Regular embeddings of the stationary tower and Woodin’s Σ_{2}^{2} maximality theorem

Richard Ketchersid  Paul B. Larson^{∗} Jindřich Zapletal^{†}

August 12, 2009

Abstract

We present Woodin’s proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all Σ_{2}^{2} sentences ϕ such that CH +ϕ holds in a forcing extension of V by a partial order in V_{δ}. We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding j: V → M with critical point ω_{1}^{V} such that M is countably closed in the forcing extension.

1 Introduction

Woodin’s Σ_{1}^{2} absoluteness theorem (see [6]) says that if δ is a measurable Woodin cardinal and ϕ is a Σ_{1}^{2} sentence which can be forced by a partial order in V_{δ}, then ϕ holds in every forcing extension by a partial order in V_{δ} which satisfies the Continuum Hypothesis. A longstanding open question (due to Steel) is whether this result extends to Σ_{2}^{2} sentences and Jensen’s principle ◇, that is, is there a large cardinal concept such that whenever δ is such a cardinal and ϕ is a Σ_{2}^{2} sentence such that ϕ + CH can be forced by a partial order in V_{δ}, then ϕ holds in every forcing extension by a partial order in V_{δ} which satisfies ◇? This paper presents a theorem of Woodin in this area, saying that if δ is a measurable Woodin cardinal, then there is a forcing extension satisfying all Σ_{2}^{2} sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V_{δ}. We present this result in a slightly extended form, adding predicates for universally Baire sets of reals.

Before presenting Woodin’s proof, we use some of the techniques from the proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then there is a homogeneous partial order which forces that there is an elementary embedding j: V → M with critical point ω_{1}^{V} such that M is countably closed in the forcing extension. The existence of such a partial order has applications in the study of Woodin’s core model induction (see [8]). For instance, Steel has shown that CH plus the existence of such a partial order implies that the Axiom of Determinacy holds in L(ℝ) and stronger models such as L(ℝ^{#}), L(ℝ^{##}), etc. The previous consistency strength upper bound for the existence of such a partial order was a superstrong cardinal (see [4] for definitions of the large cardinals used in this paper, and [6] for background on the stationary tower). This work came after we learned Woodin’s proof, but since it is simpler we present it first.

∗Research supported in part by NSF grants DMS-0401603 and DMS-0801009. The proof of Woodin’s Σ_{2}^{2} maximality theorem given in this paper is reconstructed from a conversation between Woodin and the second author at the Mathematisches Forschungsinstitut Oberwolfach Set Theory meeting in December 2005.

†Research supported in part by NSF grant DMS-0300201.

1.1 Terminology and background

We say that two partial orders are forcing-equivalent if the regular open algebras they generate are isomorphic, and that a partial order P is homogeneous if for every pair of conditions p, q in P there are conditions p′ ≤ p and q′ ≤ q such that the restrictions of P below p′ and q′ are forcing-equivalent. If P is a homogeneous partial order, then the theory (with parameters from the ground model) of every P -extension is the same, and thus computable in the ground model. We make key use of a standard forcing fact due to McAloon (Lemma 26.7 of [3] and Theorem A.0.7 of [6]), where for any cardinal γ and any set X, Coll(γ, X) is the partial order consisting of partial functions from γ to X of cardinality less than γ, ordered by inclusion.

Theorem 1.1. Any separative partial order P such that forcing with P makes P countable is forcing-equivalent to Coll(ω, P).

A regular embedding of a Boolean algebra A into a Boolean algebra B is a map π: A → B which preserves order and which maps maximal antichains of A to maximal antichains of B. Given partial order P and Q, we say that P regularly embeds into Q if the regular open algebra of P regularly embeds into the regular open algebra of Q. We make use of the following classical fact.

Theorem 1.2. If A and B are complete Boolean algebras and there is a Bname τ for a V -generic filter G ⊂ A such that [[ǎ ∈ τ]]_{B} is nonzero for every element of A, then A regularly embeds into B.

Proof. For each a ∈ A, let π(a) = [[ǎ ∈ τ]]_{B}.

The following facts appear in the appendix of [6].

Theorem 1.3. If P regularly embeds into Q, then there is a P -name τ for a partial order such that Q and P ∗ τ are forcing-isomorphic.

Theorem 1.4. If M is a model of ZFC, δ is a limit ordinal of M and x, y are sets such that {x, y} exists in a generic extension of M by a partial order in V_{δ}^{M}, then x exists in a generic extension of M[y] by a partial order in V_{δ}^{M[y]}.

We refer the reader to [6] for the definition of the stationary tower ℚ_{<δ}, as well as its basic properties. We will use the following standard facts. For the first of these, see Fact 2.7.3 of [6] and the paragraph which follows it. For the second, see Theorem 2.7.7 of [6]. For the third, see Lemma 2.7.14 of [6] and the paragraph which precedes it.

Theorem 1.5. If δ is a strongly inaccessible cardinal, then forcing with ℚ_{<δ} makes every ordinal less than δ countable.

Theorem 1.6. If δ is a Woodin cardinal, G ⊂ ℚ_{<δ} is a V -generic filter and j: V → M is the associated elementary embedding, then M is closed under ω-sequences in V [G].

Theorem 1.7. If γ is a Woodin cardinal then there is a stationary set a_{γ} consisting of countable subsets of V_{γ+1} such that for every strongly inaccessible cardinal η > γ, the inclusion map regularly embeds ℚ_{<γ} into the restriction of ℚ_{<η} to conditions b ≤ a_{γ}.

We will use the notation a_{γ} for the condition referred to in the statement of Theorem 1.7. Theorem 1.7 has a converse: for η and γ as in the statement of the theorem, a_{γ} is in the generic filter for ℚ_{<η} if and only if the restriction of the generic filter to ℚ_{<γ} is generic (see Lemma 2.7.16 of [6]).

2 Slow clubs

Suppose that M is a model of ZF, and let δ be an ordinal in M. An M-slow club through δ is a club D ⊂ δ with the property that for each limit element β of D, D intersects every club subset of β in M. When β has cofinality ω in the model containing D, the intersection requirement in the notion of slow club is nontrivial. Given a set (or class) of ordinals S, we say that a limit ordinal γ is 1-S-Mahlo if S ∩ γ is a stationary subset of γ, and, for any positive n ∈ ω, γ is (n + 1)-S-Mahlo if the set of n-S-Mahlo ordinals in S below γ is stationary. If D is an M-slow club contained in a set S in M, then every limit point of D is 1-S-Mahlo in M. For any stationary set S consisting of limit ordinals, the set of γ ∈ S which are not 1-S-Mahlo is also stationary, since for any club C ⊂ sup(S) consisting of limit ordinals, the first limit point of C in S is such a γ. This puts some limitations on methods for adding slow clubs.

2.1 Definition. Suppose that δ is a limit ordinal and S is a subset of δ. We let SC(δ, S) be the partial order consisting of triples (c, e, f) such that

• c is a finite subset of S;

• e is a finite set of closed, bounded intervals of δ disjoint from c;

• f is a regressive function whose domain is the set of α ∈ c which are not 1-S-Mahlo;

• (f(α), α) ∩ c = ∅ for each α ∈ dom(f).

Given (c, e, f), (b, d, g) in SC(δ, S), (c, e, f) ≤ (b, d, g) if b ⊂ c, d ⊂ e and g ⊂ f.

