THE ENVELOPE OF A UNDER A LOCAL HYPOTHESIS

TREVOR M. WILSON

Abstract. Given an inductive-like pointclass Γ and assuming the Ax- iom of Determinacy, Martin identified and analyzede the pointclass con- taining the norm relations of the next semiscale beyond Γ, if one exists. We show that much of Martin’s analysis can be carriede out assuming

only ZF + DCR + Det(∆Γ). This generalization requires arguments from Kechris–Woodin [10] ande e Martin [13]. The results of [10] and [13] can then be recovered as immediate corollaries of the general analysis. We also obtain a new proof of a theorem of Woodin on divergent models of AD+, as well as a new result regarding the derived model at an inde- structibly weakly compact limit of Woodin cardinals.

Introduction Given an inductive-like pointclass Γ, Martin introduced a pointclass (which we call the envelope of Γ, following [22])e whose main feature is that it con- tains the prewellorderingse of the next scale (or semiscale) beyond Γ, if such a (semi)scale exists. This work was distributed as “Notes on the nexte Suslin cardinal,” as cited in Jackson [3]. It is unpublished, so we use [3] as our reference instead for several of Martin’s arguments. Martin’s analysis used the assumption of the Axiom of Determinacy. We reformulate the notion of the envelope in such a way that many of its essen- tial properties can by derived without assuming AD. Instead we will work in the base theory ZF+DCR for the duration of the paper, stating determinacy hypotheses only when necessary. With the exception of Section 9 this paper is mainly expository. It places results of Martin, Kechris, Woodin, and others in a unified framework and in some cases substantially simplifies the original proofs. Section 1 contains background information on descriptive theory in- cluding the definition of “inductive-like pointclass.” In Section 2 we define the envelope of an inductive-like pointclass in a special case that will suffice for results in L(R) and we prove some results about it, most notably its determinacy and its closure under real quantifiers. In Sections 3 and 4 we apply these results to the envelope of the pointclass of inductive sets and

Date: May 4, 2014. 2010 Mathematics Subject Classification. Primary 03E15; Secondary 03E60. Key words and phrases. determinacy, pointclass, scale, derived model. The author gratefully acknowledges support from NSF grant DMS-1044150. 1 2 TREVOR M. WILSON then more generally to the envelopes of inductive-like in L(R). In particular we show that the Kechris–Woodin determinacy transfer theo- rem and a theorem of Martin on reflection of ordinal-definability for reals can be obtained as immediate consequences. In Section 6 we give a more general definition of the envelope for boldface inductive-like pointclasses, which is phrased in terms of Moschovakis’s notion of “companion” for such pointclasses (see Section 5.) We then show that the results of Section 2 can readily be adapted to this more general setting. In Section 7 we give an equivalent condition for the existence of semiscales that is phrased in terms of games. In the later sections these games are used to construct (under additional assumptions) semiscales on a universal Γˇ set with in the envelope of Γ. In Section 8 we give a new proofe of Woodin’s theorem on divergent modelse of AD+. Finally, in Section 9 we show that the derived model at an indestructibly weakly compact limit of Woodin cardinals satisfies “every set of reals is Suslin.” The results in this paper are mostly taken from the author’s PhD thesis [25, Ch. 3]. The author wishes to thank John Steel, who supervised this thesis work, for his guidance; Hugh Woodin, for explaining his argument for the result stated as Theorem 8.2 below; and Martin Zeman, for suggesting several corrections.

1. Pointclasses We begin by recalling some standard notions from descriptive . ω By convention we denote the ω by R and call its elements “reals.” A product space is a space of the form

X = X1 × · · · × Xn,Xi = R or Xi = ω for all i ≤ n. A pointset is a subset of a product space. A pointclass is a collection of pointsets, typically an initial segment of some complexity hierarchy for pointsets. We say that a pointclass Γ is ω-parameterized (respectively, R-parameter- ized) if for every product space X there is a Γ subset of ω×X (respectively, of R × X ) that is universal for Γ subsets of X . In this paper we consider the following types of pointclasses, which are named for their resemblance to the pointclasses IND and IND of induc- tive sets and boldface inductive sets respectively. The inductive] pointsets are those that are definable without parameters by positive elementary in- duction on R (sometimes called “absolutely” or “lightface” inductive.) By “boldface” we mean that real parameters are allowed.

Definition 1.1. A pointclass Γ is (lightface) inductive-like if it is ω-parameterized, closed under ∃R, ∀R, and recursive substitution, and has the property. THE ENVELOPE UNDER LOCAL DETERMINACY 3

Definition 1.2. A pointclass Γ is boldface inductive-like if it is R-parameterized, closed under ∃R, ∀R, and continuouse reducibility, and has the prewellorder- ing property. Note that some authors strengthen the prewellordering property to the scale property in these definitions. In this paper we will typically use the following notational conventions. By Γ and Γ we will denote an inductive-like or boldface-inductive-like pointclass respectively.e As usual we denote the dual pointclass of Γ by Γˇ = {¬A : A ∈ Γ} and the ambiguous part of Γ by ∆Γ = Γ ∩ Γ.ˇ (Note that the existence of universal sets implies non-self-duality by a diagonal argument.) Similarly for a boldface pointclass we define the dual and ambiguous pointclasses Γˇ and ∆Γ. e Givene e a lightface pointclass Γ we can define the corresponding relativiza- tions Γ(x) for x ∈ and also the boldface pointclasses Γ = S Γ(x), R x∈R Γˇ = S Γ(ˇ x), and ∆ = S ∆ (x). Naturally, if Γe is inductive-like x∈R Γ x∈R Γ thene the correspondinge boldface pointclass Γ is boldface inductive-like. Typically given a pointclass Γ we will assumee the local determinacy hypo- thesis Det(∆Γ), meaning that every two-player game of perfect information on ω with ae∆Γ payoff set is determined, although some of our results can proved undere weaker hypotheses. The prewellordering ordinal of an inductive-like pointclass Γ (or more generally of a pointclass Γ with the prewellordering property) is the supre- mum of the ordinal lengths of all ∆Γ prewellorderings of R, or equivalently the range of any regular Γ-norm one a complete Γ pointset. We will often denote the prewellordering ordinal of a pointclass by κ. The prototypical example of an inductive-like pointclass is, as noted above, the pointclass IND of inductive sets. Other examples of inductive-like 2 L(R) pointclasses include the pointclass (Σ1) and more generally the point- 2 + Σ1 under AD . Examples of boldface inductive-like pointclasses include, in addition to the boldface versions of the above lightface pointclasses, the pointclass S(κ) of κ-Suslin sets when AD holds, κ is a Suslin cardinal, and S(κ) is closed under ∀R. The following notion of countable approximation for pointsets is central to the paper. Definition 1.3 (Martin, see [3, Defn. 3.9]). Let X be a product space, let κ be an ordinal, and let (Aα : α < κ) be a sequence of subsets of X . For a pointset A ⊂ X , we say A ∈ (Aα : α < κ) if for every countable set σ ⊂ X there is an ordinal α < κ such that A ∩ σ = Aα ∩ σ. Martin then made the following definition. It is similar to the definition of “envelope” that we will use in this paper, but we will need to make a significant modification. We include the original definition below in order to provide historical context. 4 TREVOR M. WILSON

Definition 1.4 (Martin, see [3, Defn. 3.9]). Let ∆ be a pointclass and let κ be an ordinal. For a product space X and a pointsete A ⊂ X , we say A ∈ Λ(∆, κ) if A ∈ (Aα : α < κ) for some sequence (Aα : α < κ) of ∆ subsets ofe X . e If AC holds then the pointclass Λ(∆, κ) is not likely to be very useful because there will be too many well-orderede sequences of pointsets. For example, if ∆ is a nontrivial boldface pointclass (so it contains every count- able pointset)e and κ ≥ c then any pointset whatsoever is in Λ(∆, κ). To get a useful definition without assuming AD we need to require somee kind of uniformity for the sequence of pointsets (Aα : α < κ). First we deal with a special case.

Jκ(R) 2. The envelope of Σ1 e We will need some background on the fine structure of L(R) that can be found, for example, in [23, §1]. We use the Jensen hierarchy (Jα : α ∈ Ord) for L(R) but the reader would not miss much by reading “Lκ(R)” in place of “Jκ(R).” However we should note that the ordinal height of Jα(R) is ωα rather than α. By convention we allow R as an unstated parameter in our formulas. For example, by Σ1 we mean Σ1({R}). Therefore quantification over the reals always counts as bounded quantification. R Given ordinals κ and β with κ ≤ β, we say Jκ(R) ≺1 Jβ(R) if Jκ(R) is a Σ1(R ∪ {R})-elementary substructure of Jβ(R). As usual, Θ denotes the least ordinal that is not a surjective image of the reals. 1 Definition 2.1. A gap in L(R) is a maximal interval of ordinals [κ, β] such R L(R) that Jκ(R) ≺1 Jβ(R) and β ≤ Θ . The gaps partition the interval [0, ΘL(R)]. We say an ordinal κ begins a gap in L(R) if [κ, β] is a gap for some ordinal β. If κ is a limit ordinal, this means that new Σ1 facts about reals are witnessed cofinally often below κ in Jκ(R) the Jensen hierarchy. When stating results about the pointclass Σ1 we will assume without loss of generality that κ begins a gap because otherwise Jκ(R) Jα(R) Σ1 = Σ1 for some α < κ. Jκ(R) If κ begins a gap then there is a Σ1 partial surjection R 99K Jκ(R) (see Jκ(R) Jκ(R) [23, Lem. 1.11]) and therefore every Σ1 pointset is in Σ1 (x) for some real parameter x. That is, we cane replace arbitrary parameters in Jκ(R) with real parameters. If the level Jκ(R) is admissible (equivalently, if it satisfies the Σ0-collection Jκ(R) R axiom) then the pointclass Σ1 is closed under ∀ . This implies that Jκ(R) Σ1 is inductive-like, as the other clauses of the definition are easy to verify. (To verify the prewellordering property, for example, observe that

1 More precisely called a Σ1-gap THE ENVELOPE UNDER LOCAL DETERMINACY 5

Jκ(R) Jκ(R) whenever κ is a limit ordinal, a Σ1 -norm on a Σ1 set is given by mapping every element of the set to the least ordinal α < κ such that the level Jα(R) contains a witness to the relevant Σ1 property of that element.) We define a special case of the envelope as follows. (This definition is not standard; see Section 6 for remarks on other definitions.)

Definition 2.2. Let Jκ(R) be an admissible level of L(R) that begins a gap. For a pointset A, we say: Jκ(R) • A ∈ Env(Σ1 ) if A ∈ (Aα : α < κ) for some sequence of pointsets (Aα : α < κ) that is ∆1-definable over Jκ(R). Jκ(R) • A ∈ Env(Σ1 ) if A ∈ (Aα : α < κ) for some sequence of pointsets (Aα : α

Definition 2.3. Let β be an ordinal and let A be a pointset. Let p ∈ Jβ(R) be a parameter.2 Then we say: • A ∈ ODβ(p) if A is first-order definable from p and ordinal parame- ters over the structure (Jβ(R); ∈). • A ∈ OD<β(p) if A ∈ ODα(p) for some ordinal α < β.

