Long Games and σ -Projective Sets
aa r X i v : . [ m a t h . L O ] J a n LONG GAMES AND σ -PROJECTIVE SETS JUAN P. AGUILERA, SANDRA M ¨ULLER, AND PHILIPP SCHLICHT
Abstract.
We prove a number of results on the determinacy of σ -projective setsof reals, i.e., those belonging to the smallest pointclass containing the open setsand closed under complements, countable unions, and projections. We first provethe equivalence between σ -projective determinacy and the determinacy of certainclasses of games of variable length <ω (Theorem 2.4). We then give an elementaryproof of the determinacy of σ -projective sets from optimal large-cardinal hypotheses(Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of thedeterminacy of σ -projective games of a given countable length and of games withpayoff in the smallest σ -algebra containing the projective sets, from correspondingassumptions (Theorems 5.1 and 5.4). Introduction
Let ω ω denote the space of infinite sequences of natural numbers with the producttopology, i.e., the topology generated by basic (cl)open sets of the form O ( s ) = { x ∈ ω ω : x extends s } , where s ∈ ω <ω . As usual, we will refer to the elements of ω ω as reals . Given a subset A of ω ω , the payoff set , we consider the Gale-Stewart game G of length ω as follows:I x x . . . II x x . . . for x , x , . . . ∈ ω .Two players, I and II, alternate turns playing x , x , . . . ∈ ω to produce an element x = ( x , x , . . . ) of ω ω . Player I wins if and only if x ∈ A ; otherwise, Player II wins. Onecan likewise define longer games by considering subsets of ω α , where α is a countableordinal. If so, we will again regard ω α as a product of discrete spaces. A game is determined if one of players I and II has a winning strategy. A set A ⊂ ω ω is said to bedetermined if the corresponding game is.These games have been studied extensively; under suitable set-theoretic assumptions,one can prove various classes of them to be determined. One often studies the deter-minacy of pointclasses given in terms of definability (a general reference is Moschovakis[Mos09]). A pointclass central to this article is the following: Definition 1.1.
The pointclass of σ -projective sets is the smallest pointclass closedunder complements, countable unions, and projections.In this article, we consider the following classes of games, and their interplay:(1) games of fixed countable length α whose payoff is σ -projective;(2) games of variable length <α + ω whose payoff is a pointclass containing theclopen sets and contained in the σ -projective sets; Date : January 19, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Infinite Game, Determinacy, Inner Model Theory, Large Cardinal, LongGame, Mouse. Of course, one only considers unions of sets in the same space, as is usual. (3) games of countable length α with payoff in other σ -algebras.Neeman extensively studied long games and their connection to large cardinals in[Nee04]. These results are based on his earlier work in [Nee95] and [Nee02] on gamesof length ω , where he started connecting moves in long games with moves in iterationgames. He showed in [Nee04] for example that the determinacy of games with fixedcountable length and analytic payoff set follows from the existence of Woodin cardinals.Moreover, he also analyzed games of continuously coded length (see also [Nee05]) andgames of length up to a locally uncountable ordinal. In [Nee07] he even showed determi-nacy for open games of length ω , indeed for a larger class of games of length ω , fromlarge cardinals. That the determinacy of arbitrary games of length ω is inconsistent isdue to Mycielski and has been known for a long time (see [Myc64]).As for the converse, the determinacy of infinite games implies the existence of innermodels with large cardinals (cf. e.g., [Har78, KW10, Tra13, Uhl16, MSW20] and others). Summary of results . We begin in Section 2 by introducing a class of games of variablelength below ω . We call these games Γ -simple , where Γ is a pointclass. We show that σ -projective determinacy implies the determinacy of Γ-simple games of length ω whereΓ is the pointclass of all σ -projective sets. In Section 3 we introduce a class of gameswe call decoding games . These are used to show that σ -projective determinacy followsfrom simple clopen determinacy of length ω . The proof also shows directly that simple σ -projective determinacy of length ω follows from simple clopen determinacy of length ω In Section 4, we prove simple clopen determinacy of length ω (and thus σ -projectivedeterminacy of length ω ) from optimal large cardinal assumptions. The proof is level-by-level. An alternative, purely inner-model-theoretic proof of σ -projective determinacycan be found [Agu]; our proof, however, requires little inner model theory beyond thedefinition of the large-cardinal assumption. Roughly, it consists in repeatedly applying atheorem of Neeman [Nee04] to reduce a simple clopen game of length ω to an iterationgame on an extender model with many partial extenders. The difference here is thatplayers are allowed to drop gratuitously in the iteration game finitely many times totake advantage of the partial extenders in the model.Finally, in Section 5, we exhibit some additional applications of the proof in Section 4.Specifically we prove from (likely optimal) large cardinal assumptions that σ -projectivegames of length ω · θ are determined, where θ is a countable ordinal. We also prove,from a hypothesis slightly beyond projective determinacy, that games in the smallest σ -algebra containing the projective sets are determined.2. Simple games of length ω We begin by noting a result on the determinacy of games of length <ω : Theorem 2.1 (folklore) . The following are equivalent:(1) All projective games of length ω are determined.(2) All projective games of length ω · n are determined, for all n < ω .(3) All clopen games of length ω · n are determined, for all n < ω . Instead of providing a proof of Theorem 2.1, we refer the reader to the proof ofTheorem 2.4 below, an easy adaptation of which suffices (cf. Remark 2.5).Our first result is the analog of Theorem 2.1 for σ -projective games. Although theequivalence between the first two items remains true if one replaces “projective” with“ σ -projective,” the one between the last two does not. Instead, one needs to consider a Here and below, we identify the spaces ω ω , ( ω ω ) ω and ω ω × ω , as well as ω ω · n and ( ω ω ) n . In the product topology on ω ω · n . ONG GAMES AND σ -PROJECTIVE SETS 3 larger class of games that are still decided in less than ω rounds, in the sense that forany x ∈ ω ω there is n ∈ ω such that for all y ∈ ω ω , if y ↾ ω · n = x ↾ ω · n , then x is a winning run for Player I if and only if y is.Note that every game of length ω · n can be seen as a game of this form, for any n ∈ ω .For the definition below, we adapt the convention that if n ∈ ω , then a subset A of ω ω · n can be identified with { x ∈ ω ω : x ↾ ω · n ∈ A } . Definition 2.2.
Let Γ be a collection of subsets of ω ω (each A ∈ Γ identified witha subset of ω ω · n for some n ∈ ω as above). A game of length ω is Γ- simple if it isobtained as follows:(1) For every n ∈ ω , games that are decided after ω · n moves such that their payoffrestricted to sequences of length ω · n is in Γ are Γ-simple.(2) Let n ∈ ω and for each i ∈ ω let G i be a Γ-simple game. Then the game G obtained as follows is Γ-simple: Players I and II take turns playing naturalnumbers for ω · n moves, i.e., n rounds in games of length ω . Afterwards, PlayerI plays some i ∈ ω . Players I and II continue playing according to the rules of G i (keeping the first ω · n natural numbers they have already played, but not i ).(3) Let n ∈ ω and for each i ∈ ω let G i be a Γ-simple game. Then the game G obtained as follows is Γ-simple: Players I and II take turns playing naturalnumbers for ω · n moves, i.e., n rounds in games of length ω . Afterwards, PlayerII plays some i ∈ ω . Players I and II continue playing according to the rules of G i (keeping the first ω · n natural numbers they have already played, but not i ).We are mainly interested in games of length ω which are simple clopen , i.e., whichare Γ-simple for Γ = ∆ the collection of clopen sets. Let us start by noting that everysimple clopen game has in fact a payoff set which is clopen in ω ω × ω (this fact will notbe needed, but it justifies our terminology). Lemma 2.3.
Let G be a simple clopen game of length ω with payoff set B . Then B isclopen in ω ω × ω .Proof. We prove this by induction on the definition of simple clopen games. In the casethat G is a clopen game of fixed length ω · n it is clear that B is clopen. So supposethat we are given simple clopen games G i , i < ω with payoff sets B i . Moreover, supposefor notational simplicity that n = 1, i.e., the players play one round of length ω beforePlayer I plays i ∈ ω to decide which rules to follow. Inductively, we can assume thatevery B i is clopen in ω ω × ω . Let G be the game obtained by applying (2) in Definition2.2. For x ∈ ω ω × ω , write x ∗ for x ↾ ω ⌢ x ↾ [ ω + 1 , ω ) . Then x is a winning run for Player I in G iff x ∈ [ i ∈ ω (cid:8) y ∈ ω ω × ω : y ( ω ) = i ∧ y ∗ ∈ B i (cid:9) . Each B i is open, so the payoff set B of G is open. Additionally, x ∈ B if, and only if, x ∈ \ i ∈ ω (cid:8) y ∈ ω ω × ω : y ( ω ) = i ∨ y ∗ ∈ B i (cid:9) . Each B i is closed, so B is closed. Therefore, B is clopen.The argument for applying (3) in Definition 2.2 is analogous. (cid:3) The main fact about simple clopen games is that their determinacy is already equiv-alent to determinacy of Γ-simple games where Γ is the pointclass of all σ -projectivesets: JUAN P. AGUILERA, SANDRA M¨ULLER, AND PHILIPP SCHLICHT
Theorem 2.4.