The partial order SC(δ, S) has cardinality δ. Fact 2.2 below shows that if S is cofinal in δ and G ⊂ SC(δ, S) is a V -generic filter, then

C_{G} = ⋃{c | (c, e, f) ∈ G}

is an unbounded subset of δ (we call C_{G} the generic club added by SC(δ, S)). Fact 2.3 shows that C_{G} is closed. Together they show that C_{G} is a V -slow club subset of δ when S is cofinal in δ; moreover, they show that for each limit element β of C_{G}, C_{G} ∩ β intersects every cofinal subset of β ∩ S in the ground model. By Fact 2.2 and the definition of SC(δ, S), for each γ ∈ C_{G}, γ is a limit point of C_{G} if and only if γ is 1-S-Mahlo in V.

2.2 Fact. Let (c, e, f) be a condition in SC(δ, S) and let γ be any element of

S \ (⋃ e ∪ ⋃{(f(α), α) : α ∈ dom(f)}).

If γ is 1-S-Mahlo, then (c ∪ {γ}, e, f) ∈ SC(δ, S) and (c ∪ {γ}, e, f) ≤ (c, e, f). If γ is not 1-S-Mahlo, then (c ∪ {γ}, e, f ∪ {(γ, max(c ∩ γ))}) ∈ SC(δ, S) and

(c ∪ {γ}, e, f ∪ {(γ, max(c ∩ γ))}) ≤ (c, e, f).

2.3 Fact. If (c, e, f) is a condition in SC(δ, S) and γ ∈ δ \ c is a limit ordinal, then

(c, e ∪ {[max(c ∩ γ) + 1, γ]}, f) ≤ (c, e, f).

Fact 2.4 below shows that the forcing SC(δ, S) factors at each 1-S-Mahlo ordinal in S below δ. We will use this fact to demonstrate the homogeneity of various forcings considered in this paper. It also shows that if δ is a regular cardinal and 2-S-Mahlo, then SC(δ, S) preserves the regularity of δ, since, in this case, for every dense D ⊂ SC(δ, S) there will be club many γ < δ such that D ∩ SC(γ, S ∩ γ) is dense in SC(γ, S ∩ γ).

2.4 Fact. For any condition (c, e, f) ∈ SC(δ, S), and any 1-S-Mahlo α ∈ c, the partial order SC(δ, S) below (c, e, f) is isomorphic to the partial order

SC(α, S ∩ α) × SC(δ, S \ (α + 1))

below the condition

((c ∩ α, {I ∈ e | I ⊂ α}, f ∩ α^{α}), (c \ (α + 1), {I ∈ e | I ∩ α = ∅}, f \ α^{α})).

Lemma 2.5 below shows that when δ is a regular cardinal and 2-S-Mahlo, every set of ordinals of cardinality less than δ in the SC(δ, S)-extension is added by an initial segment of the partial order. It follows that forcing with SC(δ, S) makes CH hold when δ is strongly inaccessible and 2-S-Mahlo, since Lemma 2.6 implies that δ is the ω_{1} of such an extension.

Lemma 2.5. Suppose that δ is a regular cardinal, S ⊂ δ and δ is 2-S-Mahlo. Let G ⊂ SC(δ, S) be V -generic. Then for every element x of [Ord]^{<δ} in V [G], there exists a limit member γ of C_{G} such that G ∩ SC(γ, S ∩ γ) is V -generic for SC(γ, S ∩ γ), and x ∈ V [G ∩ SC(γ, S ∩ γ)].

Proof. Fix ξ < δ and let τ_{α} (α < ξ) be SC(δ, S)-names for ordinals. For each α < ξ, let T_{α} be the set of pairs (p, β) such that p ∈ SC(δ, S) and p⊩τ_{α} = β̌. Let q = (c, e, f) be a condition in SC(δ, S). Let θ be a regular cardinal greater than 2^{δ} and let Z be an elementary submodel of H(θ) such that

{δ, S, q, ⟨T_{α} : α < ξ⟩} ∈ Z,

Z ∩ δ ∈ S and Z ∩ δ is 1-S-Mahlo. Let γ = Z ∩ δ. Then (c ∪ {γ}, e, f) ≤ (c, e, f), and, by Lemma 2.4, (c ∪ {γ}, e, f) forces that the restriction of the generic filter to SC(γ, S ∩ γ) will be generic. Furthermore, for each α < ξ,

{p ∈ SC(γ, S ∩ γ) | ∃β (β̌, p) ∈ T_{α}}

is predense in SC(γ, S ∩ γ) below (c, e, f). The lemma then follows by Fact 2.4.

It follows from Lemma 2.5 that if δ is a regular cardinal and 2-S-Mahlo, then δ has uncountable cofinality in the SC(δ, S) extension. The following lemma is a sort of converse. Applying Theorem 1.1, it also shows that in many cases SC(γ, S) is forcing-equivalent to Coll(ω, γ). It follows that SC(δ, S) makes δ countable if S consists of regular cardinals and δ is a limit of 1-S-Mahlo ordinals, but not 2-S-Mahlo.

Lemma 2.6. Let γ be an ordinal, let S be a cofinal subset of γ, and suppose that γ is not a limit of 1-S-Mahlo members of S. Then forcing with SC(γ, S) makes cof(γ)^{V} countable.

Proof. Let β be the supremum of the 1-S-Mahlo members of S below δ (let β = 0 if this set is empty), and let {T_{α} : α < cof(γ)} be a partition of S into cofinal sets. The generic club given by SC(γ, S) will have ordertype ω in the interval (β, γ), and will intersect each T_{α}, inducing a surjection from ω onto cof(γ)^{V}.

The following lemma gives a homogeneity property of SC(δ, S) for suitable δ and S.

Lemma 2.7. Suppose that δ is a cardinal, and that S is a set of regular cardinals below δ such that δ is a limit of 1-S-Mahlo members of S. Let p and q be conditions in SC(δ, S). Then there exist conditions p′ ≤ p and q′ ≤ q such that the restrictions of SC(δ, S) below p′ and q′ are forcing-equivalent.

Proof. Let p = (b, d, g) and q = (c, e, f). Let γ ∈ S be 1-S-Mahlo but not a limit of 1-S-Mahlo ordinals, such that γ is larger than every member of b∪c∪⋃ d∪⋃ e.

Let p′ = (b ∪ {γ}, d, g) and let q′ = (c ∪ {γ}, e, f). Then SC(δ, S) below the condition p′ is isomorphic to

SC(γ, S ∩ γ) × SC(δ, S \ (γ + 1))

below the condition

((b, d, g), (∅, ∅, ∅))

and SC(δ, S) below the condition q′ is isomorphic to

SC(γ, S ∩ γ) × SC(δ, S \ (γ + 1))

below the condition

((c, e, f), (∅, ∅, ∅)).

By Lemma 2.6, SC(γ, S ∩ γ) below (b, d, g) and SC(γ, S ∩ γ) below (c, e, f) are both forcing-equivalent to Coll(ω, γ).

3 Slow clubs and the stationary tower

Given n ∈ ω and a cardinal δ, we say that δ is n-Mahlo-Woodin if it is n-W - Mahlo, where W denotes the class of Woodin cardinals. Recall that a stationary limit of regular cardinals is regular, so a stationary limit of Woodin cardinals is Woodin. The hypotheses of Theorem 3.1 below imply that ω_{1}^{V} is a 2-Mahlo-Woodin cardinal in M.

Our main application of slow clubs is the construction of ℚ_{<δ}^{M}-generic filters for suitable inner models M.

