which means that the set of η < ω_{1} such that g(η) is not L[A ∩ η]-generic is in J. Since M_{η} ∈ L[A ∩ η] for club many η, we are done. ⊣

Theorem 4.9 plus the result of Shelah mentioned above gives the following.

10.4 Theorem. Suppose that there exists a supercompact cardinal below infinitely many Woodin cardinals below a measurable cardnal. Then for every set of reals A in L(ℝ) there exists a ℚ_{max} condition ⟨(M, I), f⟩ such that

• A ∩ M ∈ M,

• (M, I) is A-iterable,

• ⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}, A, ∈⟩.

10. Variations 55

The proof of the following is essentially the same as for ℙ_{max}. The ℵ_{1}- density of I_{G} follows immediately from P(ω_{1}) = P(ω_{1})_{G} and the definition of ℚ_{max} (letting I_{G} and P(ω_{1})_{G} have the definitions here analogous to those used for ℙ_{max}). 10.5 Theorem. ( ZF) Suppose that for every set of reals A there exists a ℚ_{max} condition ⟨(M, I), f⟩ such that

• A ∩ M ∈ M,

• (M, I) is A-iterable,

• ⟨H(ω_{1})^{M}, A ∩ M, ∈⟩ ≺ ⟨H(ω_{1}, A, ∈⟩. Then ℚ_{max} is ω-closed and homogeneous. Furthermore, if G is an V -generic filter for ℚ_{max}, then the following hold in V [G]:

• P(ω_{1}) = P(ω_{1})_{G},

• NS_{ω_{1}} = I_{G},

• ψ_{AC},

• δ_{∼}_{2}^{1} = ω_{2},

• NS_{ω_{1}} is ℵ_{1}-dense.

To obtain the ℵ_{1}-density of NS_{ω_{1}} from the optimal hypothesis, we can use the partial order ℚ_{max}^{∗} below. Conditions in ℚ_{max}^{∗} are similar to the limit sequences used in the ℙ_{max} analysis. The utility of this approach here is that the existence of ℚ_{max}^{∗} conditions does not require the existence of a model with an ℵ_{1}-dense ideal on ω_{1}. The analyses of ℚ_{max}^{∗} and ℚ_{max} are the same, once we show that ℚ_{max}^{∗} conditions exist in suitable generality. Showing this, however, is beyond the scope of this chapter. 10.6 Definition. ℚ_{max}^{∗} is the set of pairs (⟨M_{k} : k < ω⟩, f) such that the following hold.

1. The set f is a function from ω_{1}^{M_{0}} to M_{0} in M_{0} such that for all α < ω_{1}^{M_{0}}, f(α) is a filter in Col(ω, 1 + α).

2. Each M_{k} ⊧ ZFC^{∘}.

3. Each M_{k} ∈ M_{k+1}.

4. For all k < ω, ω_{1}^{M_{k}} = ω_{1}^{M_{0}}.

5. For all k < ω, NS_{ω_{1}}^{M_{k+1}} ∩ M_{k} = NS_{ω_{1}}^{M_{k+2}} ∩ M_{k}.

6. The sequence ⟨M_{k} : k < ω⟩ is iterable.

7. For each p ∈ Col(ω, ω_{1}^{M_{0}}), {α < ω_{1}^{M_{0}} | p ∈ f(α)} ∉ NS_{ω_{1}}^{M_{1}}.

8. For each k < ω and for each a ⊆ ω_{1}^{M_{0}} such that a ∈ M_{k} \ NS_{ω_{1}}^{M_{k+1}}, there exists a p ∈ Col(ω, ω_{1}^{M_{0}}) such that

{α < ω_{1}^{M_{0}} | p ∈ f(α)} ∩ (ω_{1}^{M_{0}} \ a) ∈ NS_{ω_{1}}^{M_{k+1}}.

