MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2})

PAUL B. LARSON

Abstract. In [6], we modified a coding device from [14] and the consistency proof of Martin’s Maximum from [3] to show that from a supercompact limit of supercompact cardinals one could force Martin’s Maximum to hold while the ℙ_{max} axiom (∗) fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H(ℵ_{2}) without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result.

Keywords: Forcing Axioms; Definability; Large Cardinals

MSC2000: 03E35; 03E40, 03E55, 03E65

1. Introduction

The following maximal version of Martin’s Axiom was introduced by Foreman, Magidor and Shelah in [3].

Definition 1.1. Martin’s Maximum (MM) is the statement that if ℙ is a partial order such that forcing with ℙ preserves stationary subsets of ω_{1}, and ⟨D_{α} | α < ω_{1}⟩ is a collection of dense subsets of ℙ, then there is a filter G ⊂ ℙ meeting each D_{α}.

By convention, MM^{+} is MM with the further requirement that if τ is a ℙ-name for a stationary set, then {α < ω_{1} | ∃q ∈ G q⊩α ∈ τ} is stationary, and MM^{++} is MM^{+} but with ℵ_{1} many names for stationary subsets of ω_{1}. We let MM^{+ω} denote the version of MM^{+} with countably many names.

The ℙ_{max} axiom (∗) (Definition 5.1 of [14]) says that the Axiom of Determinacy (AD) holds in the inner model L(ℝ) and that L(P(ω_{1})) is a ℙ_{max}-generic extension

The author is supported in part by NSF grant DMS-0401603. He would like to thank David Asperó and the referee for extensive comments on earlier drafts of this paper.

2 PAUL B. LARSON

of L(ℝ) (the partial order ℙ_{max} is introduced in [14] and is not used in this paper, though Lemmas 4.4-4.6 use some of the ℙ_{max} machinery). Woodin showed [14] that MM^{++}(𝔠) (MM^{++}(κ) is MM^{++} restricted to posets of cardinality κ or less) and (∗) are independent over ZFC, assuming the consistency of a strong form of determinacy (see Theorem 10.69 in [14]; that (∗) does not imply MM^{++}(𝔠) follows from arguments in [7] but was known previously by Woodin). In [6], we used some of the ideas from Woodin’s proof that MM^{++}(𝔠) does not imply (∗) to show that MM^{+ω} does not imply (∗). In this paper we modify the argument from [6] to produce a model of MM^{+ω} in which there is a wellordering of the reals definable in H(ℵ_{2}) without parameters. This answers a question asked privately by Todorcevic shortly after the results of [6] were announced. Our interest in the question was reawakened by recent work of Asperó ([1], for instance). We note that (∗) implies that there is no wellordering of the reals definable in H(ℵ_{2}) without parameters (this follows immediately from the fact that the ℙ_{max} extension is a homogeneous extension of a model of AD containing the reals), so (∗) fails in the model in this paper also.

We will use a variant of the set of reals X_{(Code)}^{ω_{1}}(S, z) from [6], which is itself a variant of a set of reals from [14] (see Definition 10.22 of [14]). We will call our set of reals X_{(Code)}^{2}(S). As with the other variants, under (∗) this set is equal to P(ω) (we will not show this, though we will show that MM^{++} in conjunction with the existence of a Woodin cardinal below a measurable also implies that this set consists of all subsets of ω; modulo standard ℙ_{max} arguments, the proof in the ℙ_{max} context is the same). Our forcing construction is an iterated forcing which uses the construction from the consistency proof for MM from [3], adding forcings to make the set X_{(Code)}^{2}(S) code the parameter S. In the end S will be the only suitable parameter coding itself via X_{(Code)}^{2}, and will thus be definable in H(ℵ_{2}). The parameter S is a partition of ω_{1} into ℵ_{1}-many stationary sets, and there are numerous ways to define a wellordering of P(ω_{1}) in H(ℵ_{2}) from such a parameter under the assumption of MM (the axiom ψ_{AC} from [14] allows this, for instance).

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 3

We note that a ladder system on ω_{1} (under PFA, [12]) and in fact any subset of ω_{1} not constructible from a real (MM + “there exists a measurable cardinal”, [9]) can also be used as parameters defining a wellordering of the reals definable in H(ω_{2}).

2. X_{(Code)}^{2}

If x is a set of ordinals, we let ot(x) be the ordertype of x. The following definition is due to Woodin [14].

Definition 2.1. Given T ⊂ ω_{1}, T̃ is the set of β ∈ [ω_{1}, ω_{2}) such that there exist a bijection f : ω_{1} → β and a club C ⊂ ω_{1} such that for all α ∈ C, ot(f[α]) ∈ T.

If T is a subset of ω_{1}, then T̃ is the set of ordinals in the interval [ω_{1}, ω_{2}) which are necessarily in the image of T by any embedding derived from forcing with the nonstationary ideal on ω_{1} (which we denote by NS_{ω_{1}}). It is a standard fact ([7], for instance), and not too hard to show, that if A is a subset of ω_{1} added generically by initial segments, then Ã = ∅, though this can be changed in further forcing extensions.

We denote by C_{β}^{κ} the set of the ordinals less than β of cofinality κ, where κ is a regular cardinal and β is an ordinal.

Definition 2.5 below is a variant of the one used in [6], which in turn is a variant of one used in [14] to show that MM^{++}(𝔠) does not imply (∗). The real z from the definition in [6] is always the empty real, so we drop it. We also add two more conditions to the definition, making it harder for reals to enter X_{(Code)}^{2}(S).

Given an infinite x ⊂ ω we let oe(x) denote the set

{j < ω | the (j + 3)rd element of x is even}

(here we mean that the 1st element of a set is its least element). The term j + 3 in the definition ensures that for any pair n, m ∈ ω,

{oe(x) | x ∈ [ω]^{ω} ∧ {n, m} ⊂ x} = P(ω)

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(if we replaced j + 3 with j + 1, {0, 1} ⊂ x would imply that 0 ∈ oe(x) and 1 ∉ oe(x), for infinite x ⊂ ω). We note the following fact about oe, which we leave to the reader.

Lemma 2.2. If x and y are subsets of ω, x is infinite and oe(x) is infinite and co-infinite, then there exists an infinite x′ ⊂ x such that oe(x′) = y.

Let t: ω → (2^{<ω} \ {∅}) be the listing of the nonnull members of 2^{<ω} that lists shorter sequences before longer ones, and which lists sequences of the same length in lexicographical order. We let ℂ denote the set of x ⊂ ω such that for some y ∈ 2^{ω}, {t(i) : i ∈ x} = {y↾n : 0 < n < ω} (we say in this case that x codes y via t, and we let t^{∗} be the function on ℂ taking x ∈ ℂ to the real coded by x via t). Note that ℂ is a Borel set, so membership in ℂ is absolute between models of set theory containing ω_{1}. We note some additional facts about t and ℂ, left to the reader. The second part of the following lemma follows from the first.

Lemma 2.3. Let y_{0}, y_{1} and z be distinct subsets of ω.

(1) If y_{0} and y_{1} are infinite and co-infinite, then

lim_{n→∞} |t^{−1}(y_{0}↾n) − t^{−1}(y_{1}↾n)| = ∞;

(2) If a ⊂ z is nonempty and finite, oe(z) ∈ ℂ and t^{∗}(oe(z)) is infinite and co-infinite, then oe(z \ a) ∉ ℂ.

We let li(x, α) denote the least Silver indiscernible for L[x] above α, when x is a real whose sharp exists and α is an ordinal. If A is a set of reals, the A-uniform indiscernibles are the ordinals which are indiscernibles for each member of A. We say uniform indiscernibles when A is the set of all reals, and M-indiscernibles for ℝ^{M}-indiscernibles when M is a model. By convention, u_{2} denotes the second uniform indiscernible, which in our context, where the sharp of each real exists, is equal to sup{li(x, ω_{1}) : x ∈ ℝ}.

For any ordinal γ, a cofinality function for γ is a continuous, increasing cofinal function f: cof(γ) → γ. The following abuse of terminology merits it own name.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 5

Definition 2.4. If γ_{0} and γ_{1} are ordinals of cofinality ω_{1} and A_{0} ⊂ γ_{0} and A_{1} ⊂ γ_{1} are stationary in γ_{0} and γ_{1} respectively, then we say that A_{0} and A_{1} have abused stationary intersection if there exist cofinality functions f_{0} for γ_{0} and f_{1} for γ_{1} such that the set

{η < ω_{1} : f_{0}(η) ∈ A_{0} and f_{1}(η) ∈ A_{1}}

is stationary.

The choice of f_{0} and f_{1} in Definition 2.4 is irrelevant (i.e., an equivalent notion is obtained by replacing “there exist” with “for all” and removing “such that”).

The following definition is meant to be applied in the case where the sharp of every real exists.

Definition 2.5. Suppose that S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩ is a collection of pairwise disjoint stationary subsets of ω_{1}. We associate to S two subsets of P(ω),

X_{(Code)}^{2}(S) = ∪{X_{γ} : γ < ω_{2}} and Y_{(Code)}^{2}(S) = ∪{Y_{γ} : γ < ω_{2}},

where ⟨(κ_{γ}, X_{γ}, Y_{γ}) : γ < ω_{2}⟩ is the sequence generated from S as follows.

(i) Y_{0} = {∅}, X_{0} = ∅, and κ_{0} = li(∅, ω_{1}). (ii) For all nonzero γ < ω_{2}, κ_{γ} is the least Y_{γ}-uniform indiscernible η such that

k_{α} < η for all α < γ. (iii) Suppose γ is not the successor of an ordinal of cofinality ω_{1}. Then

X_{γ} = ∪{X_{α} : α < γ} and Y_{γ} = ∪{Y_{α} : α < γ}.

(iv) Suppose γ has cofinality ω_{1}. For each α < ω_{1}, let

b_{α} = oe({i < ω | S̃_{αi} ∩ κ_{γ} is stationary})

if this is defined, and ∅ otherwise. Then Y_{γ+1} = Y_{γ} ∪ {b_{α} : α < ω_{1}}.