Theorem 3.1. Suppose that M is a model of ZFC and D ⊂ ω_{1}^{V} is an M-slow club contained in the Woodin cardinals of M. Then there exists an M-generic filter for ℚ_{<ω_{1}^{V}}^{M} containing any given condition.

Proof of Theorem 3.1. Let p be a condition in ℚ_{<ω_{1}^{V}}^{M}. Removing an initial segment of D if necessary, we may assume that p ∈ ℚ_{<γ_{0}}^{M}, where γ_{0} is the least element of D. For each γ ∈ D, let G_{γ} be the set of g such that

• g is an M-generic filter for ℚ_{<γ}^{M} containing p;

• for all η ∈ D ∩ γ, g ∩ V_{η}^{M} is M-generic for ℚ_{<η}^{M}.

Since ω_{1}^{V} is a strongly inaccessible cardinal in M, G_{γ_{0}} is nonempty.

Let T be the tree on ⋃_{γ∈D} G_{γ} ordered by: g ≥ h whenever g ∈ G_{γ} and h ∈ G_{η}, for some γ, η in D, and g ∩ V_{η}^{M} = h. Theorems 1.3 and 1.7 (and the facts that ω_{1}^{V} is strongly inaccessible in M and D consists of Woodin cardinals of M) implies that every member of G has proper extensions in T. The theorem follows from the fact that T is countably closed, and the fact that the union of each uncountable branch through T is an M-generic filter for ℚ_{<ω_{1}^{V}}^{M}.

To see that T is countably closed, note that if γ is a limit point of D, then each predense subset of ℚ_{<γ}^{M} in M has predense intersection with ℚ_{<η}^{M} for club many η < γ, relative to the set of strongly inaccessible cardinals below γ, and thus with ℚ_{<η}^{M} for some η ∈ γ ∩ D. It follows that if g is a subset of ℚ_{<γ}^{M} such that g ∩ ℚ_{<η}^{M} is an M-generic filter for all η ∈ γ ∩ D, then g is also an M-generic filter. Similarly, each predense subset of ℚ_{<ω_{1}^{V}}^{M} in M has predense intersection with ℚ_{<η}^{M} for (relative) club many η < ω_{1}^{V}, and thus with ℚ_{<η}^{M} for some η ∈ D. It follows that if G is a subset of ℚ_{<ω_{1}^{V}}^{M} such that G ∩ ℚ_{<η}^{M} is an M-generic filter for all η ∈ D, then G is also an M-generic filter.

It follows from Theorem 3.1 that ℚ_{<δ} regularly embeds into any forcing which collapses δ to be ω_{1} and adds a V -slow club through the Woodin cardinals below δ. The results of the previous section show that that SC(δ, W) is such a forcing when W is the set of Woodin cardinals below a 2-Mahlo-Woodin cardinal δ.

The proof of Theorem 3.2 below is a modification of the proof of Theorem 3.1. The new element of the proof is the use of Theorem 1.4 to make sure that the reals of V [G] are all in V [H]. As in the proof of Theorem 3.2, the construction follows the generic club G exactly in order to use the fact that G is V -slow. So for each real of V [G] the construction has to wait until it reaches a sufficiently large element of W. We work with a tail of the Woodin cardinals below δ in order to apply the theorem in the proof of Theorem 3.4. We use the notion of nice names from [5] (see page 208), simply to restrict to a sufficiently large set-sized collection of names.

Theorem 3.2. Let δ be a 2-Mahlo-Woodin cardinal, let χ be an element of δ, let W denote the Woodin cardinals of V in the interval (χ, δ), and let G ⊂ SC(δ, W) be a V -generic filter. Then there exists in V [G] a V -generic filter H ⊂ ℚ_{<δ}^{V}, containing any given condition, such that V [H] contains the reals of V [G].

Proof. Let p be a condition in ℚ_{<δ}^{V} and let γ_{0} be the least γ ∈ W with p ∈ ℚ_{<γ}. Let W_{1}^{0} be the set of 1-Mahlo-Woodin cardinals in (γ_{0}, δ) which are not limits of 1-Mahlo-Woodin cardinals. Let ⟨τ_{ξ} : ξ < δ⟩ be a listing in V of all nice SC(γ, W ∩ γ)-names for reals, for all 1-Mahlo-Woodin γ < δ.

For each α < β in W_{1}^{0}, let N_{α,β} be the set of nice SC(β, W ∩ β)-names σ for which it is forced that if α and β are in C_{G}, then the realization of σ is a V -generic filter h ⊂ ℚ_{<α} such that

• h ∩ ℚ_{<γ_{0}} is a V -generic filter containing p.

• h ∩ ℚ_{<γ} is V -generic for all γ ∈ (C_{G} ∩ α) \ γ_{0}.

Fix (suppressed) wellorders of the sets N_{α,β}.

Let C^{∗} be the set of limit points of C_{G}. Working in V [G], recursively define a sequence ⟨h_{α} : α ∈ C^{∗} \ (γ_{0} + 1)⟩ such that

• h_{min(C^{∗}\(γ_{0}+1))} = ∅; • if γ is a limit element of C^{∗} \ (γ_{0} + 1), α is the least element of C^{∗} greater

than γ and β is the least element of C^{∗} greater than α, then h_{α} is the realization by G of the least element of N_{α,β} whose realization h extends h_{γ} and has the realization of τ_{ξ} in V [h], where ξ < δ is least such that

– τ_{ξ} is an SC(η, W ∩ η)-name for a real, for some η ∈ C^{∗} ∩ (γ + 1), and

– the realization of τ_{ξ} by h_{γ} is not in V [h_{γ}],

if such an ξ exists, otherwise h_{α} is the realization of the least element of N_{α,β} whose realization h extends h_{γ};

• if γ is not a limit element of C^{∗} \ (γ_{0} + 1), α is the least element of C^{∗} greater than γ and β is the least element of C^{∗} greater than α, then h_{α} is the realization of the least element of N_{α,β} which extends h_{γ};

• if α is a limit element of C^{∗} \ (γ_{0} + 1), then h_{α} = ⋃_{β∈α∩C^{∗}} h_{β}.

It follows from this construction that whenever γ is a limit element of the set C^{∗} \(γ_{0} +1), h_{γ} ∈ V [G∩SC(γ, W ∩γ)]. Let H = ⋃{h_{α} : α ∈ C^{∗} \(γ_{0} +1)}. Let E be the set of ξ < δ such that τ_{ξ} is an SC(η, W ∩ η)-name, for some ξ ∈ C^{∗}. By Lemma 2.5, every real in V [G] is the realization of τ_{ξ} for some ξ ∈ E. If ξ were the least ζ ∈ E such that the realization of τ_{ζ} were not in V [H], then, since ξ is countable in V [H] and δ is uncountable, there would be some limit element γ of C^{∗} \ (γ_{0} + 1) such that ξ is the least ζ < δ such that

• τ_{ζ} is an SC(η, W ∩ η)-name for a real, for some η ∈ C^{∗} ∩ (γ + 1), and

• the realization of τ_{ζ} by h_{γ} is not in V [h_{γ}].

Then the realization of τ_{ξ} is in V [h_{α}] by the construction above (and Theorem 1.4), where α is the least element of C^{∗} above γ.

Theorem 3.3 below is the main original result of this paper.

Theorem 3.3. Suppose that δ is a 2-Mahlo-Woodin cardinal, and let W denote the set of Woodin cardinals below δ. Then the partial order SC(δ, W) is homogeneous, and in the extension by this partial order there is an elementary embedding from V into a model M which is closed under ω-sequences in the forcing extension.