In the case where Jκ(R) is admissible we get another characterization of OD<κ. This is surely well-known, but we isolate it here in order to make an analogy later in Section 6.

Lemma 2.4. Let Jκ(R) be an admissible level of L(R) and let A be a pointset. Then the following statements are equivalent: • A ∈ OD<κ. Jκ(R) • A ∈ ∆1 (s) for some finite sequence of ordinal parameters s ∈ κ<ω. α Proof. If A ∈ OD where α < κ, say A is first-order definable over Jα(R) from a finite sequence of ordinals s0, then because the sequence of levels Jκ(R) Jκ(R) _ (Jα(R): α < κ) is ∆1 we have A ∈ ∆1 (s0 α). Jκ(R) <ω Conversely, if A ∈ ∆1 (s) where s ∈ κ then by admissibility, for any sufficiently large α < κ all the relevant witnesses to Σ1 facts about reals and Jα(R) α s have already appeared, so A is ∆1 (s). In particular, A ∈ OD .  The following result is easily seen to be a consequence of the existence of a Σ0 well-ordering of finite sequences of ordinals together with the fact that Jκ(R) the sequence of levels (Jα : α < κ) is ∆1 .

2In our applications, p will be a real or a Turing degree. 6 TREVOR M. WILSON

Lemma 2.5. Let Jκ(R) be an admissible level of L(R). Then there is a Jκ(R) <κ ∆1 sequence enumerating all OD pointsets. From Lemmas 2.4 and 2.5 we immediately get the following characteriza- tion of the envelope, which we will use many times without comment in the sequel.

Lemma 2.6. Let Jκ(R) be an admissible level of L(R) that begins a gap. Let X be a product space. For every pointset A ⊂ X the following statements are equivalent. Jκ(R) • A ∈ Env(Σ1 ). • For every countable set σ ⊂ X there is a pointset A0 ∈ OD<κ with A ∩ σ = A0 ∩ σ. Next we establish some basic closure properties of the envelope. Some of the corresponding arguments for Martin’s pointclass Λ(∆, κ) can be found in [3, Lem 3.10 and Cor. 3.11]. e

Proposition 2.7. Let Jκ(R) be an admissible level of L(R) that begins a Jκ(R) Jκ(R) gap. Then Env(Σ1 ) contains Σ1 and is closed under Boolean combi- nations, number quantification, and recursive substitution.

Jκ(R) Jκ(R) Proof. Fix a product space X . To see that Σ1 ⊂ Env(Σ1 ), let A ⊂ X be a pointset that is definable over Jκ(R) by a Σ1 formula θ. Let σ ⊂ X be countable. Because Jκ(R) is admissible the ordinal κ has uncountable cofinality, so if we choose a large enough α < κ then the ODα set 0 0 0 A ⊂ X defined by x ∈ A ⇐⇒ (Jα(R); ∈) |= θ[x] satisfies A ∩ σ = A ∩ σ. Jκ(R) Therefore A ∈ Env(Σ1 ). Jκ(R) The closure of Env(Σ1 ) under Boolean combinations is an immedi- ate consequence of the closure of OD<κ under Boolean combinations. The Jκ(R) closure of Env(Σ1 ) under number quantification is a consequence of the closure of OD<κ under number quantification, using the fact that if the pointset σ ⊂ X is countable then so is the pointset ω × σ ⊂ ω × X . The Jκ(R) closure of Env(Σ1 ) under recursive substitution follows from the clo- sure of OD<κ under recursive substitution, using the fact that the pointwise image of a countable set σ (under a recursive function) is countable.  Jκ(R) <κ Note that all results so far can be relativized from Σ1 and OD to Jκ(R) <κ Σ1 (x) and OD (x) respectively where x is a real. In particular, the Jκ(R) pointclass Env(Σ1 ) is closed under continuous reducibility. The followinge “anti-uniformization” lemma limits the extent of the enve- lope. It is essentially a localization from OD to OD<κ of an observation due to Solovay (as attributed in [23, p. 149].)

Lemma 2.8. Let Jκ(R) be an admissible level of L(R) that begins a gap. If A ⊂ R × ω is a pointset whose sections Ax (defined by Ax(n) ⇐⇒ A(x, n)) <κ Jκ(R) satisfy Ax ∈/ OD (x) for all reals x, then A/∈ Env(Σ1 ). e THE ENVELOPE UNDER LOCAL DETERMINACY 7

Jκ(R) Jκ(R) Proof. Let A ⊂ R×ω be a pointset in Env(Σ1 ), say A ∈ Env(Σ1 (x)) 0 <κ 0 where x ∈ R. Take a pointset A ∈ OD (x)e such that A ∩ σ = A ∩ σ where σ is the countable pointset {x} × ω. Then from A0 and x we can compute <κ the section Ax, so Ax ∈ OD (x).  Remark 2.9. We remark that an anti-uniformization result more along the Jκ(R) lines of Solovay’s original observation would state that, if the Π1 <κ A on R × R given by (x, y) ∈ A ⇐⇒ y∈ / OD (x) is total, then it has Jκ(R) no uniformization in Env(Σ1 ). Later we will see that if AD holds in the admissible set Jκ(R) thene this relation is indeed total and moreover Jκ(R) the pointclass Env(Σ1 ) is sufficiently closed (in particular it is closed under real quantifiers)e that these two anti-uniformization statements are equivalent.

Jκ(R) Next we proceed to establish some properties of Env(Σ1 ) that follow from the assumption of AD in Jκ(R), or equivalently by admissibility, the Jκ(R) assumption of ∆1 determinacy. In some cases, we will only need the eJκ(R) hypothesis of ∆1 determinacy. The following lemma is derived from an argument used by Kechris and Woodin to prove a determinacy transfer principle in L(R) ([10, Thm. 1.8].) It is also similar to the proof of the Kechris–Solovay theorem [9, Thm. 3.1] 1 that ∆2-determinacy implies that OD-determinacy holds in L[x] for a Turing cone of reals x. In order to state the lemma we first make a definition. For a set of reals A and a Turing ideal3 I , we say that A is determined on I if there is a strategy in I (more precisely, coded by a real in I ) for the game GA that defeats every real in I . In other words, there is a strategy σ ∈ I such that for all reals x, y ∈ I we have σ ∗ y ∈ A (if σ is a strategy for Player I) and x ∗ σ∈ / A (if σ is a strategy for Player II.)

Lemma 2.10. Let Jκ(R) be an admissible level of L(R) that begins a gap Jκ(R) and suppose that ∆1 is determined. Then there is a real t such that every OD<κ set of reals is determined on every countable Turing ideal I containing t.

Jκ Proof. Suppose not. Let (Aα : α < κ) be a ∆1 (R) sequence enumerating all OD<κ sets of reals. For a real t, define α(t) as the least ordinal α < κ such that there is a countable Turing ideal I containing t and on which the set of reals Aα is not determined. The function R → κ given by t 7→ α(t) is total Jκ(R) Jκ(R) and ∆1 , so the following game is ∆1 and is therefore determined: I x(0), z(0) x(1), z(1) ... (G) II y(0), s(0) y(1), s(1),...,

3A Turing ideal is a set of reals that is closed under recursive join ⊕ and is downward closed under Turing reducibility. 8 TREVOR M. WILSON where we say that Player I wins if and only if x ⊕ y ∈ Aα(z⊕s). Here x ⊕ y denotes the recursive join (x(0), y(0), x(1), y(1),...) of x and y, and z ⊕ s is defined similarly. Assume that Player I has a winning strategy σG in the game G. (The other case is similar.) Fix a real s coding σG. By the definition of the ordinal α(s) we may fix a countable Turing ideal I containing the real s and on which Aα(s) is not determined. Let σ be the strategy for Player I in the game GAα(s) that is obtained from σG by pretending that the real s was played alongside y by Player II in the game G, and ignoring the real z produced alongside x by Player I’s strategy σG. That is, for every finite sequence of movesy ¯ ∈ ωn we have n σ(¯y) =x ¯ ⇐⇒ σG(¯y, s  n) = (¯x, z¯) for somez ¯ ∈ ω .

This strategy σ can be computed from the real s coding σG, so it is in I . Therefore we may fix a real y ∈ I with σ ∗ y∈ / Aα(s). On the other hand, because σG is a winning strategy for player I in G there is a real z ∈ I such that σ ∗ y ∈ Aα(z⊕s). We will derive a contradiction by showing that α(z ⊕ s) = α(s). Every Turing ideal containing z ⊕ s also contains s, so α(z ⊕ s) ≥ α(s). But z ⊕ s ∈ I , so α(z ⊕ s) ≤ α(s) by the minimization in the definition of α. This is a contradiction.  Now it is a simple matter to prove determinacy of the envelope (indeed almost trivial, but we have isolated this part of the argument because it seems to make essential use of DCR whereas the lemma might be useful in a broader context.)

Proposition 2.11. Let Jκ(R) be an admissible level of L(R) that begins a Jκ(R) Jκ(R) gap and suppose that ∆1 is determined. Then Env(Σ1 ) is determined. Proof. By Lemma 2.10 we can take a real t such that every OD<κ set of reals is determined on every countable Turing ideal I containing t. Let A Jκ(R) be a set of reals in Env(Σ1 ) and suppose toward a contradiction that A is not determined. Then by a Skolem hull argument using DCR there is a countable Turing ideal I containing t and on which A is not determined. We can take a set of reals A0 ∈ OD<κ such that A ∩ I = A0 ∩ I , so the set 0 A is not determined on I either, which is a contradiction. 

Jκ(R) We can use the determinacy of Env(Σ1 ) together with the canonical <κ Jκ(R) well-ordering of OD to get a canonical well-ordering of Env(Σ1 ). A similar argument appears in the proof of Kunen’s theorem that, under AD, every (countably complete) measure on an ordinal κ < Θ is ordinal-definable (see [7, Thm. 28.20].)