The ordering on ℚ_{max}^{∗} is given by letting

(⟨N_{k} : k < ω⟩, g) < (⟨M_{k} : k < ω⟩, f)

if ⟨M_{k} : k < ω⟩ ∈ H(ω_{1})^{N_{0}} and there exists an iteration

j: ⟨M_{k} : k < ω⟩ → ⟨M_{k}^{∗} : k < ω⟩

in N_{0} such that

• j(f) = g,

• NS_{ω_{1}}^{M_{k+1}^{∗}} ∩ M_{k}^{∗} = NS_{ω_{1}}^{N_{1}} ∩ M_{k}^{∗} for all k < ω.

Condition (5) above says that the models in the sequence need not agree about stationary sets, but rather, each subset of ω_{1}^{M_{0}} in each M_{k} which is stationary in M_{k+1} is stationary in all further M_{j}’s. This extra degree of freedom is essential in constructing ℚ_{max}^{∗} conditions without presupposing the existence of ℚ_{max} conditions. Conditions (7) and (8) ensure that if

G ⊂ ⋃{P(ω_{1}^{M_{0}})^{M_{k}} \ NS_{ω_{1}}^{M_{k+1}} : k < ω}

is a ⋃{M_{k} : k < ω}-normal filter, then (letting j be the induced embedding) j(f)(ω_{1}^{M_{0}}) is a filter in Col(ω, ω_{1}^{M_{0}}) meeting every dense set in each M_{k}, and vice-versa: if g is a filter in Col(ω, ω_{1}^{M_{0}}) meeting every dense set in each M_{k}, then there is a ⋃{M_{k} : k < ω}-normal filter G contained in ⋃{P(ω_{1}^{M_{0}})^{M_{k}} \ NS_{ω_{1}}^{M_{k+1}} : k < ω} such that j(f)(ω_{1}^{M_{0}}) = g.

10.7 Remark. If Γ is a pointclass closed under continuous images such that L(Γ, ℝ) ⊧ AD_{ℝ} + “Θ is regular”, then the ℚ_{max} extension of L(Γ, ℝ) satisfies Chang’s Conjecture. However, for consistency strength reasons one cannot prove that Chang’s Conjecture holds in the ℚ_{max} extension of L(ℝ) from the assumption that AD holds in L(ℝ) (see page 651 of [37]).

The utility of the ℙ_{max} approach for manipulating ideals on ω_{1} is applied in other several ways in [37], notably to create a model in which the saturation of NS_{ω_{1}} can be destroyed without adding a subset of ω_{1}. In [?], a variation of ℙ_{max} is used to produce a model in which the saturation of NS_{ω_{1}} can be destroyed by forcing with a Suslin tree. As far as we know, these results have not been reproduced by other methods.

10. Variations 57

10.2. Conditional variations for Σ_{2} sentences

As we saw in Section 7, the ℙ_{max} extension of L(ℝ) (assuming that there exists a proper class of Woodin cardinals) satisfies all forceable Π_{2} sentences for H(ω_{2}) with parameters for NS_{ω_{1}} and sets of reals in L(ℝ). In some cases, one can fix a Σ_{2} sentence for this structure and produce a model satisfying all Π_{2} sentences forceably consistent with it (and in some cases one cannot). If ϕ is a Σ_{2} sentence of the form ∃A∀Bψ(A, B), where all quantifiers in ψ are bounded, the optimal iteration lemma for ϕ is the following statement: if

• M is a countable transitive model of ZFC^{∘},

• I is normal ideal on ω_{1}^{M} in M,

• (M, I) is iterable,

• a ∈ H(ω_{2})^{M} and H(ω_{2})^{M} ⊧ ∀bψ(a, b),

• H(ω_{2}) ⊧ ∃A∀Bψ(A, B),

• J is a normal ideal on ω_{1}, then there exists an iteration j: (M, I) → (M^{∗}, I^{∗}) of length ω_{1} such that

• I^{∗} = J ∩ M^{∗},

• H(ω_{2}) ⊧ ∀Bψ(j(a), B).