Proof. The partial order SC(δ, W) is homogeneous by Lemma 2.7. By Lemma 2.5, every countable set of ordinals in any forcing extension of V by SC(δ, W) is in a model of the form V [x] for some real in the extension. By Lemma 3.2, in any SC(δ, W) extension there is a V -generic filter H ⊂ ℚ_{<δ}^{V} such that V [H] contains all the reals of the SC(δ, W)-extension, and therefore all initial segments of the SC(δ, W)-generic filter. By Theorem 1.6, the image model M of the embedding induced by H is ω-closed in V [H], which is ω-closed in the SC(δ, W)-extension, which means that M is ω-closed in this extension.

Theorem 3.2 has the following additional corollary.

Theorem 3.4. Let δ be a 2-Mahlo-Woodin cardinal, let W denote the Woodin cardinals of V below δ, and let G ⊂ SC(δ, W) be a V -generic filter. Then every Σ_{1}^{2} sentence which can be forced by a partial order in V_{δ} holds in V [G].

Proof. Let P be a partial order in V_{δ} forcing a Σ_{1}^{2} sentence ∃X ⊂ ℝ ϕ(X), where all quantifiers in ϕ range over the reals. Let κ ∈ C_{G} be 1-Mahlo-Woodin and not a limit of 1-Mahlo-Woodin cardinals. Then by Lemma 2.6 and Fact 2.4 there is a V -generic filter h ⊂ P in V [G ∩ SC(κ, W ∩ κ)]. Let X be a set of reals satisfying ϕ in V [h]. By [2], the set of Woodin cardinals in the interval (κ, δ) is the same in V [h], V and V [G ∩ SC(κ, W ∩ κ)]. By Fact 2.4, C_{G} \ (κ + 1) is a V [h]-generic club for SC(δ, W \ (κ + 1)). By Theorem 3.2 there exists in V [G] a V [h]-generic filter H ⊂ ℚ_{<δ}^{V [h]} such that V [h][H] contains the reals of V [G]. Letting j: V [h] → M be the embedding induced by H, it follows from Theorem 1.6 that j(X) satisfies ϕ in V [G].

4 Σ_{2}^{2} maximality

In this final section we give a proof of Woodin’s Σ_{2}^{2} maximality theorem. The theorem is presented in various forms in Corollaries 4.11, 4.12 and 4.14. Most of the work goes into the proof of Theorem 4.10, which is a variant of the proof of Woodin’s Σ_{1}^{2} absoluteness theorem. The proof of Theorem 4.10 in turn requires setting up some machinery. We start by discussing symmetric extensions.

Given a strong limit cardinal δ of a ZFC model M, we take a δ-symmetric extension of M to be the least model M(ℝ^{∗}) of ZF containing M and a set of reals ℝ^{∗} with the properties that

• M(ℝ^{∗}) ∩ ℝ = ℝ^{∗};

• every member of ℝ^{∗} is generic over M by a forcing in V_{δ}^{M};

• the supremum of {ω_{1}^{L[x]} : x ∈ ℝ^{∗}} is δ.

We refer the reader to [3, 6] for more general definitions of symmetric extension. We typically denote a symmetric extension of a model M by M(ℝ^{∗}), where ℝ^{∗} is understood to be the reals of the extension. We note the following facts about δ-symmetric extensions, for a strong limit cardinal δ: (1) any two δ-symmetric extensions of M are elementarily equivalent (even with parameters from M);

(2) if M(ℝ^{∗}) is a δ-symmetric extension of M and P is a partial order in V_{δ}^{M} then M(ℝ^{∗}) is a δ-symmetric extension of an extension of M by P.

The following is Theorem 3.1.6 in [6].

Theorem 4.1. If δ is a Woodin limit of Woodin cardinals and G ⊂ ℚ_{<δ} is a V -generic filter, then V (ℝ^{V [G]}) is a δ-symmetric extension of V.

Whenever δ is a strongly inaccessible cardinal and G is V -generic for the partial order Coll(ω, <δ), V (ℝ^{V [G]}) is a δ-symmetric extension of V. Fact 2.4 and Lemmas 2.5 and 2.6 show that the same is true for SC(δ, S), when δ is a strongly inaccessible and 2-S-Mahlo, and S is a set of regular cardinals.

Given a model M of ZF, an ordinal δ ∈ M and S ⊂ δ in M, let SL(M, δ, S) be the partial order consisting of all M-generic filters for partial orders of the form SC(γ, S ∩γ)^{M}, where γ ∈ S is 1-S-Mahlo in M, ordered by end-extension. When g ∈ SL(M, δ, S) is an M-generic filter for SC(γ, S ∩ γ)^{M}, we say that the length of g is γ. Since filters for SC(δ, S) are uniquely determined by their corresponding club sets, we sometimes identify a condition g in SL(M, δ, S) with the set C_{g} ∪ {sup(C_{g})}; so each condition in SL(M, δ, S) can be identified with a closed, bounded subset of S.

The partial order SL(M, δ, S) is not ω-closed. However, it is a tree ordering, so if the set of 1-S-Mahlo γ ∈ S is cofinal in δ and δ is the ω_{1} of some SL(M, δ, S)-extension, then there are no new countable sequences of ordinals in this extension.

We let Add(1, δ) denote the forcing which adds a subset of δ by initial segments. The following lemma follows from Theorem 1.1, Fact 2.4, Lemmas 2.6 and 2.5, and genericity.

Lemma 4.2. Suppose that

• δ is a regular uncountable cardinal;

• S is a set of regular cardinals below δ and δ is 2-S-Mahlo;

• V (ℝ^{∗}) is a δ-symmetric extension of V;

Then

• if D is a V (ℝ^{∗})-generic club for SL(V, δ, S), then

– D is V -generic for SC(δ, S),

– ℝ^{∗} ⊂ V [D],

– V (ℝ^{∗})[D] = V [D];

• if (D, B) is V (ℝ^{∗})-generic for SL(V, δ, S) × Add(1, δ), then

– B is V [D]-generic for Add(1, δ),

– V (ℝ^{∗})[D][B] = V [D][B],

– V [D][B] is a generic extension of V by the partial order

SC(δ, S) ∗ Add(1, δ).

• forcing with SL(V, δ, S) over V (ℝ^{∗}) does not collapse δ.

Proof. To see that D is V -generic for SC(δ, S), let E be a dense subset of SC(δ, S) in V and let g be a condition in SL(V, δ, S). Let γ be the length of g. By Fact 2.4, SC(δ, S) below ({γ}, ∅, ∅) is isomorphic to

SC(γ, S ∩ γ) × SC(δ, S \ (γ + 1)),

and we can let E′ be the image of E (below ({γ}, ∅, ∅)) in this product. Since g is a generic filter for SC(γ, S∩γ), there is a condition (p, q) in E′ with p ∈ g. Let η > γ be 1-S-Mahlo in V with q ∈ SC(η, S ∩ (γ, η)), and let h be a V [g]-generic filter for SC(η, S ∩ (γ, η)) with q ∈ h. Then the preimage of (g, h) in SC(δ, S) is a condition in SL(V, δ, S) extending g meeting E. By genericity, then, D is V -generic for SC(V, δ, S).