The following are equivalent:(1) All σ -projective games of length ω are determined.(2) All simple σ -projective games of length ω are determined.(3) All simple clopen games of length ω are determined. The proof of σ -projective determinacy of length ω from simple clopen determinacy,i.e., (3) ⇒ (1), will take place in the next section (Proposition 3.1). For now, we contentourselves with showing that σ -projective determinacy of length ω implies simple clopendeterminacy of length ω , i.e., (1) ⇒ (3) (see Proposition 2.7), although the proof wegive easily adapts to show simple σ -projective determinacy of length ω from the samehypothesis, i.e., (1) ⇒ (2) (see Proposition 3.7). Note that (2) ⇒ (3) is obvious. It willbe convenient for the future to introduce the definition of the game rank of a simpleclopen game.To each simple clopen game G of length ω we associate a countable ordinal gr( G ),the game rank of G , by induction on the definition of simple clopen games. If G is agame of fixed length ω · n , then gr( G ) = n . If G is obtained from games G , G , . . . , andfrom an ordinal ω · n as in Definition 2.2, we letgr( G ) = sup { gr( G i ) + ω : i ∈ ω } + n. Remark . The proof of Theorem 2.4 is local. We leave the computation of the precisecomplexity bounds to the curious reader, but we mention that e.g., the proof shows theequivalence among(1) the determinacy of games of length ω which are Π n for some n ∈ ω ;(2) the determinacy of simple clopen games of length ω of rank <ω (3) the determinacy of simple games of rank <ω which are Π n for some n ∈ ω .Thus, Theorem 2.4 generalizes Theorem 2.1.The reason why we have chosen to define the game rank this way is that it willmake some arguments by induction easier later on. Let us consider some examples: ifgr( G ) = ω , then G is essentially a game in which an infinite collection of games, each ofbounded length, are given, and one of the players begins by deciding which one of themthey will play. If gr( G ) = ω + 1, then the game is similar, except that the player doesnot decide which game they will play until the first ω moves have been played. Moregenerally, a game has limit rank if and only if it begins with one player choosing oneamong a countably infinite collection of games that can be played. Remark . Let G be a simple clopen game of rank α and p be a partial play of G .Denote by G p the game that results from G after p has been played. Then G p is a simpleclopen game of rank ≤ α .Clearly, every simple clopen game has a countable rank. Now we turn to the proofof (1) ⇒ (3) in Theorem 2.4. Proposition 2.7.
Suppose that all σ -projective games of length ω are determined. Thenall simple clopen games of length ω are determined.Proof. Let us say that two games G and H are equivalent if the following hold:(1) Player I has a winning strategy in G if and only if she has one in H ; and(2) Player II has a winning strategy in G if and only if she has one in H .We prove the proposition by induction on the game rank of a simple clopen game G . In the case that the game rank is a successor ordinal, we additionally show that thegame is equivalent to a σ -projective game of length ω (this is clear in the limit case).Suppose that α is a limit ordinal and this has been shown for games of rank <α . Let α + n be the rank of G , where n ∈ ω . If n = 0, then the result follows easily: by thedefinition of game rank, the rules of G dictate that one player must begin by choosing ONG GAMES AND σ -PROJECTIVE SETS 5 one amongst an infinite sequence of games G i . If that player has a winning strategyin any one of them, then choosing that game will guarantee a win in G ; otherwise, theinduction hypothesis yields a winning strategy for the other player in each game G i andthus in G .If 0 < n , say, n = k + 1, then one argues as follows. Given a partial play p of G , wedenote by G p the game G after p has been played. Consider the following game, H :(1) Players I and II alternate ω many turns to produce a real number x .(2) Afterwards, the game ends. Player I wins if and only if ∃ x ∈ ω ω ∀ y ∈ ω ω ∃ x ∈ ω ω . . . ∀ y k ∈ ω ω Player I has a winning strategy in G p ,where p = h x, x ∗ y , . . . , x k ∗ y k i . Claim 1. H is equivalent to G . Granted the claim, it is easy to prove the proposition, for, letting p be as above, therules of G p dictate that one of the players must choose one amongst an infinite sequenceof games G i . Assume without loss of generality that this is Player I. By inductionhypothesis, the set of p such that there exists i so that Player I has a winning strategyin G i is σ -projective. Hence, the payoff set of H is σ -projective, as was to be shown.It remains to prove the claim. Thus, let 0 ≤ m ≤ k and consider the following game H m :(1) Players I and II alternate ω · ( m + 1) many turns to produce real numbers z , . . . , z m .(2) Afterwards, the game ends. Player I wins if and only if ∃ x m +1 ∀ y m +1 ∃ x m +2 . . . ∀ y k Player I has a winning strategy in the game G p ,where p = h z , . . . , z m , x m +1 ∗ y m +1 , . . . , x k ∗ y k i .Thus, H = H . By induction on q = k − m (i.e., by downward induction on m ), we showthat H m is equivalent to G . A simple modification of the argument shows that ( H m ) p is equivalent to G p , for each p ∈ ( ω ω ) l and each l ≤ m + 1; this is possible because G p is a simple clopen game of rank ≤ α + n . We will use the equivalence between ( H m ) p and G p as part of the induction hypothesis.We have shown (by the induction hypothesis for the proposition) that G is equivalentto H k . The same argument applied to G p shows that G p is equivalent to ( H k ) p for every p ∈ ( ω ω ) k +1 and thus that G p is determined. Moreover, Player I wins a run p ∈ ( ω ω ) m +1 of H m if and only if she has a winning strategy in ( H m +1 ) p and Player II wins a run p ∈ ( ω ω ) m +1 of H m if and only if Player I does not have a winning strategy in ( H m +1 ) p ;however, ( H m +1 ) p is determined for every p ∈ ( ω ω ) m +1 , as it is a σ -projective gameof length ω (this follows from an argument as right after the statement of Claim 1).Moreover, the induction hypothesis (for the claim) shows that a player has a winningstrategy in ( H m +1 ) p if and only if she has one in G p . This shows that a player has awinning strategy in H m if and only if she has one in G , as was to be shown. (cid:3) Decoding games
Our first goal in this section is to prove the following proposition, i.e., (3) ⇒ (1) inTheorem 2.4. Proposition 3.1.
Suppose simple clopen games of length ω are determined. Then σ -projective games of length ω are determined. In order to prove Proposition 3.1, we introduce a representation of σ -projective sets. Here, x ∗ y denotes the result of facing off the strategies coded by the reals x and y . JUAN P. AGUILERA, SANDRA M¨ULLER, AND PHILIPP SCHLICHT
Definition 3.2.
Fix an enumeration { A n +1 : n ∈ ω } of all basic open and basic closed sets in each ( ω ω ) k , for 1 ≤ k < ω . Suppose A ⊂ ( ω ω ) k is σ -projective. A σ -projective code [ A ] of A is defined inductively as follows:(1) If A is basic open, then [ A ] = h n i , where A = A n .(2) If A is basic closed, then [ A ] = h n i , where A = A n .(3) If A = S i A i , then [ A ] = h [ A ] , [ A ] , . . . i .(4) If A = T i A i , then [ A ] = h , [ A ] , [ A ] , . . . i .(5) If A = ( ω ω ) k \ B for some k ∈ ω , then [ A ] = h , [ B ] i .(6) If A = p [ B ], then [ A ] = h , [ B ] i .(7) If A = u [ B ], then [ A ] = h , [ B ] i .Here, u [ B ] denotes the dual of the projection, u [ B ] = { x ∈ ω ω : ∀ y ( x, y ) ∈ B } . A σ -projective code of a given set is not unique. In fact, Lemma 3.3.
Let A be σ -projective.(1) A has a σ -projective code in which no complements and no basic open sets ap-pear.(2) A has a σ -projective code in which no complements and no basic closed setsappear.Proof. Given a σ -projective code for A , one obtains, by a simple application of deMorgan’s laws, a σ -projective code in which complements are only applied to basic opensets or to basic closed sets. Since every basic open set is clopen, basic open sets can bereplaced by a countable intersection of basic closed sets in the projective code; similarly,basic closed sets can be replaced by a countable union of basic open sets. (cid:3) Definition 3.4.