Roughly, the optimal iteration lemma for ϕ says that given a countable transitive iterable model of ϕ and a fixed witness for ϕ in this model, in order to prove that there is an iteration of this model mapping this witness to a witness for ϕ is V, we need assume only that ϕ holds in V. Since this assumption is necessary, in the cases where the lemma holds, it is optimal. In [29], the optimal iteration lemma is proved for the following sentences (the first four of which are defined in [1]; we direct the reader to [29] for the other two).

• The dominating number (𝔡) is ℵ_{1}.

• The bounding number (𝔟) is ℵ_{1}.

• The cofinality of the meager ideal is ℵ_{1}.

• The cofinality of the null ideal is ℵ_{1}.

• There exists a coherent Suslin tree.

• There exists a free Suslin tree.

Given a Σ_{2} sentence ϕ as above, we can define the ℙ_{max} variation ℙ_{max}^{ϕ} as follows. Since ϕ may contradict MA_{ℵ_{1}}, we remove the requirement that the models satisfy MA_{ℵ_{1}} and ensure the uniqueness of iterations directly (alternately, we can usually replace MA_{ℵ_{1}} with ψ_{AC}). The partial order ℙ_{max}^{ϕ} is defined recursively on the ω_{1} of the selected model M.

10.8 Definition. The partial order ℙ_{max}^{ϕ} consists of all pairs ⟨(M, I), a, X⟩ such that

1. M is a countable transitive model of ZFC^{∘},

2. I ∈ M and in M, I is a normal ideal on ω_{1},

3. (M, I) is iterable,

4. a ∈ P(ω_{1})^{M} and H(ω_{2})^{M} ⊧ ∀bψ(a, b),

5. X ∈ M and X is a set (possibly empty) of pairs (⟨(N, J), b, Y ⟩, j) such that

• ⟨(N, J), b, Y ⟩ ∈ ℙ_{max}^{ϕ} ∩ H(ω_{1})^{M},

• j is an iteration of (N, J) of length ω_{1}^{M} such that j(J) = I ∩j(N)

and j(b) = a,

• j(Y) ⊆ X,

with the property that for each p ∈ ℙ_{max}^{ϕ} there is at most one j such that (p, j) ∈ X.

The order on ℙ_{max}^{ϕ} is implicit in the conditions on X:

⟨(M, I), a, X⟩ < ⟨(N, J), b, Y ⟩

if there exists a j such that (⟨(N, J), b, Y ⟩, j) ∈ X.

For ϕ as above, we have games G_{ω}^{ϕ} and G_{ω_{1}}^{ϕ} which are strictly analogous to the games G_{ω} and G_{ω_{1}} for ℙ_{max}.

For G_{ω}^{ϕ}, suppose that ⟨(N_{i}, J_{i}) : i < ω⟩ is an iterable pre-limit sequence (in the sense of ℙ_{max}) and that there exists an a ∈ P(ω_{1})^{N_{0}} such that H(ω_{2})^{N_{i}} ⊧ ∀bψ(a, b) for each i < ω. Then given a normal ideal I on ω_{1} and a set E ⊂ ω_{1}, we define G_{ω}^{ϕ}(⟨(N_{i}, J_{i}) : i < ω⟩, I, E) to be the following game of length ω_{1} where Players I and II collaborate to build an iteration of ⟨(N_{i}, J_{i}) : i < ω⟩ consisting of pre-limit sequences ⟨(N_{i}^{α}, J_{i}^{α}) : i < ω⟩ (α ≤ ω_{1}), normal ultrafilters G_{α} (α < ω_{1}) and a commuting family of embeddings j_{αβ} (α ≤ β ≤ ω_{1}), as follows. In each round α, let

Q_{α} = ⋃{P(ω_{1})^{N_{i}^{α}} \ J_{i}^{α} : i < ω}.