Furthermore, suppose the following hold.

a) For all α < ω_{1} and i < ω, S̃_{αi}∩κ_{γ} and S̃_{α(i+1)}∩κ_{γ} are not both stationary.

b) Each b_{α} ∈ ℂ.

c) li(b_{1}, ω_{1}) > κ_{γ}.

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d) The sequence ⟨li(b_{α}, ω_{1}) : α < ω_{1}⟩ is strictly increasing.

e) For all i, i′ < ω, for all α, α′ < ω_{1}, for all β ∈ C_{γ}^{ω_{1}}, if S̃_{αi} ∩ κ_{γ} is stationary in κ_{γ} and S̃_{α′i′} ∩ κ_{β} is stationary in κ_{β}, then these sets have abused stationary intersection.

Then X_{γ+1} = X_{γ} ∪ {t^{∗}(b_{0})}. Otherwise, X_{γ+1} = X_{γ}.

When the set S is clear from context, we will simply refer to κ_{γ}, X_{γ} and Y_{γ} individually as needed, sometimes using κ_{γ}^{M} (for instance) for this set as computed in a given model M. When S is not clear, we write κ_{γ}(S), etc. We will also refer to ⟨κ_{γ} : γ < ω_{2}⟩ as the κ-sequence of S, ⟨Y_{γ} : γ < ω_{2}⟩ as the Y -sequence, and so on. The sequence ⟨b_{α} : α < ω_{1}⟩ will likewise be called the b-sequence at stage γ or sometimes the S-coding at stage γ. We let ST AT (S, γ) (for some γ ≤ ω_{2}) denote the collection of stationary sets of the form {δ < ω_{1} | f(δ) ∈ S̃_{αi}}, for some β ∈ C_{γ}^{ω_{1}}, α < ω_{1}, i < ω and some cofinality function f for κ_{β}.

Condition (ive) in Definition 2.5 ensures that if we shoot a club through any stationary set of the form S̃_{αi} ∩ κ_{γ}, or any union of such stationary sets, we don’t don’t destroy the stationarity of any earlier S̃_{α′i′} ∩ κ_{β}, and therefore don’t change the b-sequence at any earlier stage. In practice, when we do shoot such a club, ST AT (S, γ) will contain all the stationary subsets of ω_{1} appearing in models from proper initial segments of our iteration so far (see Lemmas 4.7 and 4.8).

Since the definition of the S-coding at a given stage is absolute between models which agree about stationary subsets of ω_{1} (and whose ω_{2} is large enough), if M ⊂ N are models of ZFC such that NS_{ω_{1}}^{M} = NS_{ω_{1}}^{N} ∩ M, then X_{(Code)}^{2}(S)^{M} ⊂ X_{(Code)}^{2}(S)^{N} for any S ∈ M. Furthermore, if ⟨M_{α} : α ≤ β⟩ is an increasing sequence of models of ZFC which agree about stationary subsets of ω_{1}, and ω_{2}^{M_{β}} = sup{ω_{2}^{M_{α}} : α < β}, then X_{(Code)}^{2}(S)^{M_{β}} = ⋃{X_{(Code)}^{2}(S)^{M_{α}} : α < β}.

We note two more useful facts about Definition 2.5. The first follows from condition (ivc) and the fact that li(x, ω_{1}) < ω_{2} for any real x, and the second follows from conditions (ii) and (ivc).

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 7

Lemma 2.6. Suppose that M ⊂ N are models of ZFC such that ℝ^{M} = ℝ^{N} and NS_{ω_{1}}^{M} = M ∩NS_{ω_{1}}^{N}, and let S ∈ M be a partition of ω_{1}^{M} into stationary sets indexed by ω_{1}^{M} × ω. Then X_{(Code)}^{2}(S)^{M} = X_{(Code)}^{2}(S)^{N}.

Lemma 2.7. Let S be a partition of ω_{1} into stationary sets indexed by ω_{1} × ω. Suppose that α < ω_{2} is such that sup{li(x, ω_{1}) : x ∈ Y_{α}} = u_{2}. Then for all β ∈ [α, ω_{2}), X_{β} = X_{α}.

3. Standards

Definition 3.1. Given a set X, c ⊂ P_{ℵ_{1}}(X) is closed unbounded (club) if there is a function F: [X]^{<ω} → X such that c is the set of countable subsets of X closed under F. A set a ⊂ P_{ℵ_{1}}(X) is stationary if it intersects every club c ⊂ P_{ℵ_{1}}(X).

Definition 3.2. A partial order P is semi-proper if whenever p is a condition in P, X ≺ H((2^{|P |})^{+}) is countable and p, P ∈ X, there is a q ≤ p in P forcing that the realization of every P -name in X for a countable ordinal is in X.

The following statement follows from MM and implies that every forcing which preserves stationary subsets of ω_{1} is semi-proper [3].

Definition 3.3. The Weak Reflection Principle (WRP) is the statement that for all cardinals λ ≥ ω_{2} and for all stationary Z ⊂ P_{ℵ_{1}}(λ), there exists a Y ∈ P_{ℵ_{2}}(λ) containing ω_{1} such that Z ∩ P_{ℵ_{1}}(Y) is stationary in P_{ℵ_{1}}(Y).

Given a cardinal γ and a set X, Coll(γ, X) is the partial order consisting of partial functions from γ to X with domain of cardinality less than γ, ordered by inclusion. Given a cardinal γ and an ordinal η, Coll(γ, <η) is the partial order consisting of functions p from γ × η to η with domain of cardinality less than γ and the stipulation that p(α, β) ∈ β, ordered by inclusion.

We refer the reader to [5] for the definitions of the large cardinal concepts used in this paper (measurable, Woodin, (λ-)supercompact). The following well-known fact is due to Foreman, Magidor and Shelah and follows easily from the arguments in [3].

Lemma 3.4. Assume that η is a supercompact cardinal. Then WRP holds after forcing with Coll(ω_{1}, < η).

An ideal I on ω_{1} is saturated if the Boolean algebra P(ω_{1})/I has no antichains of cardinality ℵ_{2}. Such an I presaturated if whenever A ∈ I^{+} and B_{i} (i < ω) are maximal antichains in P(A)/I there exists an A′ ⊂ A in I^{+} such that for each i < ω, |{B ∈ B_{i} : B ∩ A′ ∈ I^{+}}| = ℵ_{1}. Presaturation is weaker than saturation, and saturation of NS_{ω_{1}} follows from MM(𝔠) [3] (indeed, from the restriction of MM(𝔠) to posets which do not add reals). Presaturation also implies that j(ω_{1}) = ω_{2}, where j is any generic elementary embedding derived from forcing with P(ω_{1})/I.

We will use the two following facts, due to Shelah and Woodin, respectively.

Theorem 3.5. ([13]) If δ is a Woodin cardinal, there there exists a semi-proper forcing of cardinality δ forcing that NS_{ω_{1}} is saturated.

Theorem 3.6. ([14]) If NS_{ω_{1}} is saturated and there exists a measurable cardinal, then u_{2} = ω_{2}.

As in the original consistency proof of Martin’s Maximum, we will use the following result of Laver [11].

Theorem 3.7. Let κ be a supercompact cardinal. Then there exists a function L : κ → V_{κ} such that for every set X and every cardinal λ there exists a λsupercompact embedding j : V → M such that j(L)(κ) = X.

We will be using iterations of semi-proper forcing with Revised Countable Support [13]. The following theorem from [13] (which we have rewritten in our own terminology) ensures that our forcing will be semi-proper.

Theorem 3.8. Suppose that ⟨P_{α}, Q_{∼}_{α} : α < δ⟩ is an RCS iteration, and that for every β < δ, for arbitrarily large non-limit α < β + 1, P_{β+1}/P_{α} is semi-proper, and for every α < δ, the empty condition in P_{α+n} (for some n < ω) forces that |P_{α}| = ℵ_{1}. Then P_{δ} is semi-proper.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 9

We will be taking advantage of a degree of freedom offered by Theorem 3.8: when β is a limit ordinal, the theorem does not require that Q_{β} preserve stationary subsets of ω_{1} which are in the P_{β}-extension but in no earlier P_{α}-extension. Our situation is a special case. We will have WRP (and thus that partial orders preserving stationary subsets of ω_{1} are semi-proper) in each nonlimit P_{α}-extension, so the semi-properness of each P_{β+1} will follow from the fact that for each nonlimit α < β, P_{β}/P_{α} is semi-proper and Q_{β} preserves stationary subsets of ω_{1} from P_{α}.

4. Protection and Erasure

The first half of this section, Lemmas 4.1-4.6, deals with ways to put stationary sets into ST AT (S, ω_{2}) without adding reals to X_{(Code)}^{2}(S). The basic lemma for doing this we call the Protection Lemma (Lemma 4.2 below). The proof of the Protection Lemma is based on ideas which have become standard. The lemma itself is a variation of Lemma 10.66 of [14]. For the arguments in Section 6 we will need more information about the forcings introduced here (hence Lemma 4.6).

We first note an end-extension property which is the basis for these arguments.

Lemma 4.1. Suppose that ⟨ρ_{γ} : γ ≤ η⟩ is a continuous increasing sequence of cardinals such that η < ρ_{0} and ρ_{γ} is a measurable cardinal for each nonlimit γ. Let S_{γ} (γ ≤ η) be stationary subsets of ω_{1}. Suppose that χ ≥ (2^{ρ_{η}})^{+} is a regular cardinal, and let Y be a countable elementary substructure of H(χ) containing the sequence ⟨ρ_{γ} : γ ≤ η⟩. Let ζ be any ordinal in the interval [η, ρ_{0}). Then there exists a countable Y′ ≺ H(χ) containing Y such that

• Y′ ∩ ρ_{0} end-extends Y ∩ ρ_{0};

• Y′ ∩ ζ = Y ∩ ζ;
• ot(Y′ ∩ ρ_{γ}) ∈ S_{γ} for each γ ∈ Y ∩ (η + 1).