Proposition 2.12. Let Jκ(R) be an admissible level of L(R) that begins a Jκ(R) Jκ(R) gap and suppose that ∆1 is determined. Then every pointset in Env(Σ1 ) is ordinal-definable. THE ENVELOPE UNDER LOCAL DETERMINACY 9

Jκ(R) <κ Proof. Let (Aα : α < κ) be a ∆1 enumeration of the OD sets of reals (the argument for arbitrary product spaces is similar.) For a set of reals Jκ(R) A ∈ Env(Σ1 ) and a real z, define

α(z, A) = least α such that (∀x ≤T z)(x ∈ A ⇐⇒ x ∈ Aα). Clearly α(z, A) only depends on the Turing degree of z. Let R denote the 0 0 <κ relation <, =, or >. For sets of reals A ,B ∈ OD the set {z ∈ R : α(z, A0) R α(z, B0)} is also in OD<κ, so using the characterization of the envelope in Lemma 2.6 it is easy to check that for sets of reals A, B ∈ Jκ(R) Jκ(R) Env(Σ1 ) the set {z ∈ R : α(z, A) R α(z, B)} is also in Env(Σ1 ). This set is Turing invariant and by Proposition 2.11 we have enough determinacy to use the proof of Martin’s cone theorem to show that it contains a Turing cone for exactly one of the three cases R ∈ {<, =, >}. Therefore we can Jκ(R) define a wellordering of Env(Σ1 ) by letting A ≤ B if and only if α(z, A) ≤ α(z, B) for a Turing cone of reals z. This is clearly a prewellordering. If A 6= B then α(z, A) 6= α(z, B) on a cone, so it is in fact a wellordering.  Next we will show that the envelope is closed under real quantification. We follow Martin’s argument as given in Jackson [3, Lem. 3.12] for arbitrary boldface inductive-like pointclasses under AD, checking that it works under a local determinacy hypothesis with our definition of the envelope. We remark that for reals (subsets A of the product space X = ω) the argument is essentially that of [13, Lem. 4.1]. In order to prove closure under real quantification, we first extract a lemma from the argument. To prove the lemma we seem to need AD in Jκ(R) Jκ(R) Jκ(R) (equivalently, ∆1 determinacy) and not just ∆1 determinacy. e Lemma 2.13. Let Jκ(R) be an admissible level of L(R) that begins a gap and suppose that Jκ(R) satisfies AD. For any product space X and any pointset A ⊂ X , the following statements are equivalent. Jκ(R) (1) A ∈ Env(Σ1 ). (2) For every countable set σ ⊂ X there is a cone of Turing degrees d such that for some for some pointset A0 ∈ OD<κ(d) we have A∩σ = A0 ∩ σ. Proof. The forward direction is trivial using the characterization of the en- velope in Lemma 2.6. For the converse, assume that condition (2) holds. We consider a real y as coding, in some simple way, a pair (σy, ay) where σy is a countable set of reals and ay ⊂ σy. Given a real y and a Turing degree d, we let Ay,d denote Jκ( ) 0 <κ the least (in the canonical ∆1(d) R well-ordering) pointset A ∈ OD (d) 0 such that ay = A ∩ σy, if it exists. Let C denote the set of reals y such that for a cone of Turing degrees d the pointset Ay,d is defined. Condition (2) says precisely that y ∈ C for all Jκ(R) reals y such that ay = A ∩ σy. We have C ∈ Σ1 by admissibility. 10 TREVOR M. WILSON

Given two reals y, z ∈ C, take a Turing degree d0 such that for all Tur- ing degrees d ≥T d0 the pointsets Ay,d and Az,d are both defined. By Jκ(R) ∆1 (y, z, d0) determinacy, which follows from the assumption of AD in Jκ(R), either Ay,d < Az,d for a cone of d, or Ay,d = Az,d for a cone of d, or Ay,d > Az,d for a cone of d. We can define a regular norm ϕ on C by ϕ(y) ≤ ϕ(z) if Ay,d ≤ Az,d for Jκ(R) a cone of Turing degrees d. It is easy to verify that ϕ is a Σ1 -norm. Therefore ran(ϕ) ≤ κ and a straightforward calculation using admissibility Jκ(R) shows that (the graph of) the norm ϕ is ∆1 . Jκ(R) Now we can see that A ∈ Env(Σ1 ). Let σ ⊂ X be countable. Take y ∈ R with σy = σ and ay = A ∩ σy, so that y ∈ C. Define the set Ay ⊂ X by x ∈ Ay ⇐⇒ x ∈ Ay,d for a cone of Turing degrees d. (Note that we have sufficient Turing determinacy that either x ∈ Ay,d or x∈ / Ay,d must hold for a cone of Turing degrees d.) The set Ay so defined does not depend on the particular choice of y but only on the ordinal α = ϕ(y), so we can call it Aα. We have Aα ∩ σ = A ∩ σ and the set Aα is ∆1-definable over Jκ(R) from the parameter α, so Aα ∈ <κ OD as desired.  Now it is a simple matter to prove closure under real quantification.

Proposition 2.14. Let Jκ(R) be an admissible level of L(R) that begins a Jκ(R) gap and suppose that Jκ(R) satisfies AD. Then Env(Σ1 ) is closed under real quantification. Proof. We verify the closure under ∃R—the case of ∀R will then follow using closure under complementation. Fix a product space X and consider a Jκ(R) pointset A ⊂ X × R with A ∈ Env(Σ1 ). We define the projection R Jκ(R) B ⊂ X by B = ∃ A. We will show that B ∈ Env(Σ1 ) by verifying that it satisfies condition (2) of Lemma 2.13. Let σ ⊂ X be a countable set. Fixing any sufficiently large Turing degree d, for all points x ∈ σ we have x ∈ B ⇐⇒ ∃y ≤T d ((x, y) ∈ A). Define the countable set ρ ⊂ X × R by 0 <κ 0 ρ = σ×{y ∈ R : y ≤T d}. Take A ∈ OD such that A∩ρ = A ∩ρ. Defining 0 0 0 0 <κ the set B by x ∈ B ⇐⇒ ∃y ≤T d ((x, y) ∈ A ), we have B ∈ OD (d) 0 and B ∩ σ = B ∩ σ as desired.  Next we show that the envelope produces a “gap in scales.” As usual this result is well-known under AD (see [3, Lem. 3.17 and remark following 3.18]) and we merely check that it works under our local determinacy hypothesis with our definition of the envelope. For the case where Jκ(R) is the least Jκ(R) admissible level of L(R), so that Σ1 is the pointclass of inductive sets, this result is equivalent to Martin’s [13, Cor. 4.4].

Lemma 2.15. Let Jκ(R) be an admissible level of L(R) that begins a gap Jκ(R) and suppose that Jκ(R) satisfies AD. Then there is a Π1 pointset that Jκ(R) admits no Env(Σ1 )-semiscale. e THE ENVELOPE UNDER LOCAL DETERMINACY 11

Jκ(R) Jκ(R) Proof. We claim that there is no Env(Σ1 )-semiscale on the Π1 pointset e<κ R = {(x, y) ∈ R × R : y∈ / OD (x)}. First note that the relation R is total: If there were a real x such that every <κ Jκ(R) real was in OD (x) then we would get a ∆1 (x) well-ordering of the reals, which would be in Jκ(R) by admissibility, contrary to our determinacy hypothesis. Jκ(R) Now suppose toward a contradiction that there is a Env(Σ1 )-semiscale on R. The usual construction of a uniformization from a semiscalee (by taking leftmost branches of the tree of the semiscale) can be phrased in terms of the sequence of prewellorderings rather than in terms of the semiscale itself or its tree. The envelope is projectively closed, so it can carry out this construction Jκ(R) and we get a uniformization of R in Env(Σ1 ). But then using closure under ∃R (or ∀R) we obtain a contradictione with Lemma 2.8.  Jκ(R) As we will see later, it is possible for a universal Π1 set to have a semi- Jκ(R) scale (and even a scale) each of whose prewellorderings is in Env(Σ1 ), even though the sequence of prewellorderings cannot be. Given such ae semis- cale, the sequence of prewellorderings itself must be Wadge-cofinal in the Jκ(R) pointclass Env(Σ1 ): e Proposition 2.16. Let Jκ(R) be an admissible level of L(R) that begins a ~ gap and suppose that Jκ(R) satisfies AD. Let ψ be a semiscale on a univer- Jκ(R) sal Π1 pointset such that for each i < ω the prewellordering ≤ψi is in Jκ(R) Jκ(R) Env(Σ1 ). Then every pointset in Env(Σ1 ) is continuously reducible to thee prewellordering ≤ψi for some i < ω.e

Jκ(R) Proof. If not, then by Wadge’s lemma there is a pointset in Env(Σ1 ) to which every prewellordering ≤ψi is continuously reducible, so the sequencee of prewellorderings is itself in the envelope, contradicting Lemma 2.15. Here we use the determinacy of the envelope together with its basic closure properties to see that the relevant Wadge games are determined.  3. Scales on co-inductive sets Jκ(R) The pointclass IND of inductive sets is equal to Σ1 where Jκ(R) is the least admissible level of L(R) (see [23, Cor. 2.3]) so the pointclass Env(IND) is defined by Definition 2.2. The boldface ambiguous part of IND is equal to HYP, the pointclass of boldface hyperprojective sets, and also to the class^ of pointsets in Jκ(R). (For this section we may as well take these characterizations as the definitions of IND and HYP.) If HYP determinacy holds, then Moschovakis^ [17] showed that IND has the scale^ property: every has an inductive scale. The com- plexity of scales on co-inductive sets, on the other hand, can be measured in terms of the pointclass Env(IND). ] 12 TREVOR M. WILSON

∗ As in [13] we define Σ0 to be the pointclass of sets A ∪ B where A is in- ∗ ductive and B is co-inductive. For n < ω let Πn be the class of complements ∗ ∗ ∗ of Σn sets and let Σn+1 be the class of projections of Πn sets. S ∗ Lemma 3.1. Suppose that HYP is determined. Then n<ω Σn ⊂ Env(IND). ^ Proof. This follows from the fact that Env(IND) contains IND and is closed under Boolean combinations and real quantification (Propositions 2.7 and 2.14.)  From this we easily obtain the following result. Theorem 3.2 (Martin [13, Lem. 4.1]). Suppose that HYP is determined. S ∗ ^ Let A ∈ n<ω Σn be a set of integers (a “real.”) Then A is an element of 4 CIND, the largest countable inductive set of reals.

S ∗ Jκ(R) Proof. If A ∈ n<ω Σn, then A ∈ Env(IND). In other words A ∈ Env(Σ1 ) where Jκ(R) is the least admissible level of L(R). Because A ⊂ ω this sim- <κ <κ Jκ(R) ply means that A ∈ OD . The class of OD subsets of ω is Σ1 and Jκ(R) admits a ∆1 surjection from κ. The restriction of this κ-sequence to any ordinal α < κ has countable range by AD in Jκ(R), so the entire sequence has countable range by admissibility.  We also get a “determinacy transfer” theorem that is an instance of [10, Thm. 8.1]. Theorem 3.3 (Kechris–Woodin). Suppose that HYP is determined. Then S ∗ ^ the pointclass n<ω Σn is determined. e S ∗ Proof. Relativizing the proof of Lemma 3.1 we have n<ω Σn ⊂ Env(IND), so the conclusion follows by the determinacy of the envelopee (Proposition] 2.11.)  S ∗ The determinacy of n<ω Σn is exactly the hypothesis used for Mos- chovakis’s construction ([18, Thm.e 2.2]) of scales on co-inductive sets. We can use these scales to describe the extent of the envelope. The result is well-known under AD; we check that it holds under our partial determinacy hypothesis with our definition of the envelope. Proposition 3.4. Suppose that HYP is determined. Then we have Env(IND) = S ∗ ^ ] n<ω Σn. e S ∗ Proof. We have n<ω Σn ⊂ Env(IND) by the relativization of Lemma 3.1, so by Moschovakis [18,e Thm. 2.2]] every co-inductive pointset admits a scale S ∗ whose prewellorderings are all in n<ω Σn, and therefore also in Env(IND). They must be Wadge-cofinal in Env(INDe ) by Proposition 2.16, so the] de- sired equality holds. ] 

4See [8] for the notion of “largest countable Γ set.” THE ENVELOPE UNDER LOCAL DETERMINACY 13

4. Scales in L(R) In this section we will see how the results of Section 3 can be generalized Jκ(R) from IND to the other inductive-like pointclasses Σ1 in L(R). Assume that κ begins a gap in L(R). Jκ(R) If Jκ(R) satisfies AD then by Steel [23, Thm. 2.1] every Σ1 set has a Jκ(R) Σ1 scale. If Jκ(R) is not admissible (equivalently, if it does not satisfy Jκ(R) R Jκ(R) Jκ(R) Σ0-collection) then by [23, Lem. 2.7] we have Π2 = ∀ Σ1 , Σ3 = R Jκ(R) e e e ∃ Π2 and so on, and if these pointclasses are determined then they all havee the scale property by the second periodicity theorem of Moschovakis [19, Thm. 6C.3]. Jκ(R) So we proceed to consider pointclasses of the form Σ1 where Jκ(R) is an admissible level of L(R). Such pointclasses are inductive-like. In this Jκ(R) case the complexity of scales on Π1 sets can be measured in terms of Jκ(R) Env(Σ1 ). Wee will need a bit more of the fine structure of L(R) as found, for example, Jβ (R) in [23, §1]. For an ordinal β we take the expression ρn = R, which is read th Jβ (R) as “the n projectum of Jβ(R) is R,” to mean that there is a Σn set of e Jβ (R) reals that is not in Jβ(R). This is equivalent to the existence of a Σn partial surjection R 99K Jβ(R). We remark that arbitrary parameterse from Jβ (R) Jβ(R), not only reals, are allowed in the definitions of Σn pointsets. Under certain circumstances the following lemma allowse us to replace quantification over Jβ(R) with quantification over R. The argument is stan- dard but we include it here for the reader’s convenience.