To see that ℝ^{∗} ⊂ V [D], fix x ∈ ℝ^{∗} and let g be a condition in SL(V, δ, S). Let γ be the length of g. By Fact 2.4, SC(δ, S) below ({γ}, ∅, ∅) is isomorphic to SC(γ, S ∩ γ) × SC(δ, S \ (γ + 1)). Let η < δ be the least 1-S-Mahlo cardinal in S such that the pair {g, x} is V -generic for a partial order of cardinality η. Let h be a V [g]-generic filter for SC(η, S ∩ (γ, η)) with x ∈ V [g][h]. Then the preimage of (g, h) in SC(δ, S) is a condition g′ in SL(V, δ, S) extending g with x ∈ V [g′]. By genericity, then, ℝ^{∗} ⊂ V [D].

To see that B is V [D]-generic for Add(1, δ), let (g, a) be a condition in SL(V, δ, S) × Add(1, δ), and let τ be an SC(δ, S)-name for a dense subset of Add(1, δ). By the V -genericity of D, and Lemma 2.5, whenever D^{∗} is V (ℝ^{∗})- generic for SL(V, δ, S), every real in V [D^{∗}] is in V [D^{∗} ∩ η] for some η < δ. Therefore, there is a condition g′ below g in SL(V, δ, S) such that a ∈ V [g′] and such that some extension b of a in V [g′] is forced by some condition in g′ to be in the realization of τ. Then (g′, b) is below (g, a), and the V [D]-genericity of B follows by the V (ℝ^{∗})-genericity of (D, B).

By Lemma 2.5 and the V -genericity of D for SC(δ, S), forcing with SL(V, δ, S) over V (ℝ^{∗}) does not collapse δ. This in turn implies that V (ℝ^{∗})[D] = V [D], and that forcing with SL(V, δ, S) × Add(1, δ) over V (ℝ^{∗}) does not collapse δ.

Lemmas 4.3 and 4.4 give homogeneity properties for partial orders of the form SL(V, δ, S). We will use these facts in different contexts. In some sense, the proof of Lemma 4.3 is more important that the statement of the lemma itself. Lemma 4.3 uses Corollary 26.10 of [3], which (for our purposes) says that if γ is a regular cardinal, G ⊂ Coll(ω, γ) is a V -generic filter, and x ∈ V [G] is subset of V such that γ is uncountable in V [x], then there exists a V [x]-generic filter H ⊂ Coll(ω, γ) such that V [G] = V [x][H]. (This is very similar to our Theorems 1.1 and 1.3, but not quite the same.)

Lemma 4.3. Suppose that

• M is a model of ZFC;

• δ ≤ ω_{1}^{V} is an ordinal;

• P(α)^{M} is countable for each α < δ;

• S ⊂ δ is a set of regular cardinals in M;

• δ is a limit of 1-S-Mahlo ordinals in M.

Then SL(M, δ, S) is homogeneous.

Proof. Let p, q be conditions in SL(M, δ, S) of length γ_{p} and γ_{q}, respectively. Let γ be the least 1-S-Mahlo cardinal of M above both γ_{p} and γ_{q} such that the pair {p, q} is M-generic for a partial order in V_{γ}^{M}. Since SC(γ, S \ (γ_{p} + 1)) and SC(γ, S \ (γ_{q} + 1)) are both forcing-equivalent to Coll(ω, γ), there exist by Corollary 26.10 of [3] and Lemma 2.4 conditions p′ ≤ p and q′ ≤ q of length γ such that M[p′] = M[q′]. Then the restrictions of the partial order SL(M, δ, S) below the conditions p′ and q′ are isomorphic.

Since SL(V, δ, W) × Add(1, δ) is homogeneous (in the context of Lemma 4.3), Lemma 4.4 shows that the SL(V, δ, W) × Add(1, δ)-extension of V (ℝ^{∗}) is elementarily equivalent to the same extension defined over any forcing extension of V by a partial order in V_{δ}. An analogous version of the lemma for the partial order SC(δ, W) ∗ Add(1, δ) follows from the existence of a 2-Mahlo-Woodin cardinal. We will apply the lemma in an even stronger context.

Lemma 4.4. Suppose that

• δ is a strongly inaccessible limit of 1-Mahlo-Woodin cardinals;

• V (ℝ^{∗}) is a δ-symmetric extension of V;

• P, Q are partial orders in V_{δ};

• g ⊂ P and h ⊂ Q are V -generic filters in V (ℝ^{∗});

• W_{g} is the set of Woodin cardinals of V [g] below δ;

• W_{h} is the set of Woodin cardinals of V [h] below δ;

• p is a condition in SL(V [g], δ, W_{g});

• q is a condition in SL(V [h], δ, W_{h}).

Then there exist conditions p′ ≤ p and q′ ≤ q such that SL(V [g], δ, W_{g}) below p′ and SL(V [h], δ, W_{h}) below q′ are isomorphic.

Proof. Let γ_{p} and γ_{q} be the respective lengths of p and q. Let γ be the least 1-S-Mahlo cardinal of V above both γ_{p} and γ_{q} such that the set {p, q, g, h} is V -generic for a partial order in V_{γ}. Since SC(γ, W_{g} \ (γ_{p} + 1))^{V [g]} and SC(γ, W [h] \ (γ_{q} + 1))^{V [h]} are both forcing-equivalent to Coll(ω, γ) in their respective models, there exist by Corollary 26.10 of [3] and Lemma 2.4 conditions p′ ≤ p in SL(V [g], δ, W_{g}) and q′ ≤ q in SL(V [h], δ, W_{h}) of length γ such that V [g][p′] = V [h][q′]. Then since W_{g} \γ = W_{h}\γ, p′ and q′ are as desired.

Very roughly (i.e., suppressing a few issues for a moment), the last remaining tool we need to develop for the proof of Theorem 4.10 is the ability to use a V -slow club to construct a V (ℝ^{∗})-generic filter for a partial order of the form SL(V, δ, S) × Add(1, δ) such that the corresponding extension contains a given V -generic filter for ℚ_{<δ}. To do this there needs to be a name in V (ℝ^{∗}) which gives rise to the desired filter. The filters g_{(d,b)} defined below are initial segments of the realization of this name.

If V (ℝ^{∗}) is a δ-symmetric extension of V and B is V (ℝ^{∗})-generic for Add(1, δ), then, considering consecutive ω-sequences from δ and membership (or not) in B, B lists all the members of ℝ^{∗}. We fix a recursive coding of elements of H(ω_{1}) by subsets of ω, and consider elements of H(ω_{1}) coded by consecutive ω-sequences from B in this fashion.

Suppose that δ is a limit of Woodin cardinals, and let W denote the set of Woodin cardinals below δ. Given a condition (d, b) in SL(V, δ, W)×Add(1, δ), we define a set g_{(d,b)} and an ordinal η_{(d,b)} such that either g_{(d,b)} = ∅ and η_{(d,b)} = 0 or g_{(d,b)} is a V -generic filter g_{(d,b)} in ℚ_{<η_{(d,b)}}^{V} and η_{(d,b)} ∈ d. If d is empty, so is g_{(d,b)} (so η_{(d,b)} = 0). Otherwise, η_{(d,b)} and g_{(d,b)} are defined as follows. Let g_{0} = 0 and β_{0} = 0, and, for each limit element γ of d, if g_{η} and β_{η} are defined for each η ∈ d ∩ γ, then let

g_{γ} = ⋃{g_{η} : η ∈ d ∩ γ}

and β_{γ} = sup{β_{η} : η < γ}. If g_{γ} is defined for each γ ∈ d, then g_{(d,b)} = g_{max(d)} and η_{(d,b)} = max(d). For each γ ∈ (d ∪ {0}) \ max(d), if g_{γ} and β_{γ} are defined, let γ^{+} denote the least member of d above γ. Then we choose g_{γ^{+}} and β_{γ^{+}} (or g_{(d,b)}) in the following way.