Suppose A ⊂ ( ω ω ) n for some 1 ≤ n < ω is σ -projective and fix acode [ A ] for A . The decoding game for A (with respect to [ A ]) is a game of length ω according to the following rules: In the first n rounds, which we will call the preparation ,Players I and II start by alternating turns playing ω · n natural numbers to obtain reals x , x , . . . , x n ∈ ω ω . Afterwards, they proceed according to the σ -projective code [ A ] of A via the following recursive definition:(1) If A is basic open, say A = A k , then the game is over, i.e., further moves arenot relevant. Player I wins if and only if ( x , x , . . . , x n ) ∈ A k .(2) If A is basic closed, say A = A k , then the game is over, i.e., further moves arenot relevant. Player I wins if and only if ( x , x , . . . , x n ) ∈ A k .(3) If A = S i A i , so that [ A ] = h [ A ] , [ A ] , . . . i , then Player I plays some k ∈ ω . Thegame continues from the current play (without the last move, k ) with the rulesof the decoding game with respect to [ A k ].(4) If A = T i A i , so that [ A ] = h , [ A ] , [ A ] , . . . i , then Player II plays some k ∈ ω .The game continues from the current play (without the last move, k ) with therules of the decoding game with respect to [ A k ].(5) If A = ( ω ω ) k \ B for some k ∈ ω , so that [ A ] = h , [ B ] i , then the game continuesfrom the current play with the rules of the decoding game with respect to [ B ],except that the roles of Players I and II are reversed.(6) If A = p [ B ], so that [ A ] = h , [ B ] i , then Player I plays some y ∈ ω ω in ω movesof the game, where the moves of Player II are not relevant. The game continuesfrom the current play, together with y , using the rules of the decoding gamewith respect to [ B ]. ONG GAMES AND σ -PROJECTIVE SETS 7 (7) If A = u [ B ], so that [ A ] = h , [ B ] i , then Player II plays some y ∈ ω ω in ω movesif the game, where the moves of Player I are not relevant. The game continuesfrom the current play, together with y , using the rules of the decoding gamewith respect to [ B ].Clause (5) of the preceding definition will not be used below, but we have defined itin the natural way nonetheless. Lemma 3.5.
Let A ⊂ ( ω ω ) n for some ≤ n < ω be σ -projective and fix a code [ A ] for A . A player has a winning strategy in the game with payoff set A if and only if theplayer has a winning strategy in the decoding game for A with respect to [ A ] .Proof. Assume first that Player I has a winning strategy σ in the game G A with winningset A . Then the following describes a winning strategy for Player I in the decodinggame for A with respect to [ A ]. In the first n rounds of the game, Player I follows thestrategy σ . Since σ is a winning strategy in G A , the players produce a sequence of reals( x , . . . , x n ) ∈ A . In the following rounds, Player I follows the rules of the decodinggame according to [ A ], playing witnesses to the fact that ( x , . . . , x n ) ∈ A . This meansthat, e.g., if [ A ] = h [ A ] , [ A ] , . . . i , i.e., A = S i A i , then Player I plays k ∈ ω such that( x , . . . , x n ) ∈ A k . The strategy for the other cases is defined similarly. This yields awinning strategy for Player I in the decoding game for A with respect to [ A ].Now assume that Player I has a payoff strategy σ in the decoding game for A withrespect to [ A ]. Then the restriction of σ to the first n rounds of the game is a winningstrategy for Player I in the game G A . Similarly for Player II. (cid:3) Lemma 3.6.
Let A ⊂ ( ω ω ) k be σ -projective, for some ≤ k < ω . Then, for every σ -projective code [ A ] for A in which no complements appear, the decoding game givenby [ A ] is simple clopen.Proof. Choose a code [ A ] for A in which no complements appear. We will show byinduction on the definition of simple clopen games, that the game obtained as in (6) or(7) in Definition 3.4 is simple clopen again. This will finish the proof as games obtainedby clauses (1)-(4) in Definition 3.4 are clearly simple clopen, by the definition of simpleclopen games.Let us introduce some notation for this proof. If x ∈ ω ω , we write x I for the sequenceof digits of x in even positions, and x II for the sequence of digits of x in odd positions.Thus, if x results from a run of a Gale-Stewart game, then x I is the sequence of movesof Player I and x II that of Player II.So let G be a simple clopen game with payoff set B ⊆ ( ω ω ) ω and considerthe game G P,k which is defined as follows. Players I and II start by alternatingturns playing ω · k natural numbers to define reals x , . . . , x k ∈ ω ω . Then play-ers I and II alternate moves to play some real x k +1 . Finally, Players I and II al-ternate again to produce reals z , z , . . . so that ( x , . . . , x k , x Ik +1 , z , z , . . . ) ∈ ( ω ω ) ω and we say that ( x , . . . , x k , x k +1 , z , z , . . . ) is a winning run for Player I in G P,k iff( x , . . . , x k , x I k +1 , z , z , . . . ) ∈ B . That means if B ∗ denotes the payoff set of G P,k , wehave ( x , . . . , x k , x k +1 , z , z , . . . ) ∈ B ∗ iff ( x , . . . , x k , x I k +1 , z , z , . . . ) ∈ B . We aim toshow that G P,k is simple clopen again.Suppose first that G is a clopen game of some fixed length ω · n , i.e., we consider G as a game of length ω but only the first ω · n moves are relevant for the payoff set.Fix some k ∈ ω . In particular, B is a clopen set in ω ω × ω . We can naturally write B = B × B × B for clopen sets B ⊆ ( ω ω ) k , B ⊆ ω ω and B ⊆ ( ω ω ) ω . By definition,the payoff set for the game G P,k is B × ( B ) I × B , where( B ) I = { x ∈ ω ω : x I ∈ B } , JUAN P. AGUILERA, SANDRA M¨ULLER, AND PHILIPP SCHLICHT which is clopen. Moreover, in G P,k again only the first ω · n moves are relevant, so G P,k is a clopen game of fixed length ω · n and in particular simple clopen.Now suppose that G is a simple clopen game obtained by condition (2) in Definition2.2, i.e., we are given simple clopen games G i , i < ω , and after Player I and II take turnsproducing x , . . . , x k , x k +1 , Player I plays some natural number i and the players con-tinue according to the rules of G i (keeping the moves which produced x , . . . , x k , x k +1 ).That means for every i < ω , some sequence ( x , . . . , x k , x k +1 , i, z , z , . . . ) is a winningrun for Player I in G iff ( x , . . . , x k , x k +1 , z , z , . . . ) is a winning run for Player I in G i .We can assume inductively that the games G P,ki for i < ω (obtained as above) are sim-ple clopen and we aim to show that G P,ki is simple clopen. Consider the simple clopengame G ∗ which is obtained by applying (2) in Definition 2.2 to the games G P,ki and thenatural number k + 1. Suppose Players I and II produce ( x , . . . , x k , x k +1 , i, z , z , . . . )in a run of G ∗ . Then( x , . . . , x k ,x k +1 , i, z , z , . . . ) is a winning run for Player I in G ∗ iff ( x , . . . , x k ,x k +1 , z , z , . . . ) is a winning run for Player I in G P,ki iff ( x , . . . , x k ,x I k +1 , z , z , . . . ) is a winning run for Player I in G i iff ( x , . . . , x k ,x I k +1 , i, z , z , . . . ) is a winning run for Player I in G iff ( x , . . . , x k ,x k +1 , i, z , z , . . . ) is a winning run for Player I in G P,k , where the first equivalence holds by definition of G ∗ , the second equivalence holds byinductive hypothesis, the third equivalence by choice of G , and the fourth equivalenceby definition of G P,k . Hence G P,k and G ∗ are equal and G P,k is a simple clopen game,as desired.The argument for simple clopen games obtained by condition (3) in Definition 2.2 isanalogous. (cid:3)
With Lemmata 3.5 and 3.6, Proposition 3.1 is proved. To finish the proof of Theorem2.4 one needs to show the following proposition. This is (3) ⇒ (2) and (1) ⇒ (2) inTheorem 2.4. Proposition 3.7.