Proof. By induction on η. When η is a successor ordinal, the induction step follows immediately from a standard end-extension property of measurable cardinals (Lemma 1.1.21 of [8], say). Now suppose that η is a limit ordinal, and fix Y as in the statement of the lemma. Let Z be a countable elementary submodel of

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H((2^{χ})^{+}) with Y and ⟨S_{γ} : γ ≤ η⟩ as elements and Z ∩ ω_{1} ∈ S_{η}. Let ⟨ν_{i} : i < ω⟩ be an increasing cofinal sequence in Z ∩ η (with ν_{0} = 0) and let ⟨α_{i} : i < ω⟩ be an increasing cofinal sequence in Z ∩ ω_{1}. Let Y_{0} ∈ Z satisfy the lemma with respect to Y, ζ, ⟨S_{γ} : γ ≤ 0⟩ and ⟨ρ_{γ} : γ ≤ 0⟩.

Successively choose Y_{i} (0 < i < ω) in Z such that each Y_{i+1} satisfies the lemma for Y_{i}, ρ_{ν_{i}} (in the role of ζ), ⟨S_{γ} : ν_{i} < γ ≤ ν_{i+1}⟩ and ⟨ρ_{γ} : ν_{i} < γ ≤ ν_{i+1}⟩ and satisfies ot(Y_{i+1} ∩ ρ_{ν_{i+1}}) ≥ α_{i} (which we can do by replacing S_{ν_{i}+1} with S_{ν_{i}+1} \ α_{i}, for instance). Then since the sets Y_{i} ∩ ρ_{ν_{i}} end-extend one another, the union of these sets has ordertype Z ∩ ω_{1}, so the union of {Y_{i} : i < ω} satisfies the lemma for Y, ⟨S_{γ} : γ ≤ η⟩, and ⟨ρ_{γ} : γ ≤ η⟩. □

Lemma 4.2. (Protection Lemma) Suppose that S,T are collections of stationary subsets of ω_{1}, and that the members of T are pairwise disjoint. Let π be a function from T to S, and let ⟨ρ_{γ} : γ ≤ ω_{1}⟩ be a continuous, increasing sequence of cardinals such that ρ_{γ} is a measurable cardinal for each nonlimit γ.

Then there is a (ω, ∞)-distributive semi-proper forcing in whose extension, for each T ∈ T and each γ ∈ T, ρ_{γ} ∈ S̃ if S = π(T).

Proof. Let P be the set of countable, continuous, ⊂-increasing sequences ⟨x_{ν} : ν ≤ μ⟩ ⊂ [ρ_{ω_{1}}]^{ℵ_{0}} such that for each ν ≤ μ, and each γ ∈ x_{ν} ∩ ω_{1}, if there exists a T ∈ T with γ ∈ T, then ot(x_{ν} ∩ρ_{γ}) ∈ π(T). Forcing with this partial order has the desired effect by standard arguments, almost straight from the definition of S̃.

Lemma 4.1 gives that P is semi-proper and (ω, ∞)-distributive, as follows. Fix a condition p in a countable elementary submodel Y of H((2^{ρ_{ω_{1}}} )^{+}) having P and ⟨ρ_{γ} : γ ≤ ω_{1}⟩ as members. By Lemma 4.1, there exists a countable Y′ ≺ H((2^{ρ_{ω_{1}}} )^{+}) containing Y with Y′ ∩ ω_{1} = Y ∩ ω_{1} and, for each γ ∈ Y′ ∩ ω_{1}, ot(Y′ ∩ ρ_{γ}) ∈ π(T) whenever T ∈ T and γ ∈ T. Then if p̄ is any descending ω-sequence of conditions in P ∩ Y′ extending p, such that p̄ meets every dense subset of P in Y′, then (⋃ p̄)^{⌢}⟨Y′ ∩ ρ_{ω_{1}}⟩ is a (Y, P)-semi-generic condition extending p. □

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 11

Suppose that ⟨E_{α} : α < 2^{ω_{1}}⟩ is a listing of NS_{ω_{1}}^{+} and that g is a V -generic filter for Coll(ω_{1}, 2^{ω_{1}}) (note that this forcing makes CH hold). Then the sets {γ ∈ E_{α} : g(γ) = α} (α < (2^{ω_{1}})^{V}) are pairwise disjoint and stationary. If κ is a cardinal greater than 2^{ℵ_{1}}, then, in the Coll(ω_{1}, <κ)-extension there exists a partition T of ω_{1} into stationary sets such that every stationary subset of the ground model contains a member of T modulo NS_{ω_{1}}. In such an extension, given an uncountable collection S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩ of stationary subsets of ω_{1}, a listing ⟨x_{α} : α < ω_{1}⟩ of P(ω) and reals ⟨y_{α} : α < ω_{1}⟩ such that each oe(y_{α}) = x_{α}, we let ℙ𝔽(S) be the forcing as in the Protection Lemma, with T as a partition as in the previous sentence, π: T → S any injection whose range is the set of S_{αi} such that i ∈ y_{α}, and ⟨ρ_{γ} : γ ≤ ω_{1}⟩ the closure of the first ω_{1} many measurable cardinals. Since neither of Coll(ω_{1}, <κ) and ℙ𝔽(S) add reals or destroy stationary subsets of ω_{1}, they do not add reals to X_{(Code)}^{2}(S) or change X_{(Code)}^{2}(S) or Y_{(Code)}^{2}(S) below ω_{2}^{V}. Furthermore, in this extension the ordinal κ_{ρ_{ω_{1}}} (S) is equal to ρ_{ω_{1}}, each x_{α} is in Y_{(Code)}^{2}(S) and, for every stationary E ⊂ ω_{1} in the ground model, there exists (α, i) ∈ ω_{1} × ω such that, for any cofinality function f for ρ_{ω_{1}}, the set of α ∈ ω_{1} such that f(α) ∈ S̃_{αi} is a stationary set contained modulo NS_{ω_{1}} in E.

Our forcing iterations will use the partial order Coll(ω_{1}, <μ) ∗ ℙ𝔽(S) at every successor step, where μ is the least supercompact cardinal. This and condition (ive) of Definition 2.5 will allow us to undo accidentally coded reals by destroying the stationarity of certain subsets of ω_{1} while preserving the stationarity of subsets of ω_{1} added by initial segments of the iteration. It will also help us put reals into X_{(Code)}^{2}.

The Protection Lemma allows us to code stationary subsets of ω_{1} into sets of the form S̃_{αi} ∩ κ_{γ}. In Section 5, we will need to code stationary sets of ordertype ω_{2} into sets of this form. The following modified form of the Protection Lemma does this.

Lemma 4.3. Suppose that κ is a strongly inaccessible cardinal and ⟨ρ_{α} : α ≤ κ⟩ is a continuous increasing sequence of cardinals with supremum κ such that ρ_{α} is

12 PAUL B. LARSON

measurable for each nonlimit α. Let S be a collection of stationary subsets of ω_{1}. Then there is a semi-proper (ω, ∞)-distributive partial order forcing that ρ_{κ} = ω_{2} and that S̃ ∩ C_{ρ_{κ}}^{ω} is stationary for each S in S.

Proof. This is largely the same as the proof of the Protection Lemma. Partition C_{κ}^{ω} into stationary sets E_{S} (S ∈ S). A condition in our partial order P consists of a domain F - a countable subset of κ \ ρ_{0} - and, for each η ∈ F a countable, continuous, ⊂-increasing sequence ⟨x_{ν}^{η} : ν ≤ μ⟩ ⊂ [η]^{ℵ_{0}} such that for each ν ≤ μ, and each γ ∈ x_{ν}^{η} ∩ C_{κ}^{ω} with ρ_{γ} < η, ot(x_{ν}^{η} ∩ ρ_{γ}) ∈ S, where S ∈ S is such that γ ∈ E_{S}. Stronger conditions in this forcing extend the domain and extend the sequences for each member of the domain of the weaker condition. Forcing with this partial order has the desired effect.

Lemma 4.1 gives that P is semi-proper and (ω, ∞)-distributive. Fix a condition p in a countable elementary submodel Y of H((2^{κ})^{+}) having P, S and {ρ_{α} : α < κ} as members. By Lemma 4.1, there exists a countable Y′ ≺ H((2^{κ})^{+}) containing Y with Y′ ∩ ω_{1} = Y ∩ ω_{1} and for each γ ∈ Y′ ∩ κ, ot(Y′ ∩ ρ_{γ}) ∈ S if S ∈ S and γ ∈ E_{S}. Let ⟨p_{i} : i < ω⟩ be a descending sequence of conditions in P ∩ Y′ (with p_{0} = p) meeting every dense subset of P in Y′. Then the condition consisting of domain Y′ ∩ κ and, for each η ∈ Y′ ∩ κ, the sequence p′(η)^{⌢}⟨Y′ ∩ η⟩ (where p′(η) is the union of the sequences for η in the p_{i}’s) is a (Y, P)-semi-generic condition extending p. □

Lemma 4.5 below is relatively standard. We will not apply this lemma directly, but we will use the construction in the proof of the lemma to derive an extra property of forcings of the type used in the Protection Lemma (in Lemma 4.6). Similar arguments appear in [6, 7, 14] (where one can also find the definitions of iteration and iterable). First we will give a quick proof of another standard fact, which is the key to the proof of Lemma 4.5.

Lemma 4.4. Suppose that M is a countable transitive model of ZFC in which NS_{ω_{1}} is presaturated and u_{2} = ω_{2}. Let ⟨(M_{α}, I_{α}), G_{β} : β < γ, α ≤ γ⟩ and j_{αβ}: M_{α} → M_{β}

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 13

(α ≤ β ≤ γ) make up an iteration of (M, NS_{ω_{1}}^{M}). Then j_{0γ}[ω_{2}^{M}] is cofinal in ω_{2}^{M_{γ}} and the critical sequence of j_{0,γ} is the set of all the M-uniform indiscernibles ξ in the interval [ω_{1}^{M}, ω_{1}^{M_{γ}}).