• If some consecutive ω-sequence from b above γ ∪β_{γ} codes a V -generic filter g ⊂ ℚ_{<γ^{+}}^{V} such that g ∩ ℚ_{<γ}^{V} = g_{γ}, then let g_{γ^{+}} be the first filter of this type coded by a consecutive ω-sequence from b above γ ∪ β_{γ}, and let β_{γ^{+}} be supremum of the indices of this ω-sequence.

• If there is no such consecutive ω-sequence from b above γ ∪ β_{γ}, then let g_{(d,b)} = g_{γ} and η_{(d,b)} = γ, and g_{γ^{+}} and β_{γ^{+}} are undefined.

If (d′, b′) ≤ (d, b) are conditions in SL(V, δ, W) × Add(1, δ), then g_{(d,b)} ⊂ g_{(d′,b′)} (and indeed the construction just given for (d, b) is an initial segment of the construction for (d′, b′)). The argument given in the proof of Theorem 3.1, using the fact that d is an V -slow club, shows that g_{(d,b)} is either ∅ or an V - generic filter for ℚ_{<η_{(d,b)}}^{V}. We say that (d, b) is complete if either (d, b) is the empty condition or

η_{(d,b)} = sup(b) = max(d)

and every real coded by a consecutive ω-sequence from b is in V [g_{(d,b)}]. The following lemma shows how to extend (d, b) in order to extend g_{(d,b)}.

Lemma 4.5. Suppose that

• M is a model of ZFC;

• δ is a 2-Mahlo-Woodin cardinal in M;

• ℝ^{∗} is the set of reals of V;

• M(ℝ^{∗}) is a δ-symmetric extension of M;

• W is the set of Woodin cardinals of M below δ; • (d, b) is a condition in SL(M, δ, W) × Add(1, δ);

• g is an M-generic filter for ℚ_{< max(d)}^{M} extending g_{(d,b)} such that a_{γ} ∈ g for every γ ∈ d \ η_{(d,b)}.

Then there exists a b′ extending b such that g_{(d,b′)} = g.

Proof. Clearly, if g_{(d,b)} = g, we can let b′ = b. Otherwise, η_{(d,b)} ∈ d \ {max(d)} and there is no consecutive ω-sequence from b above η_{(d,b)} ∪ {β_{(d,b)}} coding an M-generic filter g ⊂ ℚ_{<γ_{0}}^{M} such that g ∩ ℚ_{<η_{(d,b)}}^{M} = g_{η_{(d,b)}}, where γ_{0} is the least element of d above η_{(d,b)}. Let the first ω-sequence of b′ extending b above η_{(d,b)} be a real in M[g ∩ ℚ_{<γ_{1}}^{M}] coding g ∩ ℚ_{<γ_{0}}^{M}, where γ_{1} is the least element of d above γ_{0}. Then β_{γ_{0}} = sup(b′).

For each γ ≥ γ_{0} in d, let the first ω-sequence of b′ above γ ∪ β_{γ} be a real in M[g ∩ ℚ_{<γ_{2}}^{M}] coding g ∩ ℚ_{<γ_{1}}^{M}, where γ_{1} is the least member of d above γ, and γ_{2} is the least member of d above γ_{1}. Then β_{γ_{1}} = (γ ∪ β_{γ}) + ω.

For limit members γ of d above γ_{0}, β_{γ} is the supremum of {β_{η} : η < γ}. Let these be the only elements of b′ \ b.

In the context we will be working in, the complete conditions are dense.

Lemma 4.6. Suppose that δ is a 2-Mahlo-Woodin cardinal in a model M of ZFC, and M(ℝ^{∗}) is a δ-symmetric extension of M, where ℝ^{∗} is the set of reals in V. Let (d, b) be a condition in SL(M, δ, W) × Add(1, δ), where W is the set of Woodin cardinals of M below δ. Then there is complete condition (d′, b′) in SL(M, δ, W) × Add(1, δ) below (d, b).

Proof. By Theorem 3.1 and Lemma 4.5, we may assume that η_{(d,b)} = max(d). The set of reals coded by an ω-sequence from b is countable, so there is a real x constructing all such reals. Let ⟨γ_{i} : i ≤ ω⟩ be a continuous, increasing sequence of Woodin cardinals of M above max(d) ∪ sup(b) such that

1. γ_{0} is the least Woodin cardinal γ > max(d) ∪ sup(b) such that

• a ∈ ℚ_{<γ}^{M};

• the pair {g_{(d,b)}, x} exists in a generic extension of M by a partial order of cardinality less than γ;

2. γ_{ω} is the least 1-Mahlo-Woodin cardinal of M greater than γ_{0};

3. d ∪ {γ_{i} : i < ω} is M-generic for SC(γ_{ω}, W ∩ γ_{ω});

Then let d′ = d ∪ {γ_{i} : i < ω} and let b′ be a subset of γ_{ω} with the property that

• b′ end-extends b;

• the first ω-sequence of b′ above max(d) ∪ sup(b) is a real y_{0} coding an Mgeneric filter g_{0} ⊂ ℚ_{<γ_{0}} such that g_{0} ∩ V_{η_{(d,b)}}^{M} = g_{(d,b)}, a ∈ g_{0}, x ∈ M[g_{0}], and y_{0} exists in a generic extension of M by a partial order in V_{γ_{1}}^{M}; • for all i ∈ ω, the first ω-sequence of b′ above γ_{i} is a real y_{i+1} coding an

M-generic filter g_{i+1} ⊂ ℚ_{<γ_{i+1}} such that g_{i+1} ∩ V_{γ_{i}}^{M} = g_{i}, y_{i} ∈ M[g_{i+1}], and y_{i+1} exists in a generic extension of M by a partial order in V_{γ_{i+2}}^{M};

• all elements of b′ \ b are of the the form (max(d) ∪ sup(b)) + n or γ_{i} + n, for some i, n in ω.

Then (d′, b′) is the desired condition.

If (D, B) is a filter contained in SL(V, δ, W) × Add(1, δ), we let

g_{(D,B)} = ⋃{g_{(d,b)} | (d, b) ∈ (D, B)}.

Lemma 4.7 follows from Lemmas 4.2, 4.5 and 4.6.

Lemma 4.7. Suppose that δ is (in V) a 2-Mahlo-Woodin cardinal. Let V (ℝ^{∗}) be a δ-symmetric extension of V and let (D, B) be V (ℝ^{∗})-generic for

SL(V, δ, W) × Add(1, δ).

Then δ = ω_{1}^{V [D][B]}, g_{(D,B)} is a V -generic filter for ℚ_{<δ}^{V} and ℝ^{V [g_{(D,B)}]} = ℝ^{∗}.

Lemma 4.9 below is the main technical lemma for the proof of Theorem 4.10, showing that the genericity requirements for (D, B) don’t interfere with the desired result for g_{(D,B)}. Lemma 4.9 in turn requires the following technical lemma.

Lemma 4.8. Suppose that

• M is a model of ZFC;

• δ is a 2-Mahlo-Woodin cardinal in M;

• ℝ^{∗} is the set of reals in V;

• M(ℝ^{∗}) is a δ-symmetric extension of M;

• W is the set of Woodin cardinals of M below δ;

• (d, b) is a complete condition in SL(M, δ, W) × Add(1, δ),

• a is a condition in ℚ_{<δ}^{M} below a_{η_{(d,b)}}.