Jβ (R) Lemma 4.1. Let β be an ordinal and let n be an integer such that ρn = <ω R. Then for every sufficiently large finite set of ordinals s ∈ [ωβ] we have the following equalities of pointclasses.5

Jβ (R) R Jβ (R) Jβ (R)  Σn+1 (s) = ∃ Σn (s) ∧ Πn (s) Jβ (R) R Jβ (R) Σk+1 (s) = ∃ Πk (s), for all k > n. Proof. The right-to-left inclusions are trivial, so it remains to show the left- Jβ (R) Jβ (R) to-right inclusions. Because ρn = R, there is a Σn partial surjection Jβ (R) e <ω R 99K Jβ(R). There is always a Σ1 surjection [ωβ] × R → Jβ(R), so we may take the parameters defining our partial surjection R 99K Jβ(R) to be reals and ordinals. Let s ∈ [ωβ]<ω be a finite set of ordinals that is Jβ (R) sufficiently large that there is a Σn partial surjection R 99K Jβ(R) that is definable from s and a real z, saye by θ(−, −, s, z) where θ is a Σn formula. Then for any formula φ we have ∃X ∈ Jβ(R) φ(X, v) if and only if there are

5By Σ ∧ Π we denote the pointclass consisting of sets of the form A ∩ B with A ∈ Σ and B ∈ Π. 14 TREVOR M. WILSON reals x, z ∈ R such that

∃X ∈ Jβ(R) θ(x, X, s, z)&  ∀X ∈ Jβ(R) θ(x, X, s, z) =⇒ φ(X, v) . Letting φ be Πn or Πk with k > n, we obtain the desired result. 

Assuming now that Jκ(R) satisfies AD and is admissible and κ begins a Jκ(R) gap, we would like to describe Env(Σ1 ) in terms of the Jensen hierarchy. To do this we need to use a reflectione property introduced by Steel in [23].

Definition 4.2. The gap [κ, β] in L(R) is strong if Jκ(R) is admissible, and letting n < ω be least such that ρn(Jβ(R)) = R, every Σn-type realized in Jβ(R) is realized in Jα(R) for some α < β (and therefore for some α < κ by the definition of a gap.) Otherwise we say the gap is weak.

The first example of a strong gap is [κ, κ] where Jκ(R) is the first admis- Jκ(R) sible level of L(R); as mentioned previously, the pointclass Σ1 is equal to IND in this case. We allow β = ΘL(R) in the definition of a weak gap [κ, β].

Lemma 4.3. Suppose that Jκ(R) is an admissible level of L(R) satisfying AD and [κ, β∗] is a gap. Let β = β∗ if the gap is weak and β = β∗ + 1 if the <β Jκ(R) gap is strong. Then OD ⊂ Env(Σ1 ). Proof. Fix a product space X . If the gap [κ, β∗] is weak, let A ⊂ X be a pointset with A ∈ OD<β and let σ ⊂ X be a countable set. The property of being ordinal-definable over a level of the Jensen hierarchy is Σ1, so applying the definition of a gap to some real parameter coding the pair (σ, A ∩ σ) we get a pointset A0 ⊂ X <κ 0 Jκ(R) with A ∈ OD and A ∩ σ = A ∩ σ. Therefore A ∈ Env(Σ1 ) by the characterization of the envelope in Lemma 2.6. ∗ If the gap [κ, β∗] is strong, so that OD<β = ODβ , let n be the least integer β∗ such that ρn(Jβ∗ (R)) = R. Then by Lemma 4.1 the OD pointsets are generated via Boolean combinations and real quantification from pointsets Jβ∗ (R) that are Σn in ordinal parameters. (This is a generalization of the way ∗ the pointclasses Σi for i < ω are generated from IND.) The envelope is closed under Boolean combinations and real quantification, so it is enough to show that it contains the pointset A ⊂ X given by

x ∈ A ⇐⇒ Jβ∗ (R) |= θ[x, s] ∗ <ω where s ∈ [ωβ ] and θ is a Σn formula. Given a countable set σ ⊂ X , take a real z coding σ. By the definition of a strong gap the Σn-type of (z, s) in Jβ∗ (R) is realized in some level Jα(R) with α < κ. The real z is determined by its type, so there is a finite sequence <ω of ordinalss ¯ ∈ [ωα] such that the Σn-type of (z, s¯) in Jα(R) is equal to the 0 0 <κ Σn-type of (z, s) in Jβ∗ (R). Then we have A ∩ σ = A ∩ σ where A ∈ OD THE ENVELOPE UNDER LOCAL DETERMINACY 15

0 is the subset of X defined by x ∈ A ⇐⇒ Jα(R) |= θ[x, s¯]. This shows Jκ(R) that A ∈ Env(Σ1 ).  We obtain an immediate corollary regarding local ordinal-definability of reals. (The weak gap case is trivial because the property “the set of integers A is ordinal-definable over one of my levels” is Σ1, but we include it for simplicity.)

6 Theorem 4.4 (Martin ). Suppose that Jκ(R) is an admissible level of L(R) satisfying AD and [κ, β∗] is a gap. Let β = β∗ if the gap is weak and β = β∗ + 1 if the gap is strong. Let A ∈ OD<β be a set of integers (a “real.”) Then A ∈ OD<κ.

Jκ(R) Proof. By Lemma 4.3 we have A ∈ Env(Σ1 ). Because the space ω itself is a countable pointset this means A ∈ OD<κ by the characterization of the envelope in Lemma 2.6. 

For every ordinal β, every pointset A ∈ Jβ(R) is definable from param- eters over some level Jα(R) with α < β, and using a definable surjection <ω <β [ωα] × R → Jα(R) we see that A ∈ OD (x) for some real x. So by rel- ativizing Lemma 4.3 to arbitrary reals we immediately obtain the following determinacy transfer result. (The weak gap case is trivial because AD can be expressed as a Π1 statement, but we include it for simplicity.)

Theorem 4.5 (Kechris–Woodin [10]). Suppose that Jκ(R) is an admissible ∗ ∗ level of L(R) satisfying AD and [κ, β ] is a gap. Let β = β if the gap is ∗ weak and β = β + 1 if the gap is strong. Then Jβ(R) satisfies AD.

Jκ(R) Proof. Given a set of reals A ∈ Jβ(R), we have A ∈ Env(Σ1 ) by rela- tivizing Lemma 4.3, so A is determined by Proposition 2.11.e  This determinacy transfer allows us to use Steel’s construction of scales in Jκ(R) L(R) in [23] to construct scales on Π1 sets. We can then determine the Jκ(R) envelope of the pointclass Σ1 exactly. The proof from AD is well-known and we merely check that ite can be adapted to yield a proof from our local determinacy hypothesis using our definition of the envelope.

Proposition 4.6. Suppose that Jκ(R) is an admissible level of L(R) satis- fying AD and [κ, β∗] is a gap with β∗ < ΘL(R). Let β = β∗ if the gap is weak ∗ Jκ(R) and β = β + 1 if the gap is strong. Then Env(Σ1 ) = Jβ(R) ∩ ℘(R). e Jκ(R) Proof. We have Jβ(R) ∩ ℘(R) ⊂ Env(Σ1 ) by relativizing Lemma 4.3. It remains to show the reverse inclusion.e Because Jβ(R) |= AD, it follows from the scale construction in [23] (as noted in [21]) that every set of reals in

6This result is the natural adaptation of Martin [13, Lem. 4.1] to Steel’s notion of a strong gap. Steel [21, Thm. 1.14] attributes it to Martin. 16 TREVOR M. WILSON

7 Jβ(R) has a scale whose prewellorderings are all in Jβ(R). Note that κ < β (because every admissible gap of the form [κ, κ] is strong.) So in particular, Jκ(R) ~ a universal Π1 set has a scale ψ such that each prewellordering ≤ψi is in Jκ(R) Jβ(R) and hence also in Env(Σ1 ). By Proposition 2.16 the sequence of e Jκ(R) prewellorderings is Wadge-cofinal in the envelope, so Env(Σ1 ) ⊂ Jβ(R) as desired. e  5. Companions In order to generalize the definitions and results of Section 2 to arbitrary boldface inductive-like pointclasses it is helpful, though perhaps not strictly necessary, to use the notion of the companion of a pointclass introduced by Moschovakis in [16]. Recall that a structure M = (M; ∈, R~), where R~ = (R1,...,Rn) is a finite sequence of relations on M, is called admissible if the set M is transitive, nonempty, closed under pairing and union, and the structure M satisfies the ∆0-separation and ∆0-collection axiom schemas (for formulas mentioning R1,...,Rn.) Note that an admissible structure must in fact satisfy the ostensibly stronger axiom schemas of ∆1-separation and Σ1-collection. Definition 5.1 (Moschovakis [16, §9]). A companion of a boldface inductive- 8 like pointclass Γ is an admissible structure M = (M; ∈, R~), with R ∈ M, satisfying the twoe conditions M 9 Projectability: there is a ∆1 partial surjection R → M eM M Resolvability: there is a ∆1 sequence (Mα : α < Ord ) of subsets of M whose union is equale to M M and such that Γ is the pointclass of all Σ1 pointsets. e M M e In our notions of Σ1 (and ∆1 ) here we mean that any set in M is allowed as a parameter.e Companionse exist: Theorem 5.2 (Moschovakis [16, Thm. 9E.1]). Let Γ be a boldface inductive- like pointclass. Then Γ has a companion. e e For any companion M = (M; ∈, R~) of the pointclass Γ, the underlying set M consists of all sets that can be coded by well-foundede trees in ∆Γ. The pointsets in the set M are exactly the ∆Γ pointsets and Ord ∩ Meise equal to the prewellordering ordinal κ of Γ. Thee e set M can also be characterized as the smallest admissible set such thate ∆Γ ⊂ M. e e 7It is also possible to carry out the argument using results stated explicitly in [23]. To do L(R) this, let n be least such that ρn(Jβ (R)) = R (which exists because β < Θ , and is equal Jβ (R) to 1 in the strong gap case) and use the scale property for the pointclass Σn together with the fact that this pointclass is the collection of countable unions of setse in Jβ (R). 8 A Spector class on R in the terminology of [16]. 9 M By this we mean that the graph of the partial surjection is ∆1 , which implies that the M domain is Σ1 . e e THE ENVELOPE UNDER LOCAL DETERMINACY 17