Proof. This is easily proved by induction on γ, using the fact that for each α < γ, j(ω_{2}^{M_{α}}) is the least M-uniform indiscernible above ω_{2}^{M_{α}}. This in turn follows from the fact that no f: ω_{1}^{M_{α}} → ω_{2}^{M_{α}} could represent the least M-uniform indiscernible above ω_{2}^{M_{α}}, since the range of each such function is bounded by the least indiscernible above ω_{1}^{M_{α}} of some real in M. □

Lemma 4.5. Suppose that NS_{ω_{1}} is presaturated, u_{2} = ω_{2} and β is an ordinal. Suppose that ξ̄ = ⟨ξ_{α} : α ≤ β⟩ is an increasing sequence of uniform indiscernibles, and S̄ = ⟨S_{α} : α ≤ β⟩ is a sequence of stationary subsets of ω_{1}. Suppose that there exists a measurable cardinal greater than ξ_{β}. Then the set of countable x ⊂ ξ_{β} such that ot(x ∩ ξ_{α}) ∈ S_{α} for all α ∈ (β + 1) ∩ x is stationary.

Proof. Let κ > ξ_{β} be measurable and let θ > 2^{κ} be regular. Fix a function F: [ξ_{β}]^{<ω} → ξ_{β}. Let T be the tree of attempts to build a countable x ⊂ ξ_{β} closed under F and an increasing sequence of ordinals ⟨δ_{α} : α ∈ (β + 1) ∩ x⟩ such that each δ_{α} ∈ S_{α} and such that ot(x ∩ ξ_{α}) = δ_{α} for all α ≤ β (so T is a tree of height ω using some fixed set of bijections between ω and each countable ordinal). Let X be a countable elementary submodel of H(θ) with κ, F, ξ̄ and S̄ as elements. Let M be the transitive collapse of X, and for each α ∈ (β + 1) ∩ X, let ξ_{α}^{∗} and S_{α}^{∗} be the images of ξ_{α} and S_{α} respectively under this collapse. Let T^{∗} be the image of T. By standard arguments ([14], see also [10, 2]) using the presence of the measurable cardinal κ, (M, NS_{ω_{1}}^{M}) is iterable. Furthermore, each ξ_{α}^{∗} is an M-uniform indiscernible, and every M-uniform indiscernible is on the critical sequence of every sufficiently long iteration of (M, NS_{ω_{1}}^{M}), by Lemma 4.4. Let j: (M, NS_{ω_{1}}^{M}) → (N, NS_{ω_{1}}^{N}) be an iteration of (M, NS_{ω_{1}}^{M}) such that for each α ∈ (β + 1) ∩ X, ξ_{α}^{∗} ∈ j(S_{α}^{∗}). Then j(T^{∗}) has an infinite branch corresponding to the sequence ⟨ξ_{α}^{∗} : α ∈ (β + 1) ∩ X⟩ and the set j[ξ_{β}^{∗}]. This set is not necessarily in N, but since N is wellfounded, there

14 PAUL B. LARSON

must be a branch through j(T^{∗}) in N, which by elementarity means that there is a branch through T. □

Putting together the proofs of Lemma 4.5 and the Protection Lemma we have Lemma 4.6 below. The proof uses a forcing to add a partition of ω_{1} indexed by ω_{1}×ω by initial segments. Conditions in this forcing are partitions ⟨c_{αi} : (α, i) ∈ ζ × ω⟩ of a countable ordinal χ, for some countable ordinal ζ. We say that

⟨c_{αi} : (α, i) ∈ ζ × ω⟩ ≤ ⟨d_{αi} : (α, i) ∈ ξ × ω⟩

if ζ ≥ ξ and c_{αi} ∩ θ = d_{αi} for each (α, i) ∈ ξ × ω, where ⟨d_{αi} : (α, i) ∈ ξ × ω⟩ is a partition of θ. The forcing P_{0} in the statement of Lemma 4.6 is the product of this forcing with itself.

Lemma 4.6. Suppose that NS_{ω_{1}} is presaturated and u_{2} = ω_{2}, and let

⟨κ_{α} : α ≤ ω_{1} · 2⟩

be a continuous increasing sequence of cardinals such that κ_{α} is a measurable cardinal for each nonlimit α. Let S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩ and T = ⟨T_{αi} : (α, i) ∈ ω_{1} × ω⟩ be partitions of ω_{1} into stationary sets.

Let P_{0} be the forcing which adds partitions of ω_{1} ⟨A_{αi} : (α, i) ∈ ω_{1} × ω⟩ and ⟨B_{αi} : (α, i) ∈ ω_{1} × ω⟩ by initial segments.

Let P_{1} be the forcing in the P_{0}-extension consisting of all countable continuous, increasing sequences ⟨x_{γ} : γ ≤ δ⟩ ⊂ [κ_{ω_{1}·2}]^{ℵ_{0}} such that for each γ ≤ δ, each η ∈ x_{γ} ∩ ω_{1}, each α < ω_{1} and each i < ω, ot(x_{γ} ∩ κ_{η}) ∈ T_{αi} if η ∈ A_{αi} and ot(x_{γ} ∩ κ_{ω_{1}+1+η}) ∈ S_{αi} if η ∈ B_{αi} (ordered by extension).

Then P_{0} ∗ P_{1} is (ω, ∞)-distributive, preserves stationary subsets of ω_{1} and forces that for every V -indiscernible ξ not in {ω_{1}} ∪ {κ_{α} : α ≤ 2 · ω_{1}}, every stationary K ⊂ ω_{1} in the ground model, all α, β < ω_{1} and all i, j < ω, the set of countable x ⊂ κ_{ω_{1}·2} such that

x ∩ ω_{1} ∈ A_{αi} ∩ B_{βj} and ot(x ∩ ξ) ∉ K

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 15

is stationary.

Proof. Let λ be a regular cardinal greater than 2^{κ_{ω_{1}·2}}. Given a condition (ā, b̄) in P_{0}, say that a set X is (ā, b̄)-good if it is a countable elementary submodel of H(λ) with (ā, b̄) as an element and there exist (X, P_{0})-generic

(⟨a_{αi} : (α, i) ∈ (X ∩ ω_{1}) × ω⟩, ⟨b_{αi} : (α, i) ∈ (X ∩ ω_{1}) × ω⟩)

extending (ā, b̄) such that for each η ∈ X ∩ ω_{1}, each α < ω_{1} and each i < ω, ot(X ∩ κ_{η}) ∈ T_{αi} if η ∈ a_{αi} and ot(X ∩ κ_{ω_{1}+1+η}) ∈ S_{αi} if η ∈ b_{αi}.

Let E and K be stationary subsets of ω_{1}. Fix a V -indiscernible ξ not in the set {ω_{1}} ∪ {κ_{α} : α ≤ 2 · ω_{1}}, countable ordinals α, β and integers i, j < ω. It suffices to show that for every condition (ā, b̄) ∈ P_{0} the following set is stationary: the set of (ā, b̄)-good X such that

X ∩ ω_{1} ∈ a_{αi} ∩ b_{βj} ∩ E and ot(X ∩ ξ) ∉ K,

for some (⟨a_{γk} : (γ, k) ∈ (X ∩ ω_{1}) × ω⟩, ⟨b_{γk} : (γ, k) ∈ (X ∩ ω_{1}) × ω⟩) extending (ā, b̄) and witnessing that X is (ā, b̄)-good.

To see that this set is stationary, fix a function F: [H(λ)]^{<ω} → H(λ). Let T be the tree of attempts to build an (ā, b̄)-good X closed under F and a witness

(⟨a_{γk} : (γ, k) ∈ (X ∩ ω_{1}) × ω⟩, ⟨b_{γk} : (γ, k) ∈ (X ∩ ω_{1}) × ω⟩)

extending (ā, b̄) such that X ∩ ω_{1} ∈ a_{αi} ∩ b_{βj} ∩ E and ot(X ∩ ξ) ∉ K.

Let θ > 2^{λ} be regular and let Y ≺ H(θ) be countable with λ, P_{0}, (ā, b̄), S, U, α, β, E, K and F in Y. Let (⟨a_{γk} : (γ, k) ∈ (Y ∩ω_{1})×ω⟩, ⟨b_{γk} : (γ, k) ∈ (Y ∩ω_{1})×ω⟩) be a (Y, P_{0})-generic condition in P_{0} extending (ā, b̄), with Y ∩ ω_{1} ∈ a_{αi} ∩ b_{βj}.

Let M be the transitive collapse of Y, and let h: Y → M be the collapsing function. As in the proof of Lemma 4.5, (M, NS_{ω_{1}}^{M}) is iterable, and every Muniform indiscernible is on the critical sequence of every sufficiently long iteration of (M, NS_{ω_{1}}).

For each η ∈ Y ∩ω_{1}, h(κ_{η}) and h(κ_{ω_{1}+1+η}) are both M-indiscernibles, as is h(ξ). Let j: (M, NS_{ω_{1}}^{M}) → (N, NS_{ω_{1}}^{N}) be an iteration of (M, NS_{ω_{1}}^{M}) such that

16 PAUL B. LARSON

• h(ω_{1}) ∈ j(h(E));

• for each δ, η ∈ ω_{1} ∩ Y and each i < ω, h(κ_{η}) ∈ j(h(T_{δi})) if η ∈ a_{δi} and

h(κ_{ω_{1}+1+η}) ∈ j(h(S_{δi})) if η ∈ b_{δi};

• h(ξ) ∉ j(h(K)). Then j(h(T)) has an infinite branch corresponding to (⟨a_{γk} : (γ, k) ∈ (Y ∩ ω_{1}) ×

ω⟩, ⟨b_{γk} : (γ, k) ∈ (Y ∩ ω_{1}) × ω⟩) and j[h[Y ∩ V_{λ}]]. This branch is not necessarily in N, but since N is wellfounded, there must be a branch through j(h(T)) in N, which by elementarity means that there is a branch through T. □

The second half of this section shows how we can undo codings (into X_{(Code)}^{2}(S)) arising at limit stages or imposed by the partial orders given by the Laver function. The Protection Lemma is used as preparation for the Erasure Lemma below to ensure that we can undo these codings while preserving stationary subsets of ω_{1} from initial stages of the iteration.