• ḋ and ḃ are ((ℚ_{<δ}↾a)/ℚ_{<η_{(d,b)}})^{M[g_{(d,b)}]}-names such that (ḋ, ḃ) is forced to be a complete condition in SL(M, δ, W)×Add(1, δ) such that g_{(ḋ,ḃ)} = g_{(d,b)};

Then there exist continuous, increasing sequences

d^{∗} = ⟨γ_{i} : i ≤ ω⟩

and

d′ = ⟨γ′_{i} : i ≤ ω⟩,

and sets b^{∗}, b′ and g such that • γ_{0} = γ′_{0} is a Woodin cardinal of M greater than η_{(d,b)} with a ∈ ℚ_{<γ_{0}}^{M};

• γ_{ω} = γ′_{ω} is the least 1-Mahlo-Woodin cardinal of M above γ_{1};

• g is an M-generic filter contained in ℚ_{<γ_{0}} extending g_{(d,b)} with a in g;

• g decides all of ḋ and ḃ;

• d ∪ d^{∗} is M-generic for SC(γ_{ω}, W ∩ γ_{ω});

• ḋ_{g} ∪ d′ is M-generic for SC(γ_{ω}, W ∩ γ_{ω});

• M[d ∪ d^{∗}] = M[ḋ_{g} ∪ d^{∗}];

• (d∪d^{∗}, b^{∗}) is a complete condition in SL(M, δ, W)×Add(1, δ) below (d, b);

• (ḋ_{g} ∪ d′, b′) is a complete condition in SL(M, δ, W) × Add(1, δ) below (ḋ_{g}, ḃ_{g});

• g_{(d∪d^{∗},b^{∗})} = g_{(ḋ_{g}∪d′,b′)} extends g. Proof. Let γ_{0} be the least Woodin cardinal γ of M such that

• γ > η_{(d,b)};

• a ∈ ℚ_{<γ}^{M};

• the antichains deciding ḋ and ḃ are all predense in ℚ_{<γ}^{M} Let g be an M-generic filter for ℚ_{<γ_{0}}^{M} such that

• a ∈ g;

• g ∩ ℚ_{<η_{(d,b)}}^{M} = g_{(d,b)}; Let γ_{1} be the least γ > γ_{0} which is a Woodin cardinal in M such that the pair {d, ḋ_{g}} is M-generic for a partial order in V_{γ}^{M}. Let γ_{ω} be the least 1-Mahlo-Woodin cardinal of M above γ_{1}. As in the proof of Lemma 4.3, there exist sequences d^{∗} = ⟨γ_{i} : i < ω⟩ and d′ = ⟨γ′_{i} : i < ω⟩ with supremum γ_{ω} such that

• d ∪ d^{∗} is M-generic for SC(γ_{ω}, W ∩ γ_{ω});

• ḋ_{g} ∪ d′ is M-generic for SC(γ_{ω}, W ∩ γ_{ω});

• M[d ∪ d^{∗}] = M[ḋ_{g} ∪ d′]. Let g^{∗} be a generic filter for ℚ_{<γ_{ω}}^{M} extending g such that a_{γ} ∈ g^{∗} for every γ ∈ d^{∗} ∪ d′. Then by Lemma 4.5, there exist b^{∗} and b′ such that

g_{(d∪d^{∗},b^{∗})} = g_{(ḋ_{g}∪d′,b′)} = g^{∗},

as desired.

Lemma 4.9. Suppose that

• δ is a 2-Mahlo-Woodin cardinal in V;

• G ⊂ ℚ_{<δ} is a V -generic filter;

• (d, b) is a complete condition in SL(V, δ, W) × Add(1, δ);

• G ∩ V_{η_{(d,b)}} = g_{(d,b)};

• D is a dense open subset of SL(V, δ, W) × Add(1, δ) in V (ℝ^{V [G]}).

Then there exist a complete condition (d′, b′) in (SL(V, δ, W) × Add(1, δ)) ∩ D extending (d, b) such that η_{(d′,b′)} > η_{(d,b)} and G ∩ V_{η_{(d′,b′)}} = g_{(d′,b′)}.

Proof. Let η denote η_{(d,b)}. If the lemma fails, there exist a condition a in

((ℚ_{<δ}↾a_{η})/ℚ_{<η})^{V [G∩V_{η}]}

(call this forcing Q) and Q-names ḃ, ḋ and Ḋ such that a forces over the extension V [G ∩ V_{η}] that Ḋ is a dense open subset of the partial order

SL(V, δ, W) × Add(1, δ)

of V (ℝ^{V [Ġ]}) and (ḋ, ḃ) is a complete element of this partial order such that G ∩ V_{η} = g_{(ḋ_{G},ḃ_{G})} and such that for no complete condition (d^{+}, b^{+}) in

(SL(V, δ, W) × Add(1, δ)) ∩ Ḋ

are (d^{+}, b^{+}) ≤ (ḋ, ḃ) and G ∩ V_{η_{(d^{+},b^{+})}} = g_{(d^{+},b^{+})}.

Let d^{∗} = ⟨γ_{i} : i ≤ ω⟩, d′ = ⟨γ′_{i} : i ≤ ω⟩, b^{∗}, b′ and g be as in Lemma 4.8, with respect to a. Let (D, B) be a V (ℝ^{V [G]})-generic filter for

SL(V, δ, W) × Add(1, δ)

extending (d∪d^{∗}, b^{∗}), and let H = g_{(D,B)}. Then by Lemma 4.7, ℝ^{V [G]} = ℝ^{V [H]}. By the choice of a, ḋ and ḃ, there is a dense open subset D′ of the partial order SL(V, δ, W) × Add(1, δ) of V (ℝ^{V [G]}) such that for no complete condition (d^{+}, b^{+}) in

(SL(V, δ, W) × Add(1, δ)) ∩ D′

are (d^{+}, b^{+}) ≤ (ḋ_{g}, ḃ_{g}) and H ∩ V_{η_{(d^{+},b^{+})}} = g_{(d^{+},b^{+})}.

Let (D′, B′) be the filter in SL(V, δ, W) × Add(1, δ) of V (ℝ^{∗}) formed by replacing (d∪d^{∗}, b^{∗}) with (ḋ_{g} ∪d′, b′) (since M[d∪d^{∗}] = M[ḋ_{g} ∪d′], this replacement sends conditions to conditions). Then (D′, B′) is V (ℝ^{∗})-generic, and, by the final three conclusions of Lemma 4.8, g_{(D′,B′)} = H. By the genericity of (D′, B′), there exists an η′ > η such that the restriction of D′ and B′ to η′ is a complete pair (d^{+}, b^{+}) in D′, giving a contradiction.

Given a model M of ZF and an ordinal δ of M, an M-fast club through δ is a club C ⊂ δ with the property that for all limit elements β of C, C ∩ β is eventually contained in every club subset of β in M.

The proof of Theorem 4.10 uses the full stationary tower ℙ_{<κ} (see [6], though we give specific references below for the facts we need).

Theorem 4.10. Suppose that CH holds, δ is a measurable Woodin cardinal in V, and κ > δ is a Woodin cardinal. Suppose that V (ℝ^{∗}) is a δ-symmetric extension of V and (D, B) is V (ℝ^{∗})-generic for SL(V, δ, W) × Add(1, δ) as defined in V (ℝ^{∗}). Then every Σ_{2}^{2} sentence which holds in V holds in V [D][B].