Jκ(R) Remark 5.3. A companion of the pointclass Σ1 , where Jκ(R) is admis- sible and κ begins a gap in L(R), is given bye (Jκ(R); ∈, ∅). Resolvability Jκ(R) follows from the fact that the sequence of levels (Jκ(R): α < κ) is ∆1 . For projectability note that the fact that κ begins a gap implies that there Jκ(R) is a Σ1 partial surjection F : R 99K Jκ(R) (see [23, Lem. 1.11].) Then Jκ(R) we can define a ∆1 partial surjection G such that G(x) = a if and only if the least level of Jκ(R) satisfying the Σ1 fact a ∈ ran(F ) also satisfies the Σ1 fact F (x) = a. Remark 5.4. Assume that AD+ holds, there is a largest Suslin cardinal κ, and κ is a member of the Solovay sequence (θα : α < Ω). Define the pointclass Γ = S(κ) and let M = (M; ∈, R~) be a companion of Γ. Then although the underlyinge set M of the companion is definable, thee companion structure M itself is not definable, nor even ordinal-definable from any element of M. This example shows that the predicates in R~ may contribute essential Jκ(R) information to the companion structure, in contrast to the case Γ = Σ1 . e e When we are only interested in boldface Σ1 (and ∆1) definability over a companion, we may speak loosely of “the”e companione of Γ because of the following theorem. e Theorem 5.5 (Moschovakis [16, Thm. 9E.3]). Let Γ be a boldface inductive- like pointclass. If M = (M; ∈, R~) and M 0 = (M 0; ∈e, R~ 0) are companions of 0 M M 0 Γ, then M = M and moreover the Σ1 subsets of M are exactly the Σ1 subsetse of M 0. e e

M M By contrast, the notions of lightface Σ1 and ∆1 definability may depend on the choice of companion M for Γ. Next we define a type of companione for which projectability and resolv- ability are witnessed by distinguished predicates. This will give us well- behaved notions of lightface definability over the companion. We will also require the resolution sequence to have some nice properties that facilitate an analogy with the sequence of levels (Jα(R): α < κ) of Jκ(R). Definition 5.6. Let Γ be a boldface inductive-like pointclass and let κ be the prewellorderinge ordinal of Γ. We say that a companion M = (M; ∈,R0,R1,...) of Γ is good if thee predicate R0 is a partial surjection R → M and the predicatee R1 is a sequence (Mα : α < κ) satisfying S α<κ Mα = M along with the following additional properties. • Mα is transitive for all α < κ, • R ⊂ M0, and • Mα ⊂ Mβ whenever α < β < κ. Given a good companion M of Γ and an ordinal α < κ, the αth level of M th is the structure Mα = (Mα; ∈,Re0 ∩ Mα,R1 ∩ Mα,...) where Mα is the α set in the distinguished resolution sequence R1 of M . 18 TREVOR M. WILSON

M By definition every companion has a ∆1 partial surjection R → M and M a ∆1 resolution sequence (Mα : α < κe). Moreover, the sets Mα in this M 0 sequencee can be enlarged to get a ∆1 resolution sequence (Mα : α < κ) satisfying the “good” conditions above.e Then we can obtain a good compan- ion by expanding the companion by predicates for this partial surjection and M resolution sequence. Note that adding ∆1 relations as predicates to M pre- serves admissibility (we have ∆0-separatione and ∆0-collection with respect to the new predicates) and does not change the notion of Σ1-definability over M . e Note that if M is a good companion for Γ then using the distinguished M partial surjection R → M we see that every eΣ1 set (equivalently, every Γ M set) is Σ1 (x) for some real x. e e

6. The envelope in the general case In this section we generalize the definitions and results of Section 4 from Jκ(R) boldface inductive-like pointclasses Σ1 to arbitrary boldface inductive- e Jκ(R) like pointclasses Γ. Because Jκ(R) is a companion of Σ1 , a natural generalization of Definitione 2.2 is the following one. e Definition 6.1. Let Γ be a boldface inductive-like pointclass and let κ be the prewellordering ordinale of Γ. Let M denote the companion of Γ. For a e M e pointset A, we say A ∈ Env(Γ) if A ∈ (Aα : α < κ) for some ∆1 sequence of pointsets (Aα : α < κ). e e

Recall that the pointsets in M are exactly the ∆Γ pointsets. Moreover, M the definability condition (Aα : α < κ) ∈ ∆1 ise equivalent to the condi- tion that the sequences (Aα : α < κ) ande (¬Aα : α < κ) are both Γ in the codes relative to some/every Γ-norm onto κ. If AD holds then everye sequence of ∆Γ sets satisfies thise definability condition by Moschovakis’s Coding Lemma.e e10 Therefore, under the assumption of AD the pointclass 11 Env(Γ) is equal to Martin’s pointclass Λ(∆Γ, κ) (see Definition 1.4.) e e e Remark 6.2. We caution the reader that the notation “Env” and the term “envelope” are taken from Steel [22], where they were given a somewhat different definition. Jackson [4] proved that, in the intended context of AD, the pointclass Λ(∆Γ, κ) is equal to the pointclass that Steel calls Env(Γ) (which is a boldfacee e pointclass defined from a lightface inductive-like point- class Γ.) Consequently our pointclass Env(Γ) as defined in Definition 6.1 above is equal to both other pointclasses undere AD. The author does not

10Let U be a universal Γ set and use the coding lemma ([19, 7D.5]) to get a Γ choice set for the function α 7→ {(x,e y) ∈ R × R : Aα = Ux = ¬Uy}. e 11 The pointclass Λ(∆Γ, κ) is also simply equal to Λ(Γ, κ)—see [3]. However, we will not need to use this facte directly.e e THE ENVELOPE UNDER LOCAL DETERMINACY 19 know whether our pointclass Env(Γ) can be proved equal to Steel’s point- class Env(Γ) from only ZF + DCR +e Det(∆Γ). e Next we define some notions of ordinal-definability over a companion structure M that are analogous to the notions ODα and OD<κ from Defi- nition 2.3. Definition 6.3. Let Γ be a boldface inductive-like pointclass and let κ be the prewellordering ordinale of Γ. Let M be a good companion of Γ. Let A be a pointset, and let p be a parametere in M . For an ordinal α < κe, we say α,M • A ∈ OD (p) if A is first-order definable over the level Mα of M from p and ordinal parameters. • A ∈ OD<κ,M (p) if A ∈ ODα,M (p) for some α < κ. The definitions of ODα,M and OD<κ,M depend on the particular choice of good companion M for Γ, so they are mostly useful for lemmas where we temporarily fix a choice ofe good companion to do some complexity calcula- tion. The following characterization of OD<κ,M will also be useful. Lemma 6.4. Let Γ be a boldface inductive-like pointclass and let κ be the prewellordering ordinale of Γ. Let M be a good companion of Γ. Let A be a pointset. Then the followinge statements are equivalent: e • A ∈ OD<κ,M . M <ω • A ∈ ∆1 (α) for some finite sequence of ordinal parameters s ∈ κ . Proof. See Lemma 2.4.  The following result (analogous to Lemma 2.5) now follows easily. Lemma 6.5. Let Γ be a boldface inductive-like pointclass and let M be M a good companione of Γ. Then there is a ∆1 sequence enumerating all OD<κ,M pointsets. e The following characterization of the envelope (a generalization of Lemma 2.6) is immediate from the relativizations of Lemmas 6.4 and 6.5 to param- eters x ∈ R. Lemma 6.6. Let Γ be a boldface inductive-like pointclass and let κ be the prewellordering ordinale of Γ. Let M be a good companion of Γ. Let X be a product space. For everye pointset A ⊂ X the following statements are equivalent. • A ∈ Env(Γ). • There is ae real x such that for every countable set σ ⊂ X there is a pointset A0 ∈ OD<κ,M (x) with A ∩ σ = A0 ∩ σ. We may establish properties of Env(Γ) along the lines of Section 2, using <κ,M Jκ(R)e <κ Γ, M , and OD in place of Σ1 , Jκ(R), and OD respectively. In moste cases, we omit the proofs becausee they are completely analogous. 20 TREVOR M. WILSON

Proposition 6.7. Let Γ be an inductive-like pointclass. Then Env(Γ) con- tains Γ and is closed undere Boolean combinations, number quantification,e and continuouse reducibility. Proof. See Proposition 2.7.  Lemma 6.8. Let Γ be an inductive-like pointclass and let κ be the prewell- ordering ordinal ofe Γ. let M be a good companion of Γ. If A ⊂ R × ω <κ,M is a pointset whose sectionse satisfy Ax ∈/ OD (x) fore all reals x, then A/∈ Env(Γ). e Proof. See Lemma 2.8.  Next we establish some properties of Env(Γ) that follow from the assump- tion of ∆Γ determinacy. e e Lemma 6.9. Let Γ be a boldface inductive-like pointclass and suppose that ∆Γ is determined.e Let M be a good companion of Γ. Then there is a real e <κ,M e t ∈eR such that every OD set of reals is determined on every countable Turing ideal containing t.

Proof. See Lemma 2.10.  Proposition 6.10. Let Γ be a boldface inductive-like pointclass and suppose that ∆Γ is determined. Thene Env(Γ) is determined. e e Proof. eBy the relativization of Lemma 6.9, using the argument of Proposi- tion 2.11.  Proposition 6.11. Let Γ be a boldface inductive-like pointclass and suppose that ∆Γ is determined.e Then every pointset in Env(Γ) is ordinal-definable frome a pointset in Γ. e e Proof. Let M be a good companion of Γ. Let A ∈ Env(Γ), say A ∈ e eM (Aα : α < κ) where the sequence of pointsets (Aα : α < κ) is ∆1 (x). Then the argument of Proposition 2.12 shows that A is ordinal definable from M and x. Note that the companion M is definable from a Γ pointset by projectability. e  To show that the envelope is closed under real quantification, we will need the following lemma. Lemma 6.12. Let Γ be a boldface inductive-like pointclass and suppose that ∆Γ is determined.e Let M be a good companion of Γ. Let X be a product space.e e For every pointset A ⊂ X the following statementse are equivalent. • A ∈ Env(Γ). • There is ae real x such that for every countable set σ ⊂ X there is a cone of Turing degrees d such that for some pointset A0 ∈ OD<κ,M (x, d) we have A ∩ σ = A0 ∩ σ.