Let U = ⟨U_{αi} : (α, i) ∈ ω_{1} × ω⟩ be a collection of stationary, costationary subsets of ω_{1}, and fix κ ∈ C_{ω_{2}}^{ω_{1}}, B ⊂ ω_{1} × ω and a cofinality function f for κ. The Recoding Forcing ℝ𝔼ℂ(U, B, f) is the forcing to shoot a club through

ω_{1} \ ▽{γ < ω_{1} | ∃(α, i) ∈ ω_{1} × ω f(γ) ∈ Ũ_{αi} ∧ (α, i) ∈ B}

by initial segments (with a bijection from ω_{1} × ω to ω_{1} as a suppressed parameter). Since S̃_{0} ∩ S̃_{1} is empty whenever S_{0} and S_{1} are disjoint subsets of ω_{1}, this forcing preserves the stationarity of any subset of ω_{1} having stationary intersection with any set of the form {γ < ω_{1} | f(γ) ∈ Ũ_{αi}} for some (α, i) ∉ B.

Lemma 4.7. (Erasure Lemma) Suppose that S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩ is a collection of pairwise disjoint stationary subsets of ω_{1}. Fix ζ < ω_{2}, and let A_{i} (i < ω) be stationary subsets of ω_{1}. Let A be the union of {A_{i} : i < ω} and ST AT (S, ζ).

Then there is a forcing R preserving the stationarity of each member of A and forcing that X_{(Code)}^{2}(S) = ⋃{X_{β} : β < ζ}^{V}.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 17

Proof. Let γ ≥ ζ be least such that X_{γ+1} ≠ ⋃_{β<ζ} X_{β}^{V} (if there is no such γ we are done). The partial order R consists of two steps, the first of which is Coll(ω_{1}, ℝ), which adds a listing ⟨x_{α} : α < ω_{1}⟩ of all the reals of V. For each α < ω_{1}, let y_{α} be the set of i < ω such that S̃_{αi} ∩ κ_{γ} is stationary. Let f be a cofinality function for κ_{γ} and let δ ∈ [2, ω_{1}) be such that for each i < ω, if there exists (ξ, j) ∈ ω_{1} × ω such that {ν ∈ A_{i} | f(ν) ∈ S̃_{ξj}} is stationary, then there exists such a (ξ, j) with ξ < δ.

By condition (ivb) of Definition 2.5, each oe(y_{α}) is infinite and co-infinite. By Lemma 2.2, then, for each α < ω_{1} there exists an infinite y′_{α} ⊂ y_{δ+1+α} such that oe(y′_{α}) = x_{α}. By condition (iva), no y_{α} contains a consecutive pair of integers, so no y′_{α} does, either.

The second step of R is ℝ𝔼ℂ(S, B, f), where

B = ({δ} × ω) ∪ {(δ + 1 + α, i) ∈ ω_{1} × ω | α < ω_{1} ∧ i ∈ y_{δ+1+α} \ y′_{α}}.

Since the real b_{1} at stage γ is nonempty, there exists a k < ω such that S̃_{1k} ∩κ_{γ} is stationary, and by condition (ive) of Definition 2.5, S̃_{1k} ∩ κ_{γ} has abused stationary intersection with each member of ST AT (S, γ), which contains ST AT (S, ζ). Furthermore, S̃_{1k} ∩ κ_{γ} is disjoint from each S̃_{αi} ∩ κ_{γ}, (α, i) ∈ B, which means that ℝ𝔼ℂ(S, B, f) preserves the stationarity of each member of ST AT (S, ζ). Lastly, the choice of δ means that the stationarity of each A_{i} is preserved.

This forcing also makes b_{δ} = ∅ at stage γ, which since li(b_{1}, ω_{1}) > li(∅, ω_{1}) means that X_{γ+1} = ⋃_{β<γ} X_{β} in the R-extension, by condition (ivd) of Definition 2.5. It also makes b_{δ+1+α} = x_{α} at stage γ, for each α < ω_{1}. Furthermore, R does not add reals, so the reals of the R-extension are contained in the Y_{γ+1} of the R-extension, which by Lemma 2.7 means that no reals enter X_{(Code)}^{2}(S) at any stage after γ, either. □

There are many natural ways to increase the collection of stationary sets preserved in the Erasure Lemma, which can be carried directly over to stronger versions of MM^{+ω} in the main theorem. We cannot, however, preserve the stationarity of

18 PAUL B. LARSON

all the members of any given set of cardinality ℵ_{1}, however (for instance, all the stationary sets of the form S̃_{αi} ∩ κ_{γ}).

Let S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩ be a collection of pairwise disjoint stationary subsets of ω_{1}. Suppose that M[G] is a forcing extension of a model M by an iterated forcing P = ⟨P_{α}, Q_{∼}_{α} : α < η⟩ of limit length. For each α < η, let G_{α} be the restriction of G to P_{α}. We say that M[G] is (P, S)-safe if for each stationary, costationary set T ⊂ ω_{1} appearing in some M[G_{α}] (α < η), there exists a T′ ∈ ST AT (S, ω_{2})^{M[G]} such that T′ \ T ∈ NS_{ω_{1}}^{M[G]}.

Since each successor step of our iterations has the form Coll(ω_{1}, <μ) ∗ ℙ𝔽(S), the extensions by each initial segment of our iterations of limit length will be safe. The notion of safety is applied in the following corollary of Lemma 4.7, which shows that we can erase any accidental coding at limit stages of our iteration, even after an additional semi-proper forcing which may affect the coding.

Corollary 4.8. Suppose that S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩ is a collection of pairwise disjoint stationary subsets of ω_{1}. Suppose that M[G] is (P, S)-safe for some iterated forcing P = ⟨P_{α}, Q_{∼}_{α} : α < η⟩ in M of limit length. Let Q be a forcing in M[G] preserving stationary subsets of ω_{1}, and let H ⊂ Q be M[G]-generic. Let ζ be an ordinal in the interval [sup{ω_{2}^{M[G_{α}]} : α < η}, ω_{2}^{M[G][H]}), where for each α < η, G_{α} is the restriction of G to P_{α}.

Then in M[G][H] there is a forcing R preserving stationary subsets of ω_{1} from the models of the form M[G_{α}], α < η, such that if I is M[G][H]-generic for R, then X_{(Code)}^{2}(S)^{M[G][H][I]} = ⋃_{α<ζ} X_{α}^{M[G][H]}. Furthermore, R can be chosen so that the members of any fixed countable set of stationary subsets of ω_{1} in M[G][H] are stationary in M[G][H][I].

5. Coding reals

In this section we show that it is possible to force reals into X_{(Code)}^{2}(S) while preserving stationary subsets of ω_{1}.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 19

Lemma 5.1. Suppose that there exists a Woodin cardinal below a measurable cardinal, and let S be a collection of pairwise disjoint stationary subsets of ω_{1} indexed by ω_{1}×ω. Let z be a subset of ω. Then there is a partial order preserving stationary subsets of ω_{1} from V such that if V [G] is a generic extension of V by this partial order, then X_{(Code)}^{2}(S)^{V [G]} = X_{(Code)}^{2}(S)^{V} ∪ {z}.

Proof. Let S = ⟨S_{αi} : (α, i) ∈ ω_{1} × ω⟩. First force with Coll(ω_{1}, 2^{ω_{1}}) ∗ ℙ𝔽(S). Then force as in Lemma 4.3, making S̃_{αi} ∩ C_{ω_{2}}^{ω} stationary for each (α, i) ∈ ω_{1} × ω. Call this extension V^{∗}, and let ρ denote ω_{2}^{V^{∗}}. Force now with Coll(ω_{1}, ρ), and note that each S̃_{αi} ∩C_{ρ}^{ω} now has abused stationary intersection with each stationary subset of ω_{1} in V^{∗} (this will ensure that item (ive) of Definition 2.5 is satisfied at stage ρ after the following forcing, and will also ensure that the forcing ℝ𝔼ℂ(S, B, f) below does not alter X_{(Code)}^{2}(S) or Y_{(Code)}^{2}(S) below ρ). None of these partial orders adds reals, and so X_{(Code)}^{2}(S) remains the same. Furthermore, all reals are in ⋃{Y_{γ} : γ < ρ} in this extension.

Now use the Woodin cardinal and Shelah’s semi-proper forcing to make NS_{ω_{1}} saturated. Since there is a measurable cardinal in this extension, u_{2} > ρ there (any forcing preserving stationary subsets of ω_{1} and making u_{2} > ρ would suffice here). Fix in this extension a sequence of reals ⟨c_{α} : α < ω_{1}⟩ such that t^{∗}(oe(c_{0})) = z, no c_{α} contains a consecutive pair of elements of ω, each oe(c_{α}) ∈ ℂ,

li(oe(c_{1}), ω_{1}) > max{li(oe(c_{0}), ω_{1}), ρ}

and ⟨li(oe(c_{α}), ω_{1}) : α < ω_{1}⟩ is increasing. Now force with ℝ𝔼ℂ(S, B, f) (as defined before Lemma 4.7), where B = {(α, i) ∈ ω_{1} ×ω | i ∉ c_{α}}. Since ρ is the ρth uniform indiscernible of V, κ_{ρ} = ρ in this extension, and it is straightforward to check that in this extension z enters X_{(Code)}^{2}(S) at stage ρ.

Finally, we can force as in the Erasure Lemma to ensure that no reals enter X_{(Code)}^{2}(S) after stage ρ. Since the first step of our forcing was Coll(ω, 2^{ω_{1}})∗ℙ𝔽(S), the forcing from the Erasure Lemma preserves the stationarity of each stationary subset of ω_{1} from V. □

20 PAUL B. LARSON

Corollary 5.2. Suppose that MM^{++} holds and that there exists a Woodin cardinal below a measurable cardinal. Let S be a collection of pairwise disjoint subsets of ω_{1} indexed by ω_{1} × ω. Then X_{(Code)}^{2}(S) = P(ω).

The Protection and Erasure lemmas, plus Corollary 5.2, give a much simpler proof of the main result from [6] (modulo the fact that the conclusion of Corollary 5.2 follows from (∗), which follows from Lemma 5.1 and standard ℙ_{max} arguments). The argument can be easily modified to produce models of MM^{+ω} in which X_{(Code)}^{2}(S) is any desired set of reals. In the proof of our main theorem, we let X_{(Code)}^{2}(S) be a set of reals coding S.