Suppose either that simple clopen games of length ω are determinedor that σ -projective games of length ω are determined. Then simple σ -projective gamesof length ω are determined. Proposition 3.7 can be proved directly either by the method of the proof of Proposi-tion 3.1, or by that of Proposition 2.7. In the second case, one need only carry out astraightforward adaptation. In the first case, a simple σ -projective game G is reducedto the simple clopen game in which two players play the game G , producing a sequence x ∈ ( ω ω ) n for some n ∈ ω such that there is a σ -projective set A ⊂ ( ω ω ) n with theproperty that Player I wins G iff x ∈ A , no matter how the players continue playingthe rest of G . After this, the players play the decoding game for A to determine whothe winner is in a clopen way.We close this section with a useful characterization of the σ -projective sets, althoughit will not be used. Proposition 3.8 (Folklore) . A set A ⊂ R is σ -projective if and only if it belongs to L ω ( R ) .Proof. Clearly, L ω ( R ) is closed under countable sequences and P ( R ) ∩ L ω ( R ) is closedunder projections.Conversely, by induction on α < ω , one sees that L α ( R ) and the satisfaction relationfor L α ( R ) are coded by σ -projective sets of reals: ONG GAMES AND σ -PROJECTIVE SETS 9 (1) L ( R ) = R = V ω +1 is coded by itself. The satisfaction relation S for V ω +1 isgiven by S ( φ, ~a ) ↔ V ω +1 | = φ ( ~a ) , for φ a formula in the language of set theory and ~a a finite sequence of reals.Since every formula in the language of set theory is in the class Σ n for some n , S belongs to the pointclass S n<ω Σ n (i.e., the pointclass of countable unionsof projective sets) and is thus σ -projective.(2) Suppose that a σ -projective code C α for L α ( R ) has been defined and that a σ -projective satisfaction predicate S α for L α ( R ) relative to C α has been defined.Suppose that ~a ∈ ( C α ) n and φ = ∃ y ∀ y , . . . Qy m φ ( x , . . . , x n , y , . . . , y m )(where Q is a quantifier and φ is ∆ ) is a Σ m -formula with n free variables.Then, letting C φ,~aα = (cid:8) x ∈ R : ∃ y ∈ C α ∀ y ∈ C α . . . Qy m ∈ C α S α ( φ , ~a, y , . . . , y m ) (cid:9) , a σ -projective code C α +1 for L α +1 ( R ) can be defined by the disjointed union C α +1 = C α ˙ ∪ { ˆ C φ,~aα : φ is a formula and ~a ∈ ( C α ) n } , where ˆ C φ,~aα is a real number coding the set C φ,~aα . A satisfaction relation S α +1 can be defined from this: for atomic formulae, we set S α +1 ( · ∈ · , x, y ) ↔ (cid:0) x, y ∈ C α ∧ S α ( · ∈ · , x, y ) (cid:1) ∨ (cid:0) x ∈ C α ∧ ∃ φ ∃ ~a ⊂ ( C α ) lth( a ) y = ˆ C φ,~aα ∧ S α ( φ, ~a, x ) (cid:1) . For other formulae, this is done as above, so the satisfaction relation S α +1 isseen to belong to the pointclass S n<ω Σ n ( C α +1 , S α ) and is thus by using theinductive hypothesis σ -projective.(3) Suppose that a σ -projective code C α for L α ( R ) has been defined and that a σ -projective satisfaction predicate S α for L α ( R ) has been defined for every α < λ ,where λ is a countable limit ordinal. Then C λ can be defined as the disjoint(countable) union of C α +1 \ C α , for α < λ . Thus C λ is σ -projective. Thesatisfaction relation S λ can be defined as above.This completes the proof. (cid:3) Determinacy from large cardinals
In this section, we prove level-by-level that the existence of certain iterable innermodels with Woodin cardinals implies simple clopen determinacy of length ω , whichin turn implies σ -projective determinacy of length ω . A different proof of this latterresult can be found in [Agu], but the proof we give here has the advantage that itrequires almost no inner-model-theoretic background. Another advantage is that iteasily generalizes to yield further results (cf. the next section).The idea of this proof is to enhance a simple clopen game of length ω by presentingeach player with a fine-structural model that can be manipulated to obtain informationabout the game. This method of proof is due to Neeman [Nee04]; the difference inour context is that the models considered will contain many partial measures and, inaddition to taking iterated ultrapowers, we will allow the players to remove end-segmentsof their models during the game so as to make the measures total and obtain newinformation. Although this determinacy proof could have been framed directly in termsof (the decoding game for) σ -projective sets, it seems somewhat more natural to considersimple clopen games of length ω instead and proceed by induction on the game rank. E.g., it could be the tuple ( φ, ~a, α ). Before we start with the proof, let us recall the relevant inner model theoretic notionswe will need. In what follows, we will work with premice and formulae in the languageof relativized premice L pm = { ˙ ∈ , ˙ E, ˙ F , ˙ x } , where ˙ E is a predicate for a sequence ofextenders, ˙ F is a predicate for an extender, and ˙ x is a predicate for a real number overwhich we construct the premouse.For some real x , a potential x -premouse is a model M = ( J ~Eη , ∈ , ~E ↾ η, E η , x ) , where ~E is a fine extender sequence and η an ordinal or η = Ord (see Section 2 in [Ste10] fordetails). We say that such a potential x -premouse M is active iff E η = ∅ . If ν ≤ η ,we write M | ν = ( J ~Eν , ∈ , ~E ↾ ν, E ν , x ) for the corresponding initial segment of M . Apotential x -premouse M is called an x -premouse if every proper initial segment of M is ω -sound. If it does not lead to confusion, we will sometimes drop the x and just call M apremouse. Informally, an x -mouse is an iterable x -premouse. We will avoid this term asthe notion of iterability is ambiguous, but ω -iterability suffices for all our arguments. Here we say that an x -premouse M is ω -iterable if it is iterable for countable stacks ofnormal trees of length < ω (see Section 4 . smallness classes : Definition 4.1.
Let M be an ω -iterable premouse and δ ∈ M ∩ Ord.(1) We say M is of class S above δ if M is a proper class or active.(2) We say M is of class S α +1 above δ if there is some ordinal δ > δ and some N E M with δ < N ∩ Ord such that N is of class S α above δ and δ is Woodinin N .(3) Let λ be a countable limit ordinal. We say M is of class S λ above δ if λ < ω M and M is of class S α above δ for all α < λ .Moreover, we say M is of class S α if it is of class S α above 0. Example 4.2.
Let n ≥ M is called n -small if for ev-ery κ which is a critical point of an extender on the sequence of extenders of M , M | κ “there are n Woodin cardinals”. Moreover, we say M is 0-small if M is aninitial segment of L . Let M ♯n denote the unique countable, sound, ω -iterable premousewhich is not n -small, but all of whose proper initial segments are n -small, if it exists andis unique. If M ♯n exists and is unique for all n ∈ ω , then – in particular – every x ∈ ω ω has a sharp. Let N ♯ω denote the smallest active premouse extending [ n ∈ ω M ♯n . Then N ♯ω is of class S ω . If there is a premouse of class S ω +1 , then it contains N ♯ω andin fact N ♯ω is countable in it.Note that if M is a premouse of class S α , then M is aware of this, since α is countablein M . If M is of class S α and not a proper class, then M is active and we can obtaina proper class model by iterating the active extender of M and its images out of theuniverse. By convention, we shall say that this proper class model is also of class S α .We recall the definition of the game rank of a simple clopen game. If G is a game offixed length ω · n , then gr( G ) = n . If G is obtained from games G , G , . . . , and from anordinal ω · n as in Definition 2.2, we letgr( G ) = sup { gr( G i ) + ω : i ∈ ω } + n. Remark . For every simple game G and initial play p of G , let G p denote the restof the game G after p has been played. Let p , p , . . . be a sequence of initial plays of G starting with the empty sequence p = ∅ such that p i +1 end-extends p i , and either See Definition 2 . We will only need weak iterability , in the sense of [Nee04].
ONG GAMES AND σ -PROJECTIVE SETS 11 gr( G p i ) is a successor ordinal with gr( G p i ) = gr( G p i +1 ) + 1 or gr( G p i ) is a limit ordinal.Then this sequence has to be of finite length k + 1 and we can choose k large enoughsuch that gr( G p k ) = 1. Theorem 4.4.
Suppose that γ < ω and for every y ∈ ω ω there is an x ≥ T y suchthat there is a proper class x -premouse of class S γ which is a model of ZFC . Then everysimple clopen game G such that gr( G ) ≤ γ is determined. In the proof, we are going to use premice of class S γ to apply the following theoremof Neeman multiple times. Theorem 4.5 (Neeman, Theorem 2A.2 in [Nee04]) . There are binary formulae φ I and φ II in the language of set theory such that the following hold for any transitive, weaklyiterable premouse M which is a model of ZFC :Let δ < ω be a Woodin cardinal in M and let ˙ A ∈ M and ˙ B ∈ M be Col( ω, δ ) -namesfor a subset of ω ω . Then,(1) If M | = φ I [ δ, ˙ A ] , there is a strategy σ for Player I in a game of length ω such thatwhenever x is a play by σ , there is a non-dropping iterate N of M with embedding j and an N -generic g ⊂ Col( ω, j ( δ )) such that x ∈ N [ g ] and x ∈ j ( ˙ A )[ g ] .(2) If M | = φ II [ δ, ˙ B ] , there is a strategy τ for Player II in a game of length ω such that whenever x is a play by τ , there is a non-dropping iterate N of M with embedding j and an N -generic g ⊂ Col( ω, j ( δ )) such that x ∈ N [ g ] and x ∈ j ( ˙ B )[ g ] .(3) Otherwise, there is an M -generic g ⊂ Col( ω, δ ) and an x ∈ M [ g ] such that x ˙ A [ g ] and x ˙ B [ g ] . We now proceed to:
Proof of Theorem 4.4.