If α ∈ E, then Player I chooses a set A ∈ Q_{α}, and then Player II chooses a ⋃{N_{i}^{α} : i < ω}-normal filter G_{α} contained in Q_{α} with A ∈ G_{α}. If α is not in E, then Player II chooses any ⋃{N_{i}^{α} : i < ω}-normal filter G_{α} contained in Q_{α}. After all ω_{1} many rounds have been played, Player I wins if H(ω_{2}) ⊧ ∀Bψ(j_{0ω_{1}}(a), B) and if J_{i}^{ω_{1}} = I ∩ N_{i}^{ω_{1}} for each i < ω.

Similarly, given a ℙ_{max}^{ϕ} condition p = ⟨(M, I), a, X⟩, a normal ideal J on ω_{1} and a subset of ω_{1} E, we let G_{ω_{1}}^{ϕ}(p, J, E) be game of length ω_{1} where players I and II collaborate to build a descending ω_{1}-chain of conditions p_{α} = ⟨(M_{α}, I_{α}), a_{α}, X_{α}⟩ (α < ω_{1}) of ℙ_{max}^{ϕ} conditions below p as follows. In each round α, all p_{β} (β < α) have been defined. If α is a successor ordinal (or 0), Player II chooses a condition p_{α} < p_{α−1} (< p). If α is a limit ordinal, then Player I picks a condition p_{α} below each p_{β} (β < α). Then, letting j_{αβ} (α < β ≤ ω_{1}) be the induced commuting family of embeddings (and letting j be the embedding witnessing that p_{0} < p), Player I wins the game if H(ω_{2}) ⊧ ∀Bψ(j_{0ω_{1}}(j(a)), B), and if for all α < ω_{1}, j_{αω_{1}}(I_{α}) = J ∩ j_{αω_{1}}(M_{α}).

The arguments in [29] show that ♢ implies that Player I has a winning strategy in each game G_{ω}^{ϕ}(⟨(N_{i}, J_{i}) : i < ω⟩, I, E) and each game G_{ω_{1}}^{ϕ}(p, J, E) for each of the sentences listed before Definition 10.8 (typically these arguments are essentially the same as the proof of the corresponding optimal iteration lemma).

The proof of the following theorem then is a straightforward generalization of the arguments we have given for ℙ_{max}.

10.9 Theorem. Assume that AD holds in L(ℝ). Let ϕ be an Ω_{ZFC}-consistent Σ_{2} sentence for H(ω_{2}). Suppose that the optimal iteration lemma for ϕ holds, and that the following sentences are Ω_{ZFC}-consistent:

• for all iterable pre-limit sequences ⟨(N_{i}, I_{i}) : i < ω⟩ and for all normal ideals I on ω_{1}, Player I has a winning strategy in

G_{ω}^{ϕ}(⟨(N_{i}, J_{i}) : i < ω⟩, I, ω_{1});

• for all ℙ_{max}^{ϕ} conditions p and for all normal ideals J on ω_{1}, Player I has a winning strategy in G_{ω_{1}}(p, J, ω_{1}).

Let G ⊂ ℙ_{max}^{ϕ} be L(ℝ)-generic. Then in L(ℝ)[G] the following hold:

• ϕ,

• P(ω_{1}) = P(ω_{1})_{G},

• NS_{ω_{1}} = I_{G},

• NS_{ω_{1}} is saturated.

Furthermore, for every set of reals A in L(ℝ), L(ℝ)[G] satisfies every Π_{2}- sentence for the structure ⟨H(ω_{2}), NS_{ω_{1}}, A, ∈⟩ which is Ω_{ZFC}-consistent with ϕ.

The variation ℙ_{max}^{ϕ} where ϕ asserts the existence of a coherent Suslin tree is studied in [14, 18].

Bibliography

[1] Andreas Blass. Combinatorial cardinal characteristics of the continuum. In this Handbook.