Theorem 5.3. Suppose that λ is a supercompact limit of supercompact cardinals. Then there is a semi-proper partial order of cardinality λ forcing that Martin’s Maximum^{+ω} holds and forcing that there exists a partition S of ω_{1} into stationary sets (indexed by ω_{1} × ω) such that X_{(Code)}^{2}(S) = ∅. Furthermore, in this extension MM^{++} holds for all partial orders that do not add reals to X_{(Code)}^{2}(S).

Proof. Force to add a partition S = ⟨S_{αi} : α < ω_{1}, i < ω⟩ of ω_{1} by initial segments. Then each member of S is stationary and each S̃_{αi} = ∅, so X_{(Code)}^{2}(S) = ∅. Let L: λ → V_{λ} be as in Lemma 3.7.

Let P̄ = ⟨P_{α}, Q_{∼}_{α} : α < λ⟩ be a semi-proper forcing iteration satisfying the following conditions, where G_{α} denotes the restriction of the generic filter to the first α stages of the iteration.

• Q_{0} is the trivial forcing and each forcing Q_{α+1} has the form Coll(ω_{1}, <μ) ∗ ℙ𝔽(S), where μ is the least supercompact cardinal (these forcings are (ω, ∞)-distributive and preserve stationary subsets of ω_{1}, so they do not add reals to X_{(Code)}^{2}(S)).

• If α is a limit ordinal and L(α) is a pair (τ, σ), where τ is a P_{α}-name for a semi-proper forcing and σ is a P_{α} ∗ τ-name for a countable collection A of stationary subsets of ω_{1}, then the first step of Q_{α} is τ_{G_{α}}. If X_{(Code)}^{2}(S) = ∅ after forcing with τ_{G_{α}}, then Q_{α} is just this forcing. Otherwise, Q_{α} is τ_{G_{α}}

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 21

followed by a forcing as in the Erasure Lemma, preserving the stationarity of each member of A and each stationary subset of ω_{1} in each V [G_{β}] (β < α) while making X_{(Code)}^{2}(S)^{V [G_{α+1}]} = ∅.

• If α is a limit ordinal and the previous case does not apply, and X_{(Code)}^{2}(S) is not empty in V [G_{α}], then Q_{α} is a forcing as in the Erasure Lemma, preserving the stationarity of each stationary subset of ω_{1} in each V [G_{β}] (β < α) while making X_{(Code)}^{2}(S)^{V [G_{α+1}]} = ∅.

Then in the P̄-extension, Martin’s Maximum^{+ω} holds as in [3] and X_{(Code)}^{2}(S) = ∅. Furthermore, MM^{++} holds for all partial orders that do not add a real to X_{(Code)}^{2}(S). □

We note the following standard (except for the last, which follows from the argument above and the consistency proof of MM from [3]), nonoptimal facts [3, 11, 4], where for any cardinal γ, ℶ_{ω}(γ) denotes the supremum of 2^{γ}, 2^{2^{γ}}, 2^{2^{2^{γ}}}, etc.

• If λ is a supercompact cardinal, then stationarily many cardinals κ below κ are ℶ_{ω}^{+}(κ)-supercompact.

• If κ is ℶ_{ω}^{+}(κ)-supercompact, then there is a semi-proper forcing of cardinality κ forcing MM^{++}(2^{𝔠}).

• If κ is ℶ_{ω}^{+}(κ)-supercompact and a limit of supercompact cardinals, and S is a partition of ω_{1} into stationary sets indexed by ω_{1} × ω, then there is a semi-proper partial order of cardinality κ not adding reals to X_{(Code)}^{2}(S) and forcing MM^{++}(2^{𝔠}) for all partial orders not adding reals to X_{(Code)}^{2}(S).

Mixing in forcings of the form Coll(ω_{1}, <μ) ∗ ℙ𝔽(U) for supercompact cardinals μ below κ and all partitions U of ω_{1} into stationary sets indexed by ω_{1} × ω (again, these forcings don’t add reals to X_{(Code)}^{2}(S)) we have the following.

Theorem 5.4. Let S be a partition of ω_{1} into stationary sets, indexed by ω_{1} × ω. Suppose that κ is a ℶ_{ω}^{+}(κ)-supercompact cardinal and a limit of supercompact cardinals. Then there is a semi-proper partial order of cardinality κ that does not add reals to X_{(Code)}^{2}(S) and forces that Martin’s Maximum^{+ω}(2^{𝔠}) holds and MM^{++}(2^{𝔠})

22 PAUL B. LARSON

holds for all partial orders that do not add reals to X_{(Code)}^{2}(S). Furthermore, this forcing forces for all partitions U of ω_{1} indexed by ω_{1} × ω that the following statements hold.

• Y_{(Code)}^{2}(U) = P(ω);

• every stationary subset of ω_{1} contains a member of ST AT (U, ω_{2}) modulo NS_{ω_{1}}.

6. Distinguishing partitions

Our remaining goal is to show that by modifying the iteration in Theorem 5.3, we can make some partition S of ω_{1} into stationary sets have a property (definable in H(ℵ_{2})) not shared by any other partition. This plus the fact that Martin’s Maximum holds in this extension will imply that there is a wellordering of P(ω_{1}) in this extension definable in H(ℵ_{2}) without parameters. We will use U to denote partitions that we wish to distinguish from S. Lemma 6.1 gives conditions under which any given real can be forced into X_{(Code)}^{2}(U), presuming that u_{2} is greater than the κ^{∗} of the lemma, and that the sets F̃ ∩ κ^{∗} (F ∈ F) in the lemma have abused stationary intersection with the sets required in item (ive) of Definition 2.5.

Lemma 6.1. Suppose that U = ⟨U_{αi} : (α, i) ∈ ω_{1} × ω⟩ is a partition of ω_{1} into stationary sets. Suppose that κ^{∗} is a member of the κ-sequence of U of cofinality ω_{1}, and let f be a cofinality function for κ^{∗}. Suppose that d_{α} (α < ω_{1}) are subsets of ω. Suppose that F is a collection of pairwise disjoint stationary subsets of ω_{1} such that for each member F of F, F is contained in some U_{αi} and F̃ ∩ κ^{∗} is stationary. Suppose that for each α < ω_{1}, U_{αi} contains a member of F for infinitely many even integers i and for infinitely many odd integers i.

Suppose that F_{0} and F_{1} are elements of F such that, letting (α_{0}, i_{0}) and (α_{1}, i_{1}) be the unique members of ω_{1} × ω such that F_{0} ⊂ U_{α_{0}i_{0}} and F_{1} ⊂ U_{α_{1}i_{1}}, at least one of the following holds:

• α_{0} ≠ α_{1},

• |i_{0} − i_{1}| ≠ 1.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 23

Then there is an uncountable F′ ⊂ F containing {F_{0}, F_{1}} such that the partial order to shoot a club through ▽{{γ < ω_{1} | f(γ) ∈ F̃} : F ∈ F′} (under any fixed enumeration of F′) forces that for each α < ω_{1},

{i < ω|Ũ_{αi} ∩ κ^{∗} is stationary}

contains no consecutive pair of elements of ω, and

oe({i < ω|Ũ_{αi} ∩ κ^{∗} is stationary}) = d_{α}.

Furthermore, this forcing is (ω, ∞)-distributive, preserves stationarity of any subset of ω_{1} having abused stationary intersection with any of the sets F̃ ∩ κ^{∗} (F ∈ F′) and forces that Ẽ ∩ κ^{∗} is nonstationary for any E disjoint from every member of F.

Lemma 6.2 below presents a preparation forcing which in the right circumstances allows one to force a real into X_{(Code)}^{2}(U) via Lemma 6.1 without adding a real to X_{(Code)}^{2}(S), for suitable U and S.

Lemma 6.2. Suppose that there exists a cardinal κ below a Woodin cardinal below a measurable cardinal such that κ is ℶ_{ω}^{+}(κ)-supercompact and a limit of supercompact cardinals. Let S and U be partitions of ω_{1} into stationary sets, indexed by ω_{1} × ω. Then there exists a partial order P preserving stationary subsets of ω_{1} such that, letting κ^{∗} be the supremum of the first ω_{1} many measurable cardinals above κ and letting F be the collection of stationary sets of the form S_{αi} ∩ U_{βj},

• P forces each of the following statements:

– for every F ∈ F, F̃ ∩ κ^{∗} has abused stationary intersection with every member of ST AT (S, κ^{∗}) ∪ ST AT (U, κ^{∗});

– every stationary subset of ω_{1} in the ground model contains modulo NS_{ω_{1}} a member of ST AT (S, κ^{∗}) and a member of ST AT (U, κ^{∗});

– Y_{κ^{∗}}(S) ∩ Y_{κ^{∗}}(U) contains all the reals of the ground model;

– u_{2} > κ^{∗};

• P adds no reals to X_{(Code)}^{2}(S);

24 PAUL B. LARSON

• if Q̇ is a P -name for a partial order which shoots a club through a set of the form ▽{{γ < ω_{1} | f(γ) ∈ F̃} : F ∈ F′} for some uncountable F′ ⊂ F and some cofinality function f for κ^{∗}, and this forcing does not add a real to X_{(Code)}^{2}(S) at stage κ^{∗}, then there is a P ∗ Q̇-name Ṙ for a partial order preserving the stationarity of the members of ST AT (U, κ^{∗} + 1) and ST AT (S, κ^{∗}) such that X_{(Code)}^{2}(S) in the P ∗ Q̇ ∗ Ṙ-extension is the same as X_{(Code)}^{2}(S) in the ground model.

Proof. The partial order P consists of three steps. First force as in Lemma 5.4 with a partial order of cardinality κ (while preserving stationary subsets of ω_{1}) to make MM^{++}(2^{𝔠}) hold for all forcings not adding reals to X_{(Code)}^{2}(S), without adding any reals to X_{(Code)}^{2}(S). Call this extension V [G]. In V [G], NS_{ω_{1}} is saturated (since the antichain-sealing forcing adds no reals, see Lemma 2.6 and the remarks after Lemma 3.4), u_{2} = ω_{2}, and

Y_{(Code)}^{2}(S) = Y_{(Code)}^{2}(U) = P(ω).