Proof. See Lemma 2.13.  THE ENVELOPE UNDER LOCAL DETERMINACY 21

Proposition 6.13. Let Γ be a boldface inductive-like pointclass. If ∆Γ is determined, then Env(Γ)eis closed under real quantification. e e e Proof. By Lemma 6.12, using the argument of Proposition 2.14.  Next we show that the envelope produces a “gap in scales.” As usual this result is well-known under AD and we merely check that it works under our local determinacy hypothesis with our definition of the envelope.

Lemma 6.14. Let Γ be a boldface inductive-like pointclass. If ∆Γ is deter- mined, then there ise a Γˇ pointset that admits no Env(Γ)-semiscale.e e e Proof. Fixing a good companion M of Γ the relation R = {(x, y) ∈ R × <κ,M M ˇ e R : y∈ / OD (x)} is Π1 , hence Γ, and admits no Env(Γ)-semiscale by Lemma 6.8 and the argument of Lemmae 2.15. e  Given such a semiscale, the sequence of prewellorderings must be Wadge- cofinal in the pointclass Env(Γ): e Proposition 6.15. Let Γ be an inductive-like pointclass and suppose that ~ ∆Γ is determined. Let ψ ebe a semiscale on a universal Γˇ pointset such that fore each i < ω the prewellordering ≤ψi is in Env(Γ). Thene every pointset in

Env(Γ) is continuously reducible to the prewellorderinge ≤ψi for some i < ω. e Proof. By Lemma 6.14, using the argument of Proposition 2.16. 

7. Putative semiscales on Γˇ sets e Let Γ be a boldface inductive-like pointclass and assume that ∆Γ is deter- mined.e In many important cases this determinacy hypothesis impliese e that Γ has the scale property; for example, in the case Γ = IND (Moschovakis e Jκ(R) e ] [17]) and in the more general case Γ = Σ1 where Jκ(R) is admissible and κ begins a gap in L(R) (Steel [23].)e Henceforthe we assume that Γ has the scale property. e In some cases, as discussed in Sections 3 and 4, we also get scales on Γˇ sets whose prewellorderings are Wadge-cofinal in Env(Γ). However in othere cases eL(R) ˇ we cannot. For example if V = L(R) and Γ = Σ1 then Γ sets admit no scales at all, as observed by Kechris and Solovaye e (see [14].)e More generally if AD+ holds and there is a largest Suslin cardinal κ, then the pointclass Γ = S(κ) is boldface inductive-like and Γˇ sets do not admit scales. e e Remark 7.1. Although the existence of a scale on the set of reals A is equiva- lent to the existence of a semiscale on A, this equivalence is not local: Given a semiscale on a Γˇ set A whose norms are Env(Γ)-norms, the scale obtained from leftmost branchese of the tree of the semiscalee is not known in general to have the property that its norms are Env(Γ)-norms (see [3, Rmk. 3.16].) In this section we focus on semiscales becausee they will suffice for the following 22 TREVOR M. WILSON applications. However we note that a similar but more involved construc- tion, involving ideas from Jackson [5], would give us scales instead (see [25, §4.1].) The following lemma will be used to verify that certain prewellorderings we construct on Γˇ sets are in Env(Γ). e e Lemma 7.2. Let Γ be a boldface inductive-like pointclass and let κ be the prewellordering ordinale of Γ. Let ~ϕ be a Γ-scale on a Γ set U and let T be the tree of ~ϕ, so T is a treee on ω × κ. Lete ≤ be a prewellorderinge of the Γˇ set W = R \ U. Suppose that for every countable set of reals σ there ise a nonempty finite sequence of ordinals s ∈ κ<ω \ {∅} such that  12 ∀x, y ∈ W ∩ σ x ≤ y ⇐⇒ rankTx (s) ≤ rankTy (s) . Then the prewellordering ≤ is in Env(Γ). e Proof. Let M be a good companion of Γ. Take a real parameter t such that M M U ∈ Σ1 (t) and ~ϕ is a Σ1 (t)-scale. Lete σ be a countable set of reals. To show that the prewellordering ≤ is in Env(Γ) using the characterization of the envelope given in Lemma 6.6, it sufficese to find an OD<κ,M (t) set W 0 and an OD<κ,M (t) prewellordering ≤0 of W 0 such that W ∩ σ = W 0 ∩ σ and ≤ ∩ (σ × σ) = ≤0 ∩ (σ × σ). By our hypothesis, we can take a nonempty finite sequence of ordinals s ∈ κ<ω \ {∅} such that  ∀x, y ∈ W ∩ σ x ≤ y ⇐⇒ rankTx (s) ≤ rankTy (s) .

Take an ordinal α < κ that is sufficiently large that s(0) ≤ α and ϕ0(x) ≤ α 0 for all reals x ∈ U ∩ σ, and define the set U = {x ∈ U : ϕ0(x) ≤ α}. We have U ∩ σ = U 0 ∩ σ. Restricting the norms of the scale ~ϕ to the set U 0 gives 0 ~0 0 a scale on U . The tree of this restricted scale ϕ = ~ϕ  U is equal to the subtree T 0 of T defined by 0 T = {((n0, . . . , ni−1), (ξ0, . . . , ξi−1)) ∈ T : ξ0 ≤ α}. 0 0 0 0 Define the set W = R \ U = R \ p[T ], which satisfies W ∩ σ = W ∩ σ. Any node ((n0, . . . , ni−1), (ξ0, . . . , ξi−1)) ∈ T such that (ξ0, . . . , ξi−1) ex- 0 tends s must satisfy ξ0 = s(0) ≤ α, so any such node is also in T . So for every real x (not just for every real x ∈ σ, although this would be enough 0 0 for our argument) we have s ∈ Tx ⇐⇒ s ∈ Tx, and either Tx and Tx are both ill-founded below s or else they are both well-founded below s and 0 0 rankTx (s) = rankTx (s). Therefore we have ≤ ∩ (σ × σ) = ≤ ∩ (σ × σ) where ≤0 is the prewellordering of W 0 defined by 0 0 0 x ≤ y ⇐⇒ rankTx (s) ≤ rankTy (s). It suffices to show that the set W 0 and its prewellordering ≤0 have the desired complexity, namely that they are both in OD<κ,M (t). Because the

12Here we say that the rank function takes the value zero for nodes not in the tree. THE ENVELOPE UNDER LOCAL DETERMINACY 23

M 0 0 M norm ϕ0 is a Σ1 (t)-norm we have U ,W ∈ ∆1 (t, α) and in particular we 0 0 ~0 0 M have U ,W ∈ M . Moreover the restricted scale ϕ on U is a ∆1 (t, α)-scale and we have ϕ~0 ∈ M . Because M is admissible it can compute the tree T 0 0 0 M of this scale; that is, we have T ∈ M and {T } ∈ ∆1 (t, α). In particular the ordinals appearing in T 0 are bounded in κ, even though the definition of T 0 required this only for the ordinals appearing in the first coordinate. Moreover M can compute rank functions, so the prewellordering ≤0 is in M <κ,M ∆1 (t, α, s). Therefore it is in OD (t) as desired.  We now turn to some elementary observations about semiscales that do not depend on our descriptive-set-theoretic context (with Γ an inductive-like pointclass, etc.) It will be convenient to make the followinge definition. Definition 7.3. Let W be a set of reals and let C be a countable collection of pointsets. We say that C forms a semiscale on W if, whenever {xk : k < ω} is a sequence of reals in W converging to a real x, and such that for every element ≤ of C that is a prewellordering of W the ≤-equivalence class of xk is eventually constant as k → ω, we have x ∈ W . The definition of “forms a semiscale on W ” is fairly loose in that the collection C may contain elements that are not prewellorderings of W . Also, although it is required to be countable, it is not required to come with a particular enumeration in ω. Nevertheless if C forms a semiscale on W then we do indeed obtain a ~ semiscale ψ on W by taking any enumeration (≤i : i < ω) of C and defining the norm ψi : W → Ord by letting ψi(x) be the rank of x in the prewellord- ering ≤i. If one enumeration of C yields a semiscale in this manner then so does any other enumeration. Remark 7.4. In the intended application of Definition 7.3 we will have W ∈ Γˇ and C ⊂ Env(Γ). In particular, we will be able to form a semiscale out ofe prewellorderingse that are shown to be in Env(Γ) using Lemma 7.2. e Let T be the tree of a Γ-scale on a universal Γ set. If ADR holds then by a theorem of Martin ([15,e Thm. 3.1]) T is weaklye homogeneous and a standard argument (see [3, Lem. 3.15]) gives a semiscale on the Γˇ set W = R \ p[T ] whose prewellorderings are all in Env(Γ). In order to weakene the hypothesis of Martin’s theorem from ADR to ADe + Env(Γ) 6= ℘(R), Woodin ([27]) defined a game where one player plays measures frome a given countable set of measures (considered as a putative homogeneity system.) Below we describe a game that is essentially equivalent to Woodin’s game except that this player instead plays prewellorderings from a given countable set of prewellorderings (considered as a putative semiscale.) The reason for modifying the game in this way is that in some situations, such as that of Section 9, the relevant measures may not exist.13

13Another approach, taken by the author in [25, Ch. 3], is to consider putative homogeneity systems consisting of partial measures rather than total measures. 24 TREVOR M. WILSON

Definition 7.5. Let C be a countable collection of pointsets and let T be a tree on ω × κ for some ordinal κ. Define the following game between the two players B (for “branch”) and S (for “semiscale”):

C B x(0), f(0) g(0) x(1), f(1) g(1) ··· (GT ) S ≤0 ≤1 Rules. For all n < ω:

• x(n) ∈ ω, f(n) ∈ κ, ≤n ∈ C, and g(n) ∈ Ord. • ((x(0), . . . , x(n)), (f(0), . . . , f(n))) ∈ T . •≤ n is a prewellordering of R \ p[T ]. • ((x(0), . . . , x(n)), (g(0), . . . , g(n))) ∈ Tψ~ where Tψ~ is the tree of the 14 putative semiscale ψ~ on R \ p[T ] that Player S is building. The first player to deviate from these rules loses, and if both players follow the rules for all ω moves then we say that Player B wins, so the game is a closed game for Player B. C The significance of the games GT comes from the following lemma. Lemma 7.6. Let C be a countable collection of pointsets and let T be a tree on ω × κ for some ordinal κ. Then Player B has a winning strategy in C the game GT if and only if C does not form a semiscale on the set of reals R \ p[T ].