[2] Patrick Dehornoy. Progrès récents sur l’Hypothèse du continu [d’après Woodin]. (# 915), 2002-2003. Séminaire Bourbaki.

[3] Qi Feng and W. Hugh Woodin. P -points in ℚ_{max} models. Ann. Pure Appl. Logic, 119(1-3):121–190, 2003.

[4] Matthew Foreman. Ideals and generic elementary embeddings. In this Handbook.

[5] Matthew Foreman and Menachem Magidor. Large cardinals and definable counterexamples to the continuum hypothesis. Ann. Pure Appl. Logic, 76(1):47–97, 1995.

[6] 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.

[7] Noa Goldring. Woodin cardinals and presaturated ideals. Ann. Pure Appl. Logic, 55(3):285–303, 1992.

[8] Steve Jackson. Structural consequences of AD. In this Handbook.

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

[10] Thomas Jech, Menachem Magidor, William Mitchell, and Karel Prikry. Precipitous ideals. J. Symbolic Logic, 45(1):1–8, 1980.

[11] Ronald B. Jensen and Robert M. Solovay. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pages 84–104. North-Holland, Amsterdam, 1970.

[12] Yuzuru Kakuda. On a condition for Cohen extensions which preserve precipitous ideals. J. Symbolic Logic, 46(2):296–300, 1981.

[13] Akihiro Kanamori. The higher infinite. Springer Monographs in Mathematics. Springer-Verlag, Berlin, second edition, 2003. Large cardinals in set theory from their beginnings.

[14] Paul Larson. An 𝕊_{max} variation for one Souslin tree. J. Symbolic Logic, 64(1):81–98, 1999.

[15] Paul Larson. Martin’s Maximum and the ℙ_{max} axiom (∗). Ann. Pure Appl. Logic, 106(1-3):135–149, 2000.

[16] Paul Larson. Showing OCA in ℙ_{max}-style extensions. Kobe J. Math., 18(2):115–126, 2001.

[17] Paul Larson. Saturation, Suslin trees and meager sets. Arch. Math. Logic, 44(5):581–595, 2005.

[18] Paul Larson and Stevo Todorčević. Chain conditions in maximal models. Fund. Math., 168(1):77–104, 2001.

[19] Paul B. Larson. The stationary tower, volume 32 of University Lecture Series. American Mathematical Society, Providence, RI, 2004. Notes on a course by W. Hugh Woodin.

[20] Menachem Magidor. Precipitous ideals and Σ_{4}^{1} sets. Israel J. Math., 35(1-2):109–134, 1980.

[21] Donald A. Martin and Robert M. Solovay. A basis theorem for Σ_{3}^{1} sets of reals. Ann. of Math. (2), 89:138–159, 1969.

[22] Donald A. Martin and John R. Steel. The extent of scales in L(R). In Cabal seminar 79–81, volume 1019 of Lecture Notes in Math., pages 86–96. Springer, Berlin, 1983.

[23] Donald A. Martin and John R. Steel. A proof of projective determinacy. J. Amer. Math. Soc., 2(1):71–125, 1989.

[24] Yiannis N. Moschovakis. Descriptive set theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1980.

[25] Ralf Schindler and Martin Zeman. Fine structure theory. In this Handbook.

[26] Daniel Evan Seabold. Chang’s conjecture and the non-stationary ideal. J. Symbolic Logic, 66(1):144–170, 2001.

[27] Saharon Shelah. Proper and improper forcing. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, second edition, 1998.

[28] Saharon Shelah and W. Hugh Woodin. Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable. Israel J. Math., 70(3):381–394, 1990.

[29] Saharon Shelah and Jindřich Zapletal. Canonical models for ℵ_{1}- combinatorics. Ann. Pure Appl. Logic, 98(1-3):217–259, 1999.

[30] Robert M. Solovay. A model of set-theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2), 92:1–56, 1970.