We may as well assume that γ is infinite, since otherwise G hasfixed length <ω , so it is determined by the results of [Nee95] and [Nee02] (see also theintroduction of [Nee04]).Let G be a simple clopen game with gr( G ) ≤ γ , say gr( G ) = α . We assume that α isa successor ordinal, since the limit case is similar. Let r ∈ ω ω code all parameters usedin the definition of G . We may as well assume that r belongs to the Turing cone givenby the hypothesis of the theorem.To each non-terminal play p of G corresponds a game G p := the game G after p has been played.Clearly, G p is a simple game and gr( G p ) ≤ α . Given y ≥ T r , a y -premouse M , weselect δ ∈ Ord M and define formulae φ p I and φ p II , and two sets ˙ W p I and ˙ W p II , each ofwhich is either a Col( ω, δ )-name for a set of real numbers or a set of natural numbers.The definition is by induction on gr( G p ) (not on p !). Formally, these names and setsof course depend on the model M in which they are defined, so we sometimes write˙ W p I ( M ) and ˙ W p II ( M ) to make this explicit. But for simplicity and readability we willomit this whenever it does not lead to confusion.At the same time, we show that there are names, names for names, etc., for thesesets such that their interpretation with respect to generics g , . . . , g n for some n ∈ ω forcollapsing Woodin cardinals of M is a set of the same form with respect to the model M [ g ] . . . [ g n ] instead of M . More precisely, if M is a model as above and M has (atleast) n + 1 Woodin cardinals, then let P n = P ( δ , . . . , δ n ) be the forcing iteration oflength n + 1 collapsing the ordinals δ , δ , . . . , δ n to ω one after the other, where δ is As defined in Appendix A, Iterability in [Nee04]. Note that ω -iterability implies weak iterability. the least Woodin cardinal in M and δ i +1 is the least Woodin cardinal in M above δ i forall 0 ≤ i < n , i.e., P = Col( ω, δ ) and for all 0 ≤ i < n , P i +1 = P i ∗ Col( ω, δ i +1 ) ˇ , where Col( ω, δ i +1 ) ˇ ∈ M is the canonical P i -name for Col( ω, δ i +1 ). This is of courseequivalent to the product and to the Col( ω, δ n ), but we are interested in the step-by-step collapse.We consider nice P n -names ˙ p for finite sequences of reals with the property that thegame rank of G ˙ p is decided by the empty condition. By induction on the game rankof G ˙ p , we will specify for every n < ω such that M has at least n + 1 Woodin cardinalsand every such P n -name ˙ p , a P n -name ˙ B ˙ p,In in M such that whenever G is P n -genericover M , ˙ B ˙ p,In [ G ] = ˙ W p I ( M [ G ]) , where p = ˙ p [ G ]. We will call this name ˙ B ˙ p,In the good P n -name for ˙ W p I . We will alsodefine analogous names ˙ B ˙ p,II for ˙ W p II . We now proceed to the recursive definition of φ p I , φ p II , ˙ W p I , ˙ W P II , ˙ B p,In , and ˙ B p,IIn . Case . gr( G p ) = 2 (this is the base case).Let y ≥ T r and let M be any y -premouse which is a model of ZFC and of class S .Assume without loss of generality that M is minimal, in the sense that no proper initialsegment of M is a model of ZFC of class S . Let δ be the least Woodin cardinal in M ,so that M is of class S above δ , and let p ∈ M . We define the Col( ω, δ )-names for setsof reals ˙ W p I and ˙ W p II by ˙ W p I = ˙ W p I ( M ) = { ( ˙ y, q ) | q (cid:13) M Col( ω,δ ) Player I has a winning strategy in G ˇ p ⌢ ˙ y } and ˙ W p II = ˙ W p II ( M ) = { ( ˙ y, q ) | q (cid:13) M Col( ω,δ ) Player II has a winning strategy in G ˇ p ⌢ ˙ y } . Let φ p I and φ p II be the formulae given by Neeman’s theorem (Theorem 4.5) such that thefollowing hold:(1) If M | = φ p I [ ˙ W p I ], there is a strategy σ for Player I in a game of length ω suchthat whenever x is a play by σ , there is an iterate N of M , an elementaryembedding j : M → N and an N -generic h ⊂ Col( ω, j ( δ )) such that x ∈ N [ h ]and x ∈ j ( ˙ W p I )[ h ].(2) If M | = φ p II [ ˙ W p II ], there is a strategy τ for Player II in a game of length ω suchthat whenever x is a play by τ , there is an iterate N of M , an elementaryembedding j : M → N and an N -generic h ⊂ Col( ω, j ( δ )) such that x ∈ N [ h ]and x ∈ j ( ˙ W p II )[ h ].(3) Otherwise, there is an M -generic g ⊂ Col( ω, δ ) and an x ∈ M [ g ] such that x ˙ W p I [ g ] and x ˙ W p II [ g ].Now, we specify the good P n -names ˙ B ˙ p,In as claimed above, for models with enoughWoodin cardinals below δ so that P n is defined. For n = 0, suppose that M is a premousewith Woodin cardinals δ < δ and is of class S above δ and minimal above δ . Let ˙ p Note that the game rank of G p depends only on G and finitely many digits of p . For every p , φ p I will be one of three formulae (there is one possibility for the successor case and twofor the limit case). We will however use the notation φ p I to make the presentation uniform. Similarlyfor φ p II . Here and below, all names for real are assumed to be nice. A case of interest will be that of such M which are initial segments of a model N of some class S β which is minimal above δ . In such an N , δ need not be a cardinal. ONG GAMES AND σ -PROJECTIVE SETS 13 be a nice Col( ω, δ )-name for a finite sequence of reals and set˙ B ˙ p,I = { (( ˙˙ y, ˙ q ) , q ) | q (cid:13) Col( ω,δ ) ˙ q (cid:13) Col( ω,δ ) ˇ Player I has a winning strategy in G ˇ˙ p ⌢ ˙˙ y } . The good P n -names ˙ B ˙ p,In for n > B ˙ p,IIn for ˙ W p II are defined analo-gously. Case . gr( G p ) = γ + 1 for some γ ≥ y ≥ T r and let M be any y -premouse which is a model of ZFC of class S γ +1 .Assume without loss of generality that M is minimal, in the sense that no proper initialsegment of M is a model of ZFC of class S γ +1 . Let δ be the least Woodin cardinal in M , so that M is of class S γ above δ , and let p ∈ M . Then let˙ W p I = ˙ W p I ( M ) = { ( ˙ y, q ) | q (cid:13) M Col( ω ) φ ˇ p ⌢ ˙ y I [ ˙ B ] } , where ˙ B ∈ M is the good Col( ω, δ )-name with respect to ˇ p ⌢ ˙ y , so that whenever G isCol( ω, δ )-generic over M , ˙ B [ G ] = ˙ W p ⌢ y I ( M [ G ]), where p ⌢ y = (ˇ p ⌢ ˙ y )[ G ]. We also let˙ W p II = ˙ W p II ( M ) = { ( ˙ y, q ) | q (cid:13) M Col( ω,δ ) φ ˇ p ⌢ ˙ y II [ ˙ B ] } , for ˙ B analogous as above. Note that ˙ B is a good name for ˙ W p ⌢ y I or ˙ W p ⌢ y II respectivelyand hence already defined since for every real y , gr( G p ⌢ y ) = γ . Let φ p I and φ p II be theformulae given by Neeman’s theorem (Theorem 4.5) such that the following hold:(1) If M | = φ p I [ ˙ W p I ], there is a strategy σ for Player I in a game of length ω suchthat whenever x is a play by σ , there is an iterate N of M , an elementaryembedding j : M → N and an N -generic h ⊂ Col( ω, j ( δ )) such that x ∈ N [ h ]and x ∈ j ( ˙ W p I )[ h ].(2) If M | = φ p II [ ˙ W p II ], there is a strategy τ for Player II in a game of length ω suchthat whenever x is a play by τ , there is an iterate N of M , an elementaryembedding j : M → N and an N -generic h ⊂ Col( ω, j ( δ )) such that x ∈ N [ h ]and x ∈ j ( ˙ W p II )[ h ].