Furthermore, every stationary subset of ω_{1} in the ground model contains modulo NS_{ω_{1}} a member of ST AT (S, ω_{2})^{V [G]} and a member of ST AT (U, ω_{2})^{V [G]}.

Letting ⟨κ_{α} : α ≤ ω_{1} · 2⟩ be the closed increasing sequence generated by the first ω_{1} · 2 many measurable cardinals (so κ_{ω_{1}} is the κ^{∗} of the statement of the lemma), force now as in Lemma 4.6, with S as itself and F as ⟨T_{αi} : (α, i) ∈ ω_{1} ×ω⟩ (suitably reindexed). Recall that the first step of this forcing adds generic partitions of ω_{1} ⟨A_{αi} : (α, i) ∈ ω_{1} × ω⟩ and ⟨B_{αi} : (α, i) ∈ ω_{1} × ω⟩ by initial segments; the genericity of each A_{αi} implies that F̃ ∩ κ_{ω_{1}} has abused stationary intersection with every member of ST AT (S, ω_{2}^{V [G]}) ∪ ST AT (U, ω_{2}^{V [G]}), for each F ∈ F. Call this extension V [G][H]. Note that since the forcing from Lemma 4.6 does not add reals and preserves stationary subsets of ω_{1}, X_{(Code)}^{2}(S)^{V [G][H]} = X_{(Code)}^{2}(S)^{V}. Note also that for every stationary K ⊂ ω_{1} in V [G], the only V [G]-indiscernibles in K̃ are in the set {ω_{1}} ∪ {κ_{α} : α ≤ ω_{1} · 2}. It follows that κ_{ω_{1}} is the only ρ ∈ C_{κ_{ω_{1}·2}}^{ω_{1}} is greater than ω_{2}^{V [G]} for which there exists (α, i) ∈ ω_{1} × ω such that either of S̃_{αi} ∩ ρ

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 25

and Ũ_{αi} ∩ ρ is stationary. We will see that these last two facts are preserved by the remaining forcings in this proof, which will preserve the fact that no real enters either X_{(Code)}^{2}(U) or X_{(Code)}^{2}(S) at any stage in the interval (ω_{2}^{V [G]}, κ_{ω_{1}·2}) other than κ_{ω_{1}}. Note also then that ST AT (S, κ_{ω_{1}}) = ST AT (S, ω_{2}^{V [G]}) and ST AT (U, κ_{ω_{1}}) = ST AT (U, ω_{2}^{V [G]})

Finally, force with any partial order preserving stationary subsets of ω_{1} making u_{2} > κ_{ω_{1}}. This is the last step of P. Call this extension V [G][H][I]. Since this last forcing preserves stationary subsets of ω_{1}, κ_{ω_{1}} and κ_{ω_{1}·2} remain on the κ-sequences of both S and U in V [G][H][I], and, for each α < κ_{ω_{1}·2} + 1, X_{α}(S)^{V [G][H]} = X_{α}(S)^{V [G][H][I]}.

Now suppose that we force over V [G][H][I] with a forcing Q of the form ▽{{γ < ω_{1} | f(γ) ∈ F̃} : F ∈ F′} for some uncountable F′ ⊂ F and some cofinality function f for κ_{ω_{1}}, and that this forcing does not add a real to X_{(Code)}^{2}(S) at stage κ_{ω_{1}}. This partial order shoots a club through a subset of ω_{1} containing some A_{αi} modulo NS_{ω_{1}}. Now force as in the Erasure Lemma (the forcing Ṙ of this lemma) to make no real enter X_{(Code)}^{2}(S) at stage κ_{ω_{1}·2} or later. This partial order wellorders the reals in ordertype ω_{1} and then shoots a club through a subset of ω_{1} containing some B_{βj} modulo NS_{ω_{1}}. Since B_{βj} has stationary intersection with each A_{γk}, this second forcing preserves the stationarity of each A_{γk} left stationary by forcing with the realization of Q, and thus preserves the stationarity of each member of ST AT (U, κ_{ω_{1}} + 1).

By Lemma 4.6, no V [G]-indiscernible in κ_{ω_{1}·2}\{κ_{α} : α < ω_{1}·2} is forced by Q∗Ṙ into K̃ for any stationary K ⊂ ω_{1} in V [G]. To see this, fix a condition in Q ∗ Ṙ deciding the values of (α, i) and (β, j) as in the previous paragraph, fix such an indiscernible ξ and a name τ for a function supposedly witnessing that ξ is forced into K̃, and choose a countable elementary submodel X of a suitable H(θ) with this condition and τ in X, X ∩ ω_{1} ∈ A_{αi} ∩ B_{βj} and ot(X ∩ ξ) ∉ K (the set of such X is stationary in V [G][H] by Lemma 4.6 and genericity, and remains stationary in V [G][H][I] since ξ has cardinality ℵ_{1} in V [G][H]). Then any X-generic filter for

26 PAUL B. LARSON

Q ∗ Ṙ naturally defines a Q ∗ Ṙ condition witnessing that τ is not a name for a function witnessing that ξ ∈ K̃.

It follows that in the Q ∗ Ṙ extension, the b-sequences for both S and U still consist of the empty real at every stage in the interval (ω_{2}^{V [G]}, κ_{ω_{1}·2}) other than κ_{ω_{1}}. In particular, no real enters X_{(Code)}^{2}(S) at any stage γ in the interval (κ_{ω_{1}}, κ_{ω_{1}·2}) either in this final extension. □

Fixing a coding of elements of H(ω_{1}) by reals, for each partition U = ⟨U_{αi} : (α, i) ∈ ω_{1} × ω⟩ of ω_{1} into stationary sets, let E_{U} be the set of x ⊂ ω such that x codes in this fixed coding a set of the form ⟨U_{αi}∩β : α < β⟩, for some β < ω_{1}. Given subsets A, B of ω_{1}, we say A =^{∗} B if A △ B ∈ NS_{ω_{1}} and A ⊂^{∗} B if A \ B ∈ NS_{ω_{1}}.

Lemma 6.3 below allows one to distinguish between S and U in the forcing extension in the main theorem.

Lemma 6.3. Suppose that there exists a cardinal λ below a Woodin cardinal below a measurable cardinal such that λ is ℶ_{ω}^{+}(λ)-supercompact and a limit of supercompact cardinals.

Let S = ⟨S_{αi} : α < ω_{1}, i < ω⟩ and U = ⟨U_{αi} : α < ω_{1}, i < ω⟩ be two partitions of ω_{1} into stationary sets. Then there is a partial order preserving stationary subsets of ω_{1} which forces no real outside of E_{S} into X_{(Code)}^{2}(S) but forces a real outside of E_{U} into X_{(Code)}^{2}(U).

Proof. We break into two cases, the second of which has four subcases. In the first case, where U is essentially the same as S, we can force a real in E_{S} \ E_{U} into both X_{(Code)}^{2}(S) and X_{(Code)}^{2}(U). In the second case, where U and S are significantly different, we can force a real not in E_{U} into X_{(Code)}^{2}(U) without adding any real to X_{(Code)}^{2}(S).

The first (close) case is when S_{αi} =^{∗} U_{αi} for all (α, i) ∈ ω_{1} × ω. In this case we first force as in Theorem 5.4. Then since u_{2} = ω_{2} and NS_{ω_{1}} is saturated in this extension, for each pair S_{αi}, U_{αi} there is a real in Y_{(Code)}^{2}(S) ∩ Y_{(Code)}^{2}(U) whose indiscernibles are disjoint from S_{αi} △ U_{αi} (see Theorem 3.16 of [14]) and thus from

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 27

S̃_{αi} △ Ũ_{αi}. Then each set of the form S̃_{αi} △ Ũ_{αi} is disjoint from the κ-sequences of both S and U above some ordinal below the ω_{2} of this extension. It follows that in all forcing extensions preserving stationary subsets of ω_{1}, the S-codings and U-codings are the same at all stages after the ω_{2} of this extension.

Let β < ω_{1} be large enough so that ⟨S_{αi} ∩ β : (α, i) ∈ β × ω⟩ and ⟨U_{αi} ∩ β : (α, i) ∈ β ×ω⟩ are distinct, and let z be a real coding ⟨S_{αi} ∩β : (α, i) ∈ β ×ω⟩ in our fixed coding. Then forcing as in Lemma 5.1 adds z and only z to both X_{(Code)}^{2}(S) and X_{(Code)}^{2}(U).

The second case is when S_{αi} △ U_{αi} is stationary for some (αi) ∈ ω_{1} × ω (we do not, however, fix such a pair (α, i)). In this case, we first force with P as in Lemma 6.2, and fix κ^{∗} as in the statement of the lemma. Note that κ^{∗} is on the κ-sequences of both S and U in this extension, and, for every stationary set F of the form S_{αi} ∩ U_{βj}, F̃ has abused stationary intersection with every member of ST AT (S, κ^{∗}) and ST AT (U, κ^{∗}), and that every stationary subset of ω_{1} in the ground model contains modulo NS_{ω_{1}} a member of each of these sets.