[31] Robert M. Solovay. The independence of DC from AD. In Cabal Seminar 76–77 (Proc. Caltech-UCLA Logic Sem., 1976–77), volume 689 of Lecture Notes in Math., pages 171–183. Springer, Berlin, 1978.

[32] John R. Steel. An outline of inner model theory. In this Handbook.

[33] John R. Steel and Robert Van Wesep. Two consequences of determinacy consistent with choice. Trans. Amer. Math. Soc., 272(1):67–85, 1982.

[34] John R. Steel and Philip D. Welch. Σ_{3}^{1} absoluteness and the second uniform indiscernible. Israel J. Math., 104:157–190, 1998.

[35] Stevo Todorčević. Partition problems in topology, volume 84 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1989.

[36] W. Hugh Woodin. Some consistency results in ZFC using AD. In Cabal seminar 79–81, volume 1019 of Lecture Notes in Math., pages 172–198. Springer, Berlin, 1983.

[37] W. Hugh Woodin. The axiom of determinacy, forcing axioms, and the nonstationary ideal, volume 1 of de Gruyter Series in Logic and its Applications. Walter de Gruyter & Co., Berlin, 1999.

[38] W. Hugh Woodin. The continuum hypothesis. I. Notices Amer. Math. Soc., 48(6):567–576, 2001.

[39] W. Hugh Woodin. The continuum hypothesis. II. Notices Amer. Math. Soc., 48(7):681–690, 2001.

[40] W. Hugh Woodin. The Ω conjecture. In Aspects of complexity (Kaikoura, 2000), volume 4 of de Gruyter Ser. Log. Appl., pages 155–169. de Gruyter, Berlin, 2001.

[41] W. Hugh Woodin. Correction: “The continuum hypothesis. II”. Notices Amer. Math. Soc., 49(1):46, 2002.

[42] W. Hugh Woodin. Set theory after Russell: the journey back to Eden. In One hundred years of Russell’s paradox, volume 6 of de Gruyter Ser. Log. Appl., pages 29–47. de Gruyter, Berlin, 2004.

[43] W. Hugh Woodin. The cardinals below |[ω_{1}]^{<ω_{1}}|. Ann. Pure. Appl. Logic, 140(1-3):161–232, 2006.

Index

A-closed model, 47 A(G), 38 A_{G}, 29 I_{G}, 29 S_{p}^{f}, 53 Y_{Coll}(I), 53 ♢_{ω}(ω_{2}), 46 Ω-conjecture, 50 Ω-logic, implication, 48 Ω_{T} -validity, 47 Ω_{ZFC}-consistency, 48 Θ, 44

δ_{∼}_{2}^{1}, 6

δ-weakly homogeneous tree, 23 κ-universally Baire set of reals, 38 <δ-weakly homogeneously Suslin set

of reals, 23 G((M, I), J, B), 17 G_{ω_{1}}(p, J, B), 30 G_{ω}(⟨(N_{i}, J_{i}) : i < ω⟩, I, B), 21 P(ω_{1})_{G}, 29 NS_{ω_{1}}, 6 Col(α, β), 10 Col(α, <β), 10 o.t., 35 ZFC^{∘}, 7

AD_{ℝ}, 46

Chang’s Conjecture, 46 countably complete tower, 22

DC, 44

homogeneity of ℙ_{max}, 17 iterable pre-limit sequence, 20 iterate of a pre-limit sequence, 20 iteration of a pre-limit sequence, 19

ladder system, 30 limit sequence, 18

Martin’s Maximum^{++}(𝔠), 46 Martin’s Maximum^{+β}, 43

normal ideal, 7

normal ultrafilter for a sequence, 19

optimal iteration lemma, 57

pre-condition, 14 precipitous ideal, 7 presaturated, 37

semi-saturated ideal on ω_{2}, 46

universally Baire set of reals, 47

weakly homogeneously Suslin set of

reals, 23 weakly presaturated ideal, 46