(3) Otherwise, there is an M -generic g ⊂ Col( ω, δ ) and an x ∈ M [ g ] such that x ˙ W p I [ g ] and x ˙ W p II [ g ].Now, we specify the good P n -names ˙ B ˙ p,In . For n = 0, suppose M is a premouse withWoodin cardinals δ < δ and that M is of class S γ above δ and minimal above δ . Let˙ p ∈ M be a nice Col( ω, δ )-name for a finite sequence of reals. Set˙ B ˙ p,I = { (( ˙˙ y, ˙ q ) , q ) | q (cid:13) Col( ω,δ ) ˙ q (cid:13) Col( ω,δ ) ˇ φ ˇ˙ p ⌢ ˙˙ y I [ ˙ B ] } , where ˙ B is a good P ( δ , δ )-name for ˙ W p I . That means ˙ B is such that whenever G is P ( δ , δ )-generic over M , ˙ B [ G ] = ˙ W p ⌢ y I ( M [ G ]), where p ⌢ y = (ˇ˙ p ⌢ ˙˙ y )[ G ]. The P n -names˙ B ˙ p,In for n > B ˙ p,IIn for ˙ W p II are defined analogously. Case . gr( G p ) = λ is a limit ordinal and the rules of G dictate that, after p , it is PlayerI’s turn.Let y ≥ T r and let M be any y -premouse which is a model of ZFC and of class S λ . Let p ∈ M . Then let ˙ W p I = ˙ W p I ( M ) be the set of all k ∈ ω such that there is an η < M ∩ Ordsuch that M | η is an active y -premouse of class S gr( G p⌢k ) which is minimal, in the sensethat no proper initial segment thereof is of class S gr( G p⌢k ) , and, if we let M ∗ be theresult of iterating the active extender of M | η out of the universe, then M ∗ | = φ p ⌢ k I [ δ ∗ , ˙ W p ⌢ k I ( M ∗ )] . This makes sense because for all k ∈ ω , gr( G p ⌢ k ) < gr( G p ), so the set ˙ W p ⌢ k I ( M ∗ )has been defined for all M ∗ as above. Moreover, ˙ W p I belongs to M , since the formulae φ p ⌢ kI ( ˙ W p ⌢ k I [ M ∗ ]), together with their parameters ˙ W p ⌢ k I ( M ∗ ), are definable uniformlyin p , gr( G p ), and η , as can be shown inductively by following this construction and theproof of Theorem 4.5 (cf. the proof of [Nee04, Theorem 1E.1]).We let φ p I [ ˙ W p I ( M )] be the formula asserting that ˙ W p I is non-empty. Similarly, let˙ W p II = ˙ W p II ( M ) be the set of all k ∈ ω such that whenever η < M ∩ Ord is such that M | η is an active y -premouse of class S gr( G p⌢k ) which is minimal, then M ∗ | = φ p ⌢ k II [ ˙ W p ⌢ k II ( M ∗ )] , for M ∗ defined as above. We let φ p II [ ˙ W p II ( M )] be the formula asserting that ˙ W p II is equalto ω . As before, the set ˙ W p II belongs to M .The good P n -names ˙ B ˙ p,In and ˙ B ˙ p,IIn are defined as before, for premice M which areof class S λ above finitely many Woodin cardinals. Case . gr( G p ) = λ is a limit ordinal and the rules of G dictate that, after p , it is PlayerII’s turn.Let y ≥ T r and let M be any y -premouse which is a model of ZFC and of class S λ .Let p ∈ M . Then let ˙ W p I ( M ) be the set of all k ∈ ω such that there is an η < M ∩ Ordsuch that M | η is an active y -premouse of class S gr( G p⌢k ) which is minimal, in the sensethat no proper initial segment of M | η is of class S gr( G p⌢k ) , and, if we again let M ∗ bethe result of iterating the active extender of M | η out of the universe, then M ∗ | = φ p ⌢ k I [ ˙ W p ⌢ k I ( M ∗ )] . Again, ˙ W p I is in M . We let φ p I [ ˙ W p I ( M )] be the formula asserting that ˙ W p I is equal to ω .Similarly, let ˙ W p II = ˙ W p II ( M ) be the set of all k ∈ ω such that whenever η < M ∩ Ord issuch that M | η is an active y -premouse of class S gr( G p⌢k ) which is minimal, then M ∗ | = φ p ⌢ k II [ ˙ W p ⌢ k II ( M ∗ )] , for M ∗ defined as above. We let φ p II [ ˙ W p II ( M )] be the formula asserting that ˙ W p II isnon-empty.The good P n -names ˙ B ˙ p,In and ˙ B ˙ p,IIn are defined as before, for premice M which areof class S λ above finitely many Woodin cardinals.This completes the definition of the names ˙ W p I , ˙ W p II , ˙ B ˙ p,In , ˙ B ˙ p,IIn , and the formulae φ p I and φ p II . Continuing with the proof of Theorem 4.4, we prove a technical claim whichshows that these names behave well under elementary embeddings. Claim 1.
Suppose gr( G ) is a successor ordinal, let M be a proper class premouse whichis a model of ZFC of class S gr( G ) and minimal, in the sense that no proper initial segmentof M is a model of ZFC of class S gr( G ) . Let j : M → N be an elementary embedding.Let p ∈ M be a finite sequence of reals in M . Then j ( ˙ W p I ( M )) = ˙ W p I ( N ) and, analogously, j ( ˙ W p II ( M )) = ˙ W p II ( N ) . Proof.
Let δ denote the least Woodin cardinal of M . We will in fact show that whenever˙ p ∈ M is a P n -name for a finite sequence of reals such that gr( G ˙ p ) is decided by theempty condition, we have j ( ˙ B ˙ p,In ) = ( ˙ B j ( ˙ p ) ,In ) N , and j ( ˙ B ˙ p,IIn ) = ( ˙ B j ( ˙ p ) ,IIn ) N , Recall that the structure M includes a predicate for its extender sequence. Since gr( G ) is a successor ordinal and M is minimal, M has a Woodin cardinal. ONG GAMES AND σ -PROJECTIVE SETS 15 i.e., if ˙ B ∈ M is a good P n -name such that whenever G is P n -generic over M ,˙ B [ G ] = ˙ W ˙ p [ G ]I ( M [ G ]) , then j ( ˙ B ) ∈ N is a good j ( P n )-name such that whenever H is j ( P n )-generic over N , j ( ˙ B )[ H ] = ˙ W j ( ˙ p )[ H ]I ( N [ H ])and similarly for ˙ B ˙ p,IIn .This will yield the claim if applied to n = 0, as, by definition,˙ W p I ( M ) = { ( ˙ y, q ) : q (cid:13) M Col( ω,δ ) φ ˇ p ⌢ ˙ y I [ ˙ B ˇ p ⌢ ˙ y,I ] } . Hence, j ( ˙ W p I ( M )) = { ( ˙ y, q ) : q (cid:13) N Col( ω,j ( δ )) φ ˇ p ⌢ ˙ y I [ j ( ˙ B ˇ p ⌢ ˙ y,I )] } = { ( ˙ y, q ) : q (cid:13) N Col( ω,j ( δ )) φ ˇ p ⌢ ˙ y I [( ˙ B j (ˇ p ⌢ ˙ y ) ,I ) N ] } = ˙ W p I ( N ) , and similarly for ˙ B ˙ p,IIn .By the definition of simple games, gr( G p ) depends only on G , the length of p , andfinitely many values of p (those in which the players determine the subgames played).Thus, if ˙ p is a P n -name in M for a finite sequence of reals such that (cid:13) M P n gr( G ˙ p ) = γ + 1for some countable ordinal γ and if ˙ y is a P ( δ , . . . , δ n , δ )-name for a real, then (cid:13) M P ( δ ,...,δ n ,δ ) gr( G ˇ˙ p ⌢ ˙ y ) = γ. The claim is proved simultaneously for all premice M and all ˙ p ∈ M , by induction onthe game rank of gr( G p ). We prove it for the names ˙ B ˙ p,In for Player I; the other part issimilar. We proceed by cases:First suppose that γ = 2 and let n < ω and ˙ p ∈ M be a P n -name for a finite sequenceof reals such that (cid:13) M P n gr( G ˙ p ) = 2. Let ˙ B = ˙ B ˙ p,In ∈ M , so that whenever G is P n -genericover M , ˙ B [ G ] = ˙ W ˙ p [ G ]I ( M [ G ]) . By definition, (cid:13) M . . . (cid:13) ˙ B = { ( ˙ y, q ) | q (cid:13) Col( ω,δ ) Player I has a winning strategy in G ˇ˙ p ⌢ ˙ y } . By elementarity, (cid:13) N . . . (cid:13) j ( ˙ B ) = { ( ˙ y, q ) | q (cid:13) Col( ω,j ( δ )) Player I has a winning strategy in G j (ˇ˙ p ) ⌢ ˙ y } . But this implies that j ( ˙ B ) is a good j ( P n )-name such that whenever H is j ( P n )-genericover N , j ( ˙ B )[ H ] = ˙ W j ( ˙ p )[ H ]I ( N [ H ]) . This finishes the case γ = 2.If γ = β + 1, where β >