The second case has four subcases, which will see are exhaustive. Fix a sequence ⟨d_{α} : α ∈ ω_{1}⟩ such that

• each d_{α} is an infinite subset of ω;

• no d_{α} contains consecutive elements of ω;

• each oe(d_{α}) ∈ ℂ;

• ⟨li(oe(d_{α}), ω_{1}) : α < ω_{1}⟩ is increasing;

• li(oe(d_{1}), ω_{1}) > κ^{∗};

• each t^{∗}(oe(d_{α})) is infinite and co-infinite;

• t^{∗}(oe(d_{0})) ∈ P(ω) \ E_{U}. It suffices to see that at this point there is a suitable choice of F_{0}, F_{1} and F as in

Lemma 6.1 which adds t^{∗}(oe(d_{0})) to X_{(Code)}^{2}(U) at stage κ^{∗} without adding any real to X_{(Code)}^{2}(S) at stage κ^{∗}. Suitable here means that F̃_{0} ∩κ^{∗} and F̃_{1} ∩κ^{∗} both being stationary, and Ẽ ∩ κ^{∗} being nonstationary for each E ⊂ ω_{1} having nonstationary intersection with each member of F, implies that no real enters X_{(Code)}^{2}(S) at stage

28 PAUL B. LARSON

κ^{∗}, while for each α < ω_{1}, U_{αi} will have stationary intersection with some member of F for infinitely even i and infinitely many i odd, which means that it is still possible to force any desired real into X_{(Code)}^{2}(U) (in particular, t^{∗}(oe(d_{0}))). Then by Lemma 6.2 we can further force to ensure that no reals enter X_{(Code)}^{2}(S) at a stage after κ^{∗}, either. We will do just this in the four subcases below, with F as a subset of the collection of stationary sets of the form S_{αi} ∩ U_{βj}, for α, β < ω_{1} and i, j < ω.

In the first subcase, suppose that there exist α, β, γ < ω_{1}, i, j, k < ω such that S_{αi} ∩U_{βj} and S_{α(i+1)} ∩U_{γk} are both stationary and one of the following cases holds:

• β ≠ γ,

• |j − k| ≠ 1.

Then let F_{0} = S_{αi} ∩ U_{βj}, F_{1} = S_{α(i+1)} ∩ U_{γk} and F be all stationary sets of the form S_{δn} ∩ U_{ρm}, for m, n ∈ ω, δ, ρ ∈ ω_{1}.

For the rest of the proof we assume that the first subcase does not apply, and so there exists a function g: ω_{1} → ω_{1} such that whenever S_{αi} ∩ U_{βj} and S_{αi} ∩ U_{γk} are stationary, β = γ = g(α) and |k − j| ∈ {0, 2}.

In the second subcase, suppose that there exists α^{∗} < ω_{1} such that one of the two following statements holds.

• ⋃{S_{α^{∗}i} : i < ω even} fails to contain (modulo NS_{ω_{1}}) all but finitely many of the sets U_{g(α^{∗})j} for j odd and fails to contain (modulo NS_{ω_{1}}) all but finitely many of the sets U_{g(α^{∗})j} for j even.

• ⋃{S_{α^{∗}i} : i < ω odd} fails to contain (modulo NS_{ω_{1}}) all but finitely many of the sets U_{g(α^{∗})j} for j odd and fails to contain (modulo NS_{ω_{1}}) all but finitely many of the sets U_{g(α^{∗})j} for j even.

In the first (subsub)case, let F be all stationary sets of the form S_{δn} ∩U_{ρm} where n, m ∈ ω, δ, ρ ∈ ω_{1} such that either n is not even or δ ≠ α^{∗}, and let F_{0} and F_{1} both be any one element of F. In the second case, let F be all stationary sets of the form S_{δn} ∩ U_{ρm} where n, m ∈ ω, δ, ρ ∈ ω_{1} such that either n is not odd or δ ≠ α^{∗}, and let F_{0} and F_{1} both be any one element of F.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 29

Assuming that the first two subcases do not apply, g is injective.

In the third subcase, suppose that the first two subcases do not apply, and two distinct set S_{αi} and S_{αk} both intersect some U_{g(α)j} stationarily. Since the first subcase does not apply, S_{αi} \ U_{g(α)j} intersects at most one set of the form U_{g(α)n} stationarily (fix n if it exists; n must be either j − 2 or j + 2, though we will not use this fact). Let F_{0} = S_{αi} ∩ U_{g(α)j}, F_{1} = S_{αk} ∩ U_{g(α)j} and F be all stationary sets of the form S_{δp} ∩ U_{ρm} (excluding S_{αi} ∩ U_{g(α)n} if n as above exists). If the forcing corresponding to each F′ ⊂ F as in Lemma 6.1 adds a real to X_{(Code)}^{2}(S) at stage κ^{∗}, then fix one such F′ and force instead with F′ \ {S_{αi} ∩ U_{g(α)j}}. This codes the same real into X_{(Code)}^{2}(U) at stage κ^{∗}, but it cannot be that both forcings add a real to X_{(Code)}^{2}(S), since the set oe({m < ω | S̃_{αm} ∩ κ^{∗} is stationary}) cannot be in ℂ in both cases, by Lemma 2.3.

If the first three cases do not apply, then for each α < ω_{1} there is a permutation e_{α}: ω → ω such that for each i < ω, S_{αi} =^{∗} U_{g(α)e_{α}(i)}. It follows easily from the fact that |e_{α}(i) − e_{α}(i + 1)| = 1 for all α < ω_{1}, i < ω that each e_{α} is the identity function. If g is not the identity function, ⟨li(d_{g^{−1}(β)}, ω_{1}) : β < ω_{1}⟩ is not increasing, so we can let F be all stationary sets of the form S_{αi} ∩ U_{βj} and let F_{0} and F_{1} both be any one of these sets. If g is the identity then we are in the first case of the proof. □

7. The main theorem

In Theorem 7.1 we modify the iteration from the proof of Theorem 5.3 to make X_{(Code)}^{2}(S) = E_{S}, which makes S definable in the H(ℵ_{2}) of the extension without parameters. The third condition on the iteration is probably subsumed by the second one, but we include it to make certain.

Undoubtedly there are many other ways to make the parameter S definable.

Theorem 7.1. Assume that there exists a supercompact limit of supercompact cardinals. Then there is a semi-proper partial order forcing Martin’s Maximum^{+ω} and forcing the existence of a partition S of ω_{1} into uncountably many stationary sets

30 PAUL B. LARSON

such that S is definable in H(ℵ_{2}) without parameters. Furthermore, in this extension, Martin’s Maximum^{++} holds for partial orders that don’t add reals not in E_{S} to X_{(Code)}^{2}(S),

Proof. Let λ be a supercompact limit of supercompact cardinals. Force to add a partition S = ⟨S_{αi} : α < ω_{1}, i < ω⟩ be a partition of ω_{1} of ω_{1} by initial segments. Then each member of S is stationary and each S̃_{αi} = ∅, so X_{(Code)}^{2}(S) = ∅.

Let L: λ → V_{λ} be as in Lemma 3.7. Let P̄ = ⟨P_{α}, Q_{∼}_{α} : α < λ⟩ be a semi-proper forcing iteration satisfying the

following conditions, where G_{α} denotes the restriction of the generic filter to the first α stages of the iteration.

• Q_{0} is the trivial forcing and each forcing Q_{α+1} has the form Coll(ω_{1}, <μ) ∗ ℙ𝔽(S).

• If α is a limit ordinal and L(α) is a triple (0, τ, σ), where τ is a P_{α}-name for a semi-proper forcing and σ is a P_{α} ∗τ-name for a countable collection A of stationary subsets of ω_{1}, then the first step of Q_{α} is τ_{G_{α}}. If X_{(Code)}^{2}(S) ⊂ E_{S} after forcing with τ_{G_{α}}, then Q_{α} is just this forcing. Otherwise, Q_{α} is τ_{G_{α}} followed by a forcing as in the Erasure Lemma, preserving the stationarity of each member of of A and each stationary subset of ω_{1} in each V [G_{β}] (β < α) while making X_{(Code)}^{2}(S)^{V [G_{α+1}]} = ⋃{X_{(Code)}^{2}(S)^{V [G_{β}]} : β < α}.

• If α is a limit ordinal and L(α) is a triple (1, τ, σ), where τ is a P_{α}-name for a partition of ω_{1} into stationary sets (indexed by ω_{1} × ω) and σ is P_{α}-name for a real, then if there exists a forcing preserving stationary subsets of ω_{1} and forcing σ_{G_{α}} into X_{(Code)}^{2}(τ_{G_{α}}) in whose extension X_{(Code)}^{2}(S) ⊂ E_{S}, then Q_{α} is such a forcing.

• If α is a limit ordinal and the previous two cases do not apply, and X_{(Code)}^{2}(S) is not a subset of E_{S} in V [G_{α}], then Q_{α} is a forcing as in the Erasure Lemma, preserving the stationarity of each stationary subset of ω_{1} in each V [G_{β}] (β < α) while making X_{(Code)}^{2}(S)^{V [G_{α+1}]} = ⋃{X_{(Code)}^{2}(S)^{V [G_{β}]} : β < α}.

MARTIN’S MAXIMUM AND DEFINABILITY IN H(ℵ_{2}) 31

Then in the P̄-extension, Martin’s Maximum^{+ω} holds, and Martin’s Maximum^{++} holds for partial orders that don’t add reals not in E_{S} to X_{(Code)}^{2}(S), by the original consistency proof for Martin’s Maximum in [3]. By construction, X_{(Code)}^{2}(S) ⊂ E_{S} in this extension, and X_{(Code)}^{2}(S) ⊃ E_{S} by Lemma 5.1. By Lemma 6.3, in this extension S is the unique partition U of ω_{1} into stationary sets indexed by ω_{1} × ω such that X_{(Code)}^{2}(U) = E_{U}. □

We note that the following question is still open.

Question 7.2. Does Martin’s Maximum^{++} imply (∗)?

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

[5] A. Kanamori, The Higher Infinite, Springer-Verlag, Berlin, 2003

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[9] P.B. Larson, The nonstationary ideal in the ℙ_{max} extension, J. Symbolic Logic, to appear [10] P.B. Larson, Forcing over models of determinacy, Handbook of Set Theory, Foreman,

Kanamori, Magidor (eds.), to appear

[11] R. Laver, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel J. Math. 29 (1978), 385-388

[12] J.T. Moore, Set mapping reflection, J. Math. Logic 5 (2005), no. 1, 87–97

[13] S. Shelah, Proper and Improper Forcing, Springer-Verlag, Berlin, 1998

[14] W.H. Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, Walter de Gruyter & Co., Berlin, 1999

32 PAUL B. LARSON

Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056, USA

E-mail address: larsonpb@muohio.edu
URL: http://www.users.muohio.edu/larsonpb/