Proof. Fix an enumeration (≤i : i < ω) of all of the pointsets in C that are prewellorderings of R \ p[T ] and let ψ~ denote the corresponding putative semiscale on R \ p[T ]. Now suppose that Player B has a winning strategy. By feeding the se- quence of prewellorderings (≤i : i < ω) into the strategy, we obtain a real x that is in the projection of T as witnessed by f ∈ κω but is also in the projection of the tree of the putative semiscale ψ~ as witnessed by g ∈ Ordω. Therefore ψ~ is not a semiscale on R \ p[T ]. For the converse, if the putative semiscale ψ~ on R\p[T ] is not a semiscale, ~ then this is witnessed by a convergent sequence of reals xk → x (mod ψ) ω such that x ∈ p[T ] and xk ∈/ p[T ] for all k < ω. Let f ∈ κ witness x ∈ p[T ]. C Then Player B has a winning strategy in GT : Regardless of the moves of Player S we play out the coordinates of x and f one-by-one as x(n) and f(n), and in response to a prewellordering ≤ played by Player S on turn n we let g(n) be the eventual ordinal rank, as k → ω, of the real xk in that prewellordering ≤.  C Note that the game GT is a closed game on a wellordered set, so by the Gale–Stewart theorem it is determined and we could rephrase the lemma as “Player S has a winning strategy if and only if C forms a semiscale.” More- over Player B’s moves are ordinals, so by the proof of the Gale–Stewart

14 More precisely, there is a real xn ∈ R \ p[T ] such that for all i ≤ n we have xn(i) = x(i) and g(i) is the rank of xn in the prewellordering ≤i. THE ENVELOPE UNDER LOCAL DETERMINACY 25 theorem, if there is a winning strategy for Player B then there is a canon- C ical winning strategy FT . The main point of considering these games is C that these strategies FT give us a canonical way to generate failures of the semiscale property in the cases where it fails.

8. Application to divergent models of AD+ We will use Proposition 6.10 (determinacy of the envelope) and Lemma 7.6 (criterion for the existence of semiscales,) to give a simple proof of the following theorem on divergent models of AD+. The original proof, which is unpublished, used a different argument involving Sacks forcing, similar to the proof of the derived model theorem (see [28].) We say that models M0 + and M1 of AD with R ∪ Ord ⊂ M0,M1 are divergent if neither M0 ∩ ℘(R) nor M1 ∩ ℘(R) contains the other. By Wadge’s Lemma, this implies that no model of determinacy contains both M0 and M1. + Theorem 8.1 (Woodin). If M0 and M1 are divergent models of AD with R ∪ Ord ⊂ M0,M1, then the model M = L(M0 ∩ M1 ∩ ℘(R)) satisfies AD+ + DC + “every set of reals is Suslin.”

Notice that M ∩ ℘(R) = M0 ∩ M1 ∩ ℘(R) because both M0 and M1 are closed under constructibility. Also, AD+ holds in M because it is downward absolute to any transitive model that contains all the reals. By a standard argument the pointclass M ∩℘(R) is closed under countable sequences in V (in the sense of any natural coding.) This is because by divergence and Wadge’s lemma, both M0 and M1 have surjections R → M ∩ ℘(R). A countable sequence from M ∩ ℘(R) is coded by a real relative to each surjection, and both models contain all the reals, so the sequence is in both models. Therefore cof(ΘM ) > ω, which implies that DC holds in M by Solovay [20]. To prove the theorem, it remains to show that M satisfies the statement “every set of reals is Suslin.” Under AD+ the set of Suslin cardinals is closed below Θ (see [11, Thm. 7.2]) so it suffices to show that M has no largest Suslin cardinal. Let κ be a Suslin cardinal of M. By the Coding Lemma the models M0, M1, and M have the same subsets of κ. Trees on ω × κ are essentially subsets of κ, so the models M0, M1, and M have the same κ-Suslin sets of reals. We define the pointclass Γ by e Γ = S(κ)M0 = S(κ)M1 = S(κ)M . e Recall that under AD, for any Suslin cardinal κ the pointclass S(κ) is R-parameterized, closed under ∃R and Wadge reducibility, and has the scale property. If Γ is not closed under ∀R then we can get scales beyond Γ (namely on ∀eRΓ) by Moschovakis’s second periodicity theorem, in whiche case the model eM has a Suslin cardinal greater than κ as desired. So we now assume that Γ is closed under ∀R, in which case Γ is a boldface inductive-like pointclass. Moreovere its prewellordering ordinale is κ itself. 26 TREVOR M. WILSON

Notice that Env(Γ)M0 and Env(Γ)M1 are both contained in Env(Γ)V be- cause membership ine Env(Γ) is absolutee between transitive models contain-e ing Γ. All three envelopes aree determined by Proposition 6.10 and are closed undere continuous reducibility and Boolean combinations, so by Wadge’s Lemma we may assume without loss of generality that Env(Γ)M0 ⊂ Env(Γ)M1 . (This observatione is the newe ingredient in the proof.) M It follows that Env(Γ) 0 6= ℘(R) ∩ M0 because otherwise M0 and M1 would not be divergent.e Therefore, working in the model M0 we may use the theorem stated below to get a semiscale ψ~ ∈ M0 on a universal Γˇ set whose prewellorderings are in the pointclass Env(Γ)M0 . e Assuming the existence of such a semiscale, it ise a simple matter to com- plete the proof. Because Env(Γ)M0 is contained in Env(Γ)M1 it is contained in the intersection model M, soe the prewellorderings ofeψ~ are all in M. By closure of M ∩ ℘(R) under countable sequences, the semiscale ψ~ itself is in M. So in M our universal Γˇ set is Suslin. It cannot be κ-Suslin in M because it is not in Γ, so M muste have a Suslin cardinal above κ as desired. e Theorem 8.2 (Woodin [27]). Assume AD. Let Γ be a boldface inductive- like pointclass. If Env(Γ) 6= ℘(R) then every Γˇ sete has a semiscale whose prewellorderings are in Env(e Γ). e e Remark 8.3. As previously mentioned, a similar result was first proved by Martin under the hypothesis of ADR. Jackson [5] showed that “semiscale” can be strengthened to “scale” in the conclusion of Theorem 8.2. For the sake of completeness, and also to illustrate ideas which will be used later in Section 9, we will give a proof of Theorem 8.2 using the games C GT of Section 7. Proof of Theorem 8.2. If Env(Γ) 6= ℘(R) then by Wadge’s lemma there is a surjection π : R → Env(Γ). Soe we have a fine, countably complete measure

U on ℘ω1 (Env(Γ)), namelye the pushforward of Martin’s cone measure on

℘ω1 (R). The existencee of such a measure is the only consequence of AD beyond ∆Γ determinacy that we will need for the proof. Let κebe the prewellordering ordinal of Γ and let T on ω × κ be the tree of a Γ-scale ~ϕ on a universal Γ set. It sufficese to show that there is a semiscale eψ~ on the universal Γˇ set Re\p[T ], each of whose prewellorderings is in Env(Γ). Assume toward ae contradiction that there is no such semiscale. C Then bye Lemma 7.6, Player B has a winning strategy in the game GT for every countable set C ⊂ Env(Γ). For such a set C we let F C denote the C canonical winning strategy fore Player B in the game GT . We define a sequence of prewellorderings (≤n : n < ω) of R \ p[T ], each of which is a member of Env(Γ), by recursion on n. We will obtain a contradiction by showing that fore U-almost every set C ∈ ℘ω1 (Env(Γ)) this e THE ENVELOPE UNDER LOCAL DETERMINACY 27

C sequence of prewellorderings is a legal play for Player S in the game GT that defeats Player B’s “winning” strategy F C. Let n < ω. Once we have defined the prewellorderings ≤0,..., ≤n−1, we have these prewellorderings as elements of C for U-almost every set C by C fineness, so we may regard them as moves for Player S in the game GT and the strategy F C gives us moves xC(0), f C(0), gC(0), . . . , xC(n), f C(n) for Player B in response. That is, B xC(0), f C(0) gC(0) ··· xC(n), f C(n) S ≤0 C is the corresponding partial run of the game GT according to the strategy F C. Define the nonempty finite sequence of ordinals C C C sn = f (0), . . . , f (n). Then we can define the prewellordering ≤n of R \ p[T ] by ∗ C C  y ≤n z ⇐⇒ ∀U C rankTy (sn) ≤ rankTz (sn) . Here (as elsewhere) we use the convention that assigns the rank zero to nodes that are not in the tree. This prewellordering ≤n is easily seen to be in Env(Γ) by Lemma 7.2 and the countable completeness of U.

Oncee we have our sequence of prewellorderings (≤n : n < ω), let ψ~ de- note the corresponding putative semiscale on the set R \ p[T ]. Because the measure U is fine and countably complete we have {≤n : n < ω} ⊂ C for U-almost every set C ∈ ℘ω1 (Env(Γ)). For such a set C, the sequence C (≤n : n < ω) is a legal play for Playere S in the game GT . Using countable completeness twice, we obtain a real x that is played as xC for almost all sets C. Momentarily considering any set C that is in all the measure-one sets considered above, we see that x ∈ p[T ] (as witnessed by the branch f C) and also that x is in the projection of the tree of ψ~ (as witnessed by the branch gC.) So x is a counterexample to the semiscale property for ψ~ in the sense that x ∈ p[T ] and we can find a sequence of reals ~ (xk : k < ω) with xk ∈/ p[T ] for all k < ω and with xk → x (mod ψ). Using countable completeness and the definition of the prewellorderings ≤n, we can take a set C in all the measure-one sets considered above and with the additional property that for every n < ω the ordinal rank of the C node sn in the tree Txk is eventually constant as k → ω. Let C h(n) = lim rankTx (s ). k→ω k C C For any n < ω we have sn ∈ Tx by the rules for Player B, so we have sn ∈ Txk for all sufficiently large k because xk → x. That is, eventually we are not in the trivial case where we have defined the rank to be zero because the node is not in the tree. Therefore, for sufficiently large k the ordinals h(n) C C and h(n + 1) are the ordinal ranks of the node sn and its successor sn+1 28 TREVOR M. WILSON in a well-founded tree, so h(n) > h(n + 1) and we get a strictly decreasing sequence of ordinals, a contradiction. 

9. Application to derived models If δ is a supercompact cardinal and G ⊂ Col(ω, <δ) is a V -generic fil- V [G] ter, then the symmetric submodel V (R ) of the generic extension V [G] satisfies “every tree on ω × Ord is weakly homogeneous” and therefore sat- isfies “every Suslin set of reals is co-Suslin” (Woodin; see [15, Thm. 3.2].) This implies that the derived model at δ satisfies “every Suslin set of reals is co-Suslin,” which, because of AD+, is equivalent to “every set of reals is Suslin.” If one wishes only to obtain a derived model satisfying “every set of reals is Suslin” it suffices to assume the weaker hypothesis that δ is a limit of Woodin cardinals and of <δ-strong cardinals (see [24] for a proof in the case of the “old” derived model.) In this section we weaken the hypothesis of supercompactness in a differ- ent direction, namely to the hypothesis of indestructible weak compactness. This is a “weakening” in the sense that indestructible weak compactness can be forced from a supercompact cardinal using the Laver preparation [12] (because it follows from indestructible supercompactness) and it is un- known whether this relative consistency result can be reversed, although this is conjectured in [1]. The least weakly compact cardinal can be indestruc- tibly weakly compact ([2, Thm. 3.11],) so indestructibly weakly compact cardinals have very weak reflection properties.

Theorem 9.1. Assume that ZFC holds, δ is a limit of Woodin cardinals, and δ is weakly compact after forcing with Col(δ, δ+). Then the derived model D(V, δ) satisfies AD+ + DC + “every set of reals is Suslin.”