1, let again n < ω , ˙ p ∈ M be a P n -name for a finite sequenceof reals such that (cid:13) M P n gr( G ˙ p ) = β + 1 and let ˙ B = ˙ B ˙ p,In ∈ M , so that whenever G is P n -generic over M , ˙ B [ G ] = ˙ W ˙ p [ G ]I ( M [ G ]) . This means that (cid:13) M . . . (cid:13) ˙ B = { ( ˙ y, q ) | q (cid:13) Col( ω,δ ) φ ˇ˙ p ⌢ ˙ y I [ ˙ C ] } , where ˙ C ∈ M is a good P ( δ , . . . , δ n , δ )-name such that whenever g is P ( δ , . . . , δ n , δ )-generic over M , ˙ C [ g ] = ˙ W (ˇ˙ p ⌢ ˙ y )[ g ]I ( M [ g ]) . By inductive hypothesis, and using that (cid:13) M P ( δ ,...,δ n ,δ ) gr( G ˇ˙ p ⌢ ˙ y ) = β < gr( G ˇ˙ p ), wehave that j ( ˙ C ) ∈ N is a good P ( j ( δ ) , . . . , j ( δ n ) , j ( δ ))-name such that whenever h is P ( j ( δ ) , . . . , j ( δ n ) , j ( δ ))-generic over N , j ( ˙ C )[ h ] = ˙ W j (ˇ˙ p⌢ ˙ y )[ h ]I ( N [ h ]) . Since, by elementarity, (cid:13) N . . . (cid:13) j ( ˙ B ) = { ( ˙ y, q ) | q (cid:13) Col( ω,j ( δ )) φ j (ˇ˙ p ⌢ ˙ y )I [ j ( ˙ C )] } , it follows that j ( ˙ B ) is a good j ( P n )-name such that for every j ( P n )-generic H over N , j ( ˙ B )[ H ] = ˙ W j ( ˙ p )[ H ]I ( N [ H ]) , as desired.If γ is a limit ordinal, let again n < ω , ˙ p ∈ M be a P n -name for a finite sequence ofreals such that (cid:13) M P n gr( G ˙ p ) = γ . Let ˙ B = ˙ B ˙ p,In ∈ M , so that whenever G is P n -genericover M , ˙ B [ G ] = ˙ W ˙ p [ G ]I ( M [ G ]) . This means that (cid:13) M . . . (cid:13) ˙ B = { k ∈ ω | there is an initial segment M ∗ minimal of class S gr( G ˙ p⌢k ) which satisfies φ ˙ p ⌢ k I [ ˙ C ] } , where ˙ C ∈ M ∗ is a good P n -name such that whenever g is P n -generic over M ∗ ,˙ C [ g ] = ˙ W ( ˙ p ⌢ k )[ g ]I ( M ∗ [ g ]) . By inductive hypothesis, j ( ˙ C ) ∈ N | ( j ( M ∗ ) ∩ Ord) is a good j ( P n )-name such thatwhenever h is j ( P n )-generic over N | ( j ( M ∗ ) ∩ Ord), j ( ˙ C )[ h ] = ˙ W j ( ˙ p ⌢ k )[ h ]I ( N | ( j ( M ∗ ) ∩ Ord)[ h ]) . Since, by elementarity, (cid:13) N . . . (cid:13) j ( ˙ B ) = { k ∈ ω | there is an initial segment N ∗ minimal of class S gr( G ˙ p⌢k ) which satisfies φ j ( ˙ p ⌢ k )I [ j ( ˙ C )] } , it follows that j ( ˙ B ) is a good j ( P n )-name such that for every j ( P n )-generic H over N , j ( ˙ B )[ H ] = ˙ W j ( ˙ p )[ H ]I ( N [ H ]) , as desired.The argument for ˙ W p II ( M ) is analogous. This completes the proof of the claim. (cid:3) Now we turn to the proof of the following claim, from which the theorem follows.Recall that α = gr( G ). Claim 2.
Let M be an r -premouse which of ZFC of class S α but has no proper initialsegment of class S α . Let δ denote the least Woodin cardinal in M . Then(1) If M | = φ ∅ I [ ˙ W ∅ I ] , then Player I has a winning strategy in G .(2) If M | = φ ∅ II [ ˙ W ∅ II ] , then Player II has a winning strategy in G .(3) M | = φ ∅ I [ ˙ W ∅ I ] ∨ φ ∅ II [ ˙ W ∅ II ] .Proof. If M is active, then we may identify it with the proper class sized iterated ultra-power by its top extender and its images, so we may assume that M is a model of ZFC .Note that M always has a Woodin cardinal, as α is a successor ordinal. We first prove(3). Suppose that M = φ ∅ I [ ˙ W ∅ I ] and M = φ ∅ II [ ˙ W ∅ II ]. We inductively construct a run ofthe game G by its initial segments p m together with reals y m , y m -premice M p m which ONG GAMES AND σ -PROJECTIVE SETS 17 are proper class models of ZFC and ordinals δ p m . Let p be the empty play, y = r , and M = M . Inductively, suppose that p m , y m , and M p m have been defined and that M p m = φ p m I [ ˙ W p m I ] and M p m = φ p m II [ ˙ W p m II ] , and let δ p m be the least Woodin cardinal of M p m , if it exists.If gr( G p m ) = γ + 1 is a successor ordinal for some γ ≥
2, then, by Neeman’s theorem(Theorem 4.5), there is an M p m -generic g m ⊂ Col( ω, δ p m ) and a y ∈ M p m [ g m ] such that y / ∈ ˙ W p m I [ g m ] and y / ∈ ˙ W p m II [ g m ]. This means that M p m [ g m ] = φ p ⌢m y I [ ˙ W p ⌢m y I ] and M p m [ g m ] = φ p ⌢m y II [ ˙ W p ⌢m y II ] . Let p m +1 = p ⌢m y , M p m +1 = M p m [ g m ]. Here M p m [ g m ] can be rearranged as a y m +1 -premouse for some real y m +1 coding y m and g m as g m collapses a cutpoint of the y m -premouse M p m . We will always consider M p m +1 as such a y m +1 -premouse.Suppose that gr( G p m ) is a limit ordinal and the rules of G dictate that after p m , itis Player I’s turn. By inductive hypothesis we have M p m = φ p m I [ ˙ W p m I ], so there is noactive initial segment M p m | ν , ν ∈ Ord, of M p m of class S gr( G p⌢mk ) which is minimal andsatisfies φ p ⌢m k I [ ˙ W p ⌢m k I ], for any k ∈ ω . Moreover, M p m = φ p m II [ ˙ W p m II ]. Hence, there issome k ∈ ω and an active initial segment of M p m of class S gr( G p⌢m k ) which is minimaland does not satisfy φ p ⌢m k II [ ˙ W p ⌢m k II ], where δ ∗ is defined analogous as above. Fix such a k and such an active initial segment M p m | η of M p m and let M p m +1 be the proper classmodel resulting from iterating the active extender of M p m | η out of the universe. Set p m +1 = p ⌢m k and y m +1 = y m . Then, M p m +1 = φ p m +1 I [ ˙ W p m +1 I ] and M p m +1 = φ p m +1 II [ ˙ W p m +1 II ] . The case that gr( G p m ) is a limit ordinal and the rules of G dictate that it is Player II’sturn after p m is similar.Finally, suppose that gr( G p m ) = 2. By Neeman’s theorem (Theorem 4.5), there issome M p m -generic g m ⊂ Col( ω, δ p m ) and some y ∈ M m [ g m ] such that y / ∈ ˙ W p m I [ g m ] and y / ∈ ˙ W p m II [ g m ]. This means that(1) M p m [ g m ] | = “Player I does not have a winning strategy in G p ⌢m y ”, and(2) M p m [ g m ] | = “Player II does not have a winning strategy in G p ⌢m y ”.However, gr( G p ⌢m y ) = 1, so G p ⌢m y is a clopen game of length ω . This is a contradiction,as M p m [ g m ] is a model of ZFC , so it certainly satisfies clopen determinacy. This proves(3).We now prove (1); the proof of (2) is similar.Let M be as in the statement of the claim and suppose M | = φ ∅ I [ ˙ W ∅ I ]. We will describea winning strategy σ for Player I in G (in V ) as a concatenation of strategies σ m fordifferent rounds of G . Playing against arbitrary moves of Player II, we will for m ≥ p m for initial segments of G according to σ together with • reals y m such that y m +1 ≥ T y m and y = r , • y m -premice M p m which are proper class models of ZFC of class S gr( G pm ) , noneof whose initial segments are of class S gr( G pm ) , and such that p m ∈ M p m , • premice N p m together with iteration embeddings j m : M p m → N p m , • if M p m has a Woodin cardinal and δ p m is the least Woodin cardinal in M p m ,Col( ω, j ( δ p m ))-generics g p m over N p m .In case M p m does not have a Woodin cardinal, we will let g p m be undefined and let M p m +1 = N p m . In the other case, we let M p m +1 = N p m [ g p m ]. Finally, we will stop theconstruction after finitely many steps when gr( G p m ) = 1.Let p be the empty play and M p = M . We will inductively argue that M p m | = φ p m I [ ˙ W p m I ( M p m )] , where ˙ W p m I ( M p m ) is the Col( ω, δ p m )-name or set of natural numbers in M p m definedabove.Assume inductively that p n and M p n with M p n | = φ p n I [ ˙ W p n I ( M p n )] are already con-structed for all n ≤ m and that gr( G p m ) ≥
2. To construct p m +1 and M p m +1 wedistinguish the following cases.Assume first that gr( G p m ) = γ + 1 for some γ ≥