Proof. Let ∃X⊂ℝ ∀Y ⊂ℝ ϕ(X, Y) be a Σ_{2}^{2} sentence which holds in V. Any two models of the form V [D][B] are elementarily equivalent, so it suffices to show that ∃X⊂ℝ ∀Y ⊂ℝ ϕ(X, Y) holds in some such model.

Let a be a condition in ℙ_{<κ} such that

• a forces that H ∩ V_{δ} will be V -generic for ℚ_{<δ}, where H ⊂ ℙ_{<κ} is the generic filter;

• a forces that j(ω_{1}) = δ, where j is the embedding induced by H;

• a forces that P(δ)^{V} has cardinality ℵ_{1} in V [H];

• a forces that there exists a V -fast club contained in the 1-Mahlo-Woodin cardinals of V below δ.

The existence of such an a from a measurable Woodin cardinal is shown in [1] and the proof of Theorem 3.2.1 in [6], modulo the fact that any normal measure on δ concentrates on the 1-Mahlo-Woodin cardinals of V below δ (1-Mahlo-Woodin cardinals themselves are not mentioned in [1, 6]). Let H ⊂ ℙ_{<κ} be a V -generic filter with a ∈ H. Let j: V → M be the induced embedding. Then G = H ∩V_{δ} is V -generic for ℚ_{<δ}. Let j′: V → M′ be the embedding induced by G. Let C be the V -fast club added by H. Let ζ be the least strongly inaccessible cardinal of V above δ. Let ℝ^{∗} be the reals of M. Since CH holds in V and j(ω_{1}) = δ, ℝ^{∗} = j(ℝ) = j′(ℝ), so ℝ^{∗} is also the set of reals in M′. Then V_{ζ}[G] is in M, and V_{ζ}(ℝ^{∗}) is a symmetric extension of V_{ζ}.

Since C ∈ M (see Theorem 2.5.8 of [6]) and C is a V -slow club through the 1-Mahlo-Woodin cardinals of V below δ, M can construct filters in SL(V, δ, W) × Add(1, δ) below any condition and meeting any ℵ_{1} many dense sets in V (ℝ^{∗}). Working in M, construct a V_{ζ}(ℝ^{∗})-generic filter (D, B) for SL(V_{ζ}, δ, W) × Add(1, δ) such that g_{(D,B)} = G. Since measurable Woodin cardinals are 2-Mahlo-Woodin, this can be done by Lemma 4.9, using C to guarantee genericity at limit states, and using the fact that P(δ)^{V} has cardinality ℵ_{1} in M to ensure genericity of the final filter.

Now let X_{0} be a set of reals in V such that V ⊧ ∀Y ⊂ℝ ϕ(X_{0}, Y). Then j′(X_{0}) = j(X_{0}), so j(X_{0}) ∈ V [G] ⊂ V [D][B], and M ⊧ ∀Y ⊂ ℝ ϕ(j(X_{0}), Y). Since P(δ)^{V [D][B]} ⊂ M, V [D][B] ⊧ ∀Y ⊂ℝ ϕ(j(X_{0}), Y), and thus the sentence ∃X⊂ℝ∀Y ⊂ℝ ϕ(X, Y) holds in V [D][B].

It suffices in the statement of Theorem 4.10 (and the corollaries below) to let δ be a full Woodin cardinal (in the terminology of [1]) and let κ be a Woodin cardinal. The full Woodin cardinals constitute a measure one set for any normal measure on a measurable Woodin cardinal.

By Lemma 4.4, we get that all Σ_{2}^{2} sentences holding in any extension by a partial ordering in V_{δ} satisfying CH hold in V [D][B].

Corollary 4.11. Suppose that δ is a measurable Woodin cardinal in V, and κ > δ is a Woodin cardinal. Suppose that V (ℝ^{∗}) is a δ-symmetric extension of V and (D, B) is V (ℝ^{∗})-generic for SL(V, δ, W)×Add(1, δ) as defined in V (ℝ^{∗}), where W is the set of Woodin cardinals of V below δ. Then if ϕ is a Σ_{2}^{2} sentence and CH + ϕ holds in a forcing extension of V by a partial order in V_{δ}, then ϕ holds in V [D][B].

By Lemma 4.2, we get the following.

Corollary 4.12. Suppose that δ is a measurable Woodin cardinal in V, and κ > δ is a Woodin cardinal. Let (D, B) be V -generic for SC(δ, W) ∗ Add(1, δ), where W is the set of Woodin cardinals of V below δ. Then if ϕ is a Σ_{2}^{2} sentence and CH + ϕ holds in a forcing extension of V by a partial order in V_{δ}, then ϕ holds in V [D][B].

Theorem 4.10 and Corollary 4.11 continue to hold when a predicate for a universally Baire set of reals is added to the language. Showing this requires only that, in the proof of Theorem 4.10, if A is universally Baire set of reals in V, then j(A) is equal to the reinterpretation of A in V [D][B]. This in turn follows from the following theorem of Steel (proofs appear in [1, 6]).

Theorem 4.13. Let λ be a strongly inaccessible cardinal and let T be a a λ^{+}- weakly homogeneous tree. If S is the Martin-Solovay tree for the complement of the projection of T and k is an elementary embedding derived from forcing with ℚ_{<λ} then the corresponding generic embedding k: V → M satisfies k(S) = S.

Corollary 4.14. Suppose that δ is a measurable Woodin cardinal, A is set of reals such that A and ℝ \ A are δ^{+}-weakly homogeneously Suslin and κ > δ is a Woodin cardinal. Suppose that (D, B) is V -generic for SL(V, δ, W) ∗ Add(1, δ), where W is the set of Woodin cardinals of V below δ. Then every Σ_{2}^{2}-sentence with an additional predicate for A which can be forced to hold by a partial order in V_{δ} holds in V [D][B].

It is not possible to add a predicate for NS_{ω_{1}} to the language in Theorem 4.10. One way to see this is given in [7].

A natural question is whether the forcing Add(1, δ) is necessary to achieve Σ_{2}^{2}-maximality.

References

[1] I. Farah, P.B. Larson, Absoluteness for universally Baire sets and the uncountable I, Set Theory and its Applications, Quaderni di Matematica 17 (2006), 47-92 [2] J.D. Hamkins, W.H. Woodin, Small forcing creates neither strong nor

Woodin cardinals, Proc. Amer. Math. Soc. 128 (2000) 10, 3025-3029

[3] T. Jech, Set Theory, Springer-Verlag, Berlin, 2003

[4] A. Kanamori, The Higher Infinite. Large cardinals in set theory from their beginnings, 2nd edition, Springer Monographs in Mathematics, 2003

[5] K. Kunen, Set Theory. An introduction to independence proofs, North Holland, 1983

[6] P.B. Larson, The Stationary Tower. Notes on a course by W. Hugh Woodin, American Mathematical Society University Lecture Series, vol. 32, 2004

[7] P.B. Larson, S. Shelah, The stationary set splitting game, Math. Logic Quarterly 54 (2008) 2, 187-193

[8] R. Schindler, J. Steel, The core model induction, book in preparation

Department of Mathematics and Statistics Miami University Oxford, Ohio 45056 USA ketchero@muohio.edu larsonpb@muohio.edu

Department of Mathematics University of Florida Gainesville, FL 32611 zapletal@math.ufl.edu