Remark 9.2. In terms of relative consistency, Theorem 9.1 was anticipated by [6, Theorem 0.4], which uses inner model theory to give a a model of large cardinal axioms of slightly higher consistency strength than AD+ + DC + “every set of reals is Suslin” from the same degree of indestructible weak compactness. This inner model theoretic approach also has the advantage that it does not need the assumption that δ is a limit of Woodin cardinals, whereas Theorem 9.1 needs the Woodin cardinals in order to make use of the derived model theorem. (A hybrid “descriptive inner model theoretic” proof should be possible as well, using the core model induction to build the model of determinacy instead of using the derived model theorem.) However, any attempt to use inner model theory or descriptive inner 2 model theory to prove that every set of reals—or even every Π1 set of reals— is Suslin in the derived model D(V, δ) itself rather than in the derived model of a mouse or in a submodel of D(V, δ) would probably require a full solution to the mouse set conjecture. THE ENVELOPE UNDER LOCAL DETERMINACY 29

Before proving Theorem 9.1 we give some background on derived models. For more information, see [24]. Assume that δ is a limit of Woodin cardinals. Let G ⊂ Col(ω, <δ) be a V -generic filter and define the set of reals ∗ [ RG = R ∩ V [G  α]. α<δ ∗ V [G] In the context of the theorem we have RG = R because δ is inaccessible, ∗ but the notation RG is well-established for the general case so we will use it. We say that a tree T is δ-absolutely complemented if there is a tree T˜ such that p[T˜] = R \ p[T ] in every generic extension of V by a poset of size less than δ. Let uBδ denote the pointclass of δ-absolutely Baire sets of reals; that is, the sets of reals given by projections p[T ] for δ-absolutely complemented trees T . Because δ is a limit of Woodin cardinals this is equal to the pointclass Hom<δ of <δ-homogeneously Suslin sets of reals by work of Martin, Solovay, Steel, and Woodin. Absolutely complementing trees give us a canonical way to extend un- iversally Baire sets of reals to generic extensions (which agrees with the canonical extensions given by homogeneity systems.) Accordingly we define the pointclass ∗  ∗ HomG = p[T ] ∩ RG : ∃α < δ (T ∈ V [G  α]&

V [G  α] |= “T is δ-absolutely complementing”) . ∗ Note that HomG is the pointclass of sets that are both Suslin and co-Suslin ∗ in the symmetric submodel V (RG) of V [G]. Woodin’s derived model theorem in its most general form [26, Thm. 31] (see [28] for a proof) says that there is a transitive model D(V, δ, G), called the derived model of V at δ by G that is maximal under inclusion subject to the conditions + • D(V, δ, G) |= ZF + AD + V = L(℘(R)) and ∗ ∗ • L(RG) ⊂ D(V, δ, G) ⊂ V (RG). ∗ Moreover, Woodin showed that HomG ⊂ D(V, δ, G). The pointclass ∗ HomG consists exactly of the sets of reals that are both Suslin and co- Suslin in D(V, δ, G); one direction follows from a theorem of Steel (applied in all intermediate models V [G  α]) which says that the pointclass Hom<δ has the scale property. If δ is regular (and therefore inaccessible) then the ∗ pointclass HomG has uncountable Wadge cofinality, so the derived model D(V, δ, G) satisfies DC. Because the forcing Col(ω, <δ) is homogeneous, the theory of the derived model does not depend on G. Accordingly, we will drop G from the notation and speak of “the” derived model D(V, δ) when appropriate. Remark 9.3. The term “derived model” (or sometimes “old derived model”) ∗ ∗ has also been used to refer to the submodel L(R , Hom ) of D(V, δ). Because AD+ is downward absolute to transitive models containing all the reals, ∗ ∗ the “old” derived model theorem, which says that L(R , Hom ) satisfies 30 TREVOR M. WILSON

AD+, can be seen as a special case of the derived model theorem. The ∗ ∗ models L(R , Hom ) and D(V, δ) agree to the extent that in each model the pointclass of sets that are Suslin and co-Suslin is exactly Hom∗. However, it ∗ ∗ is possible to have L(R , Hom ) ( D(V, δ) because the model D(V, δ) may contain sets of reals that are not constructible from its Suslin co-Suslin sets. If the derived model D(V, δ) satisfies “every set of reals is Suslin,” then ∗ ∗ so does the submodel L(R , Hom ), and similarly for DC, so Theorem 9.1 ∗ ∗ implies the corresponding result for the model L(R , Hom ). Under AD+ the set of Suslin cardinals is closed below Θ, so to show that D(V, δ) satisfies “every set of reals is Suslin” it suffices to show that it has no largest Suslin cardinal. Let κ be a Suslin cardinal of D(V, δ). Then the pointclass Γ = S(κ)D(V,δ) e is R-parameterized, closed under ∃R and continuous reducibility, and has the scale property. If Γ is not closed under ∀R then we can get scales beyond Γ (namely on ∀RΓ)e by second periodicity, in which case the model M has a Susline cardinal greatere than κ as desired. So assume that Γ is closed under ∀R, in which case Γ is boldface inductive- like. Moreover itse prewellordering ordinal is κ itself. e Now assume, as in the hypothesis of Theorem 9.1, that our limit of Woodin cardinals δ is weakly compact after forcing with Col(δ, δ+). Forcing with + ∗ ∗ Col(δ, δ ) does not change R or Hom because it does not change Vδ. More- over, forcing with Col(δ, δ+)V over the model V [G] does not change Env(Γ). It does not shrink Env(Γ) because the membership of a set of reals in Env(eΓ) is absolute between transitivee models of set theory containing Γ and havinge the same reals. On the other hand, it does not enlarge Γ becausee the forcing is homogeneous and every set of reals in Env(Γ) is ordinal-definablee from a set in Γ by Proposition 6.11. e + + Thee cardinality of Env(Γ) in V [G] is at most δ = ω2 = c by Wadge’s Lemma because Env(Γ) ise a determined pointclass by Proposition 6.10. So passing from V to V Col(e δ,δ+) we may assume that δ is weakly compact and that Env(Γ) has cardinality δ = ω1 = c in V [G]. We now use the following lemma toe construct a semiscale on a universal Γˇ set. Lemma 9.4. Assume that ZFC holds, δ is weaklye compact, and G ⊂ Col(ω, <δ) ∗ is a V -generic filter. Let Γ ∈ V (RG) be a boldface inductive-like pointclass such that ∆Γ is determined.e If Env(Γ) has size δ = ω1 in V [G], then every e ˇ e ∗ pointset in Γ has a semiscale in V (RG) whose prewellorderings are all in Env(Γ). e e Assuming that the lemma holds, so that every Γˇ pointset has a semiscale ∗ in V (RG), we can easily complete the proof of the theorem.e (We remark that the fact that the prewellorderings are in Env(Γ) will not be needed, although it would be needed for an argument that usede the core model induction THE ENVELOPE UNDER LOCAL DETERMINACY 31 rather than the derived model theorem.) By the standard construction of the tree of a semiscale, every set of reals in Γ is co-Suslin as well as Suslin in ∗ ∗ V (R ). This means that Γ ⊂ Hom , or equivalentlye that every set of reals in Γ is Suslin and co-Susline in D(V, δ). But a universal Γˇ set of reals cannot be eκ-Suslin in D(V, δ), so D(V, δ) must have a Suslin cardinale above κ, as desired. It remains to prove the lemma.

Proof of Lemma 9.4. By Proposition 6.11 every set of reals in Env(Γ) is ∗ ordinal-definable from a set of reals in Γ, so we have Env(Γ) ∈ V (R ).e Take − <δ a transitive model M |= ZFC with Vδe⊂ M, M ⊂ M,e and |M| = δ. By ∗ our hypothesis on the size of Env(Γ) we may arrange that Env(Γ) ∈ M(R ) and also that M[G] |= Env(Γ) =eω1. e Becase δ is weakly compacte we can take an elementary embedding j : M → N with N transitive and crit(j) = δ. Now we can take a V [G]- generic filter H ⊂ Col(ω,

C be the corresponding partial run of the game Gˆ(T ) according to the strategy F . Define the finite sequence of ordinals n+1 sn = (f(0), . . . , f(n)) ∈ j(κ) ∗ and define the prewellordering ≤n of R \ U by

y ≤n z ⇐⇒ rankˆ(T )y (sn) ≤ rankˆ(T )z (sn). (We use the convention that assigns the rank zero to nodes that are not in the tree.) Using Lemma 7.2 and the elementary of ˆ, it is easy to check that the prewellordering ≤n is indeed in Env(Γ). Once we have the sequence of moves (≤en : n < ω), we may continue with ∗ the argument. Because Env(Γ) ∈ M(R ) the restriction ˆ  Env(Γ) depends V [H] only on the set R and note on the generic filter H. So by the homogeneitye of the forcing and the definability of the canonical winning strategy F , both the real x and the sequence of moves (≤n : n < ω) are in V [G]. Because the model M is closed under <δ-sequences in V , its generic extension M[G] is closed under countable sequences in V [G], and we have x ∈ M[G] and (≤n : n < ω) ∈ M[G]. ∗ Letting ψ~ be the putative semiscale on the set R \U corresponding to the ~ sequence of prewellorderings (≤n : n < ω), we have ψ ∈ M[G] as well. In the model N[H] the real x witnesses that ˆ(ψ~), which is the putative semiscale from the sequence of prewellorderings (ˆ(≤n): n < ω), is not a semiscale on ∗ the set ˆ(R \U). This is because x was played by the strategy F in response to this sequence of prewellorderings, and F is a winning strategy for Player B. Therefore by the elementarity of ˆ, in M[G] the real x witnesses that ψ~ is ∗ not a semiscale on the set R \ U, meaning that we have a sequence of reals ∗ ~ (xk : k < ω) with xk ∈ R \U for each k < ω and with xk → x (mod ψ), but x ∈ U. By the definition of ψ~ this means that for every n < ω the ordinal rank of the node sn in the tree ˆ(T )xk is eventually constant as k → ω, so we can define

h(n) = lim rankˆ(T ) (sn). k→ω xk

For any n, note that we have sn ∈ ˆ(T )x by the rules for Player B and we have xk → x, so we have sn ∈ ˆ(T )xk for all sufficiently large k. That is, eventually we are not in the trivial case where the rank is defined to be zero because the node is not in the tree. Therefore, for sufficiently large k the ordinals h(n) and h(n + 1) are the ordinal ranks of the node sn and its successor sn+1 in a well-founded tree, so h(n) > h(n + 1) and we get a strictly decreasing sequence of ordinals. This contradiction completes the proof of the lemma.  The reader can easily verify that the proofs of Lemma 9.4 and Theo- rem 9.1 respectively can be adapted to yield the following results. See [25, THE ENVELOPE UNDER LOCAL DETERMINACY 33

Prop. 3.8.1] for a proof using a more complicated method involving partial measures instead of prewellorderings.15 Lemma 9.5. Assume that ZFC holds, δ is a cardinal that is δ+-strongly ∗ compact, and G ⊂ Col(ω, <δ) is a V -generic filter. Let Γ ∈ V (RG) be a boldface inductive-like pointclass such that ∆Γ is determined.e Then every ˇ ∗ e e set in Γ has a semiscale in V (RG) whose prewellorderings are all in Env(Γ). e e Theorem 9.6. Assume that ZFC holds, δ is a limit of Woodin cardinals, and δ is δ+-strongly compact. Then the derived model D(V, δ) satisfies AD+ + DC + “every set of reals is Suslin.”

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