2. Since M p m | = φ p m I [ ˙ W p m I ( M p m )] , it follows from Neeman’s theorem (Theorem 4.5) that there is a strategy σ m for PlayerI in a game of length ω such that whenever x is a play by σ m , there is an iterate N p m of M p m , an elementary embedding j : M p m → N p m and an N p m -generic h ⊂ Col( ω, j ( δ p m ))such that x ∈ N p m [ h ] and x ∈ j ( ˙ W p m I ( M p m )[ h ]. Hence, by Claim 1, x ∈ ˙ W p m I ( N p m )[ h ].By definition, N p m [ h ] | = φ p ⌢m x I [ ˙ W p ⌢m x I ( N p m [ h ])] . Let p m +1 = p ⌢m x , M p m +1 = N p m [ h ], and g p m = h . Note that p m +1 ∈ M p m +1 and asbefore N p m [ h ] can be rearranged as a y m +1 -premouse for some real y m +1 coding y m and h . Again, we will always consider M p m +1 as such a y m +1 -premouse.Now suppose that gr( G p m ) = λ is a limit ordinal and the rules of G dictate that,after p m , it is Player I’s turn. By assumption, M p m | = φ p m I [ ˙ W p m I ( M p m )]. So M p m | = ˙ W p m I ( M p m ) = ∅ . Hence, there is a natural number k and an ordinal η such that M p m | η is an active initialsegment of M p m of class S gr( G p⌢m k ) and is minimal, and, if we let N p m be the result of iter-ating the active extender of M p m | η out of the universe, then N p m | = φ p ⌢m k I [ ˙ W p ⌢m k I ( N p m )] . Let p m +1 = p ⌢m k , y m +1 = y m , and M p m +1 = N p m . Moreover, let σ m be the strategywhich tells Player I to play k .For the other limit case suppose that gr( G p m ) = λ is a limit ordinal and the rulesof G dictate that, after p m , it is Player II’s turn. In this case we can let σ m = ∅ asPlayer I is not playing, i.e., we only have to react to what Player II is playing in thenext round. Suppose that Player II plays some natural number k and let p m +1 = p ⌢m k .Since M p m | = φ p m I [ ˙ W p m I ( M p m )], we have M p m | = ˙ W p m I ( M p m ) = ω. In particular, k ∈ ˙ W p m I ( M p m ). Thus there is an ordinal η such that M p m | η is anactive initial segment of M p m , minimal of class S gr( G p⌢m k ) , and if we let N p m denotethe result of iterating the active extender of M p m | η out of the universe, then N p m | = φ p ⌢m k I [ ˙ W p ⌢m k I ( N p m )] . Let M p m +1 = N p m , and y m +1 = y m .Finally, assume that gr( G p m ) = 2. Since M p m | = φ p m I [ ˙ W p m I ( M p m )], by Neeman’stheorem (Theorem 4.5), there is a strategy σ m for Player I in a game of length ω such that whenever x is a play by σ m , there is an iterate N p m of M p m , an elementaryembedding j : M p m → N p m and an N p m -generic h ⊂ Col( ω, j ( δ p m )) such that x ∈ N [ h ]and x ∈ j ( ˙ W p m I ( M p m ))[ h ]. Then, by Claim 1, x ∈ ˙ W p m I ( N p m ))[ h ]. Therefore, N p m [ h ] | = Player I has a winning strategy in G p ⌢m x . We let p m +1 = p ⌢m x and stop the construction. Since gr( G p ⌢m x ) = 1, G p ⌢m x is a clopengame of length ω . As N p m [ h ] is a proper class model of ZFC , we can use absoluteness ofwinning strategies for clopen games of length ω to obtain that Player I has a winningstrategy in G p ⌢m x in V . Let σ m +1 be a strategy for Player I witnessing this.This process describes a winning strategy σ for Player I in G by concatenating thestrategies σ i for 1 ≤ i ≤ m + 1, as desired. (cid:3) This finishes the proof of Theorem 4.4. (cid:3)
ONG GAMES AND σ -PROJECTIVE SETS 19 Further applications
In this section, we present some additional applications of the proof of Theorem 4.4.Since the proofs are similar, we simply sketch the differences.5.1.
Longer games.
We begin by noting that the results presented so far generalize tolonger games. In particular:
Theorem 5.1.
Let θ be a countable ordinal. Suppose that for each α < ω and each y ∈ R there is some x ≥ T y and an x -premouse M of class S α above some λ belowwhich there are θ Woodin cardinals in M . Then σ -projective games of length ω · θ aredetermined. The theorem is a consequence of a more general result akin to Theorem 4.4, namely,the determinacy of simple σ -projective games of length ω · ( θ + ω ), in the following sense: Definition 5.2.
Let Γ be a collection of subsets of ω ω · θ + ω (with each A ∈ Γ identifiedwith a subset of ω ω · θ + ω · n for some n ∈ ω as in Definition 2.2). A game of length ω · θ + ω is Γ- simple if it is obtained as follows:(1) For every n ∈ ω , games in Γ of fixed length ω · θ + ω · n are Γ-simple.(2) Let n ∈ ω and for each i ∈ ω let G i be a Γ-simple game. Then the game G obtained as follows is Γ-simple: Players I and II take turns playing naturalnumbers for ω · θ + ω · n moves, i.e., θ + n rounds in games of length ω . Afterwards,Player I plays some i ∈ ω . Players I and II continue playing according to therules of G i (keeping the first ω · θ + ω · n natural numbers they have alreadyplayed).(3) Let n ∈ ω and for each i ∈ ω let G i be a Γ-simple game. Then the game G obtained as follows is Γ-simple: Players I and II take turns playing naturalnumbers for ω · θ + ω · n moves, i.e., θ + n rounds in games of length ω . Afterwards,Player II plays some i ∈ ω . Players I and II continue playing according to therules of G i (keeping the first ω · θ + ω · n natural numbers they have alreadyplayed).The notion of game rank in this context is defined as before: if G is a game of fixedlength ω · θ + ω · n , then gr( G ) = n . If G is obtained from games G , G , . . . , and froman ordinal ω · θ + ω · n as in Definition 5.2, we letgr( G ) = sup { gr( G i ) + ω : i ∈ ω } + n. Theorem 5.3.
Let θ be a countable ordinal. Suppose that for each α < ω and each y ∈ R there is some x ≥ T y and an x -premouse M of class S α above some λ belowwhich there are θ Woodin cardinals in M . Then, simple σ -projective games of length ω · ( θ + ω ) are determined.Proof Sketch. The theorem is proved like Theorem 4.4: first, by arguing as in Proposi-tion 3.7, one sees that is suffices to prove determinacy for simple clopen games of length ω · ( θ + ω ). Let G be a game of successor rank α definable from a parameter x . Given y ≥ T x and an active y -premouse M of class S α , one defines sets ˙ W p I ( M ) and ˙ W p II ( M )and formulae φ p I and φ p II by induction on gr( G p ) as in the proof of Theorem 4.4, providedgr( G p ) < gr( G ).The difference is as follows: in the proof of Theorem 4.4, gr( G ) = gr( G p ) occurs when p = ∅ ; here, it happens when p is a θ -sequence of reals. Instead of defining ˙ W p I in thiscase, we define only ˙ W ∅ I and ˙ W ∅ II . These are names for sets of θ + 1-sequences of realsand are defined as in Case 2 in the proof of Theorem 4.4; namely,˙ W ∅ I = (cid:8) ( ˙ p, q ) | q (cid:13) M Col( ω,δ ) φ ˙ p I [ ˙ W ˙ p I ] (cid:9) , where ˙ p is a name for a θ + 1-sequence of reals, δ is the least Woodin cardinal of M above λ , and δ ∗ is the second Woodin cardinal of M above λ , if it exists, and ω ,otherwise. (Note that δ exists since, by assumption, α is a successor ordinal). ˙ W ∅ II is defined similarly. Once these names have been defined, one applies Theorem 2A.2of [Nee04] (the general version of Theorem 4.5) so that (using the fact that M has θ Woodin cardinals below λ ) one obtains formulae φ I and φ II with parameters ˙ W ∅ I and˙ W ∅ II , such that one of the following holds:(1) If M | = φ I [ ˙ W ∅ I ], there is a strategy σ for Player I in a game of length ω · θ + ω such that whenever ~x is a play by σ , there is a non-dropping iterate N of M with an embedding j and an N -generic g ⊂ Col( ω, j ( δ )) such that ~x ∈ N [ g ] and ~x ∈ j ( ˙ W ∅ I )[ g ].(2) If M | = φ II [ ˙ W ∅ II ], there is a strategy τ for Player II in a game of length θ · ω + ω such that whenever ~x is a play by τ , there is a non-dropping iterate N of M with an embedding j and an N -generic g ⊂ Col( ω, j ( δ )) such that ~x ∈ N [ g ] and ~x ∈ j ( ˙ W ∅ II )[ g ].(3) Otherwise, there is an M -generic g ⊂ Col( ω, δ ) and an ~x ∈ M [ g ] such that ~x ˙ W ∅ I [ g ] and ~x ˙ W ∅ II [ g ].An argument as in the proof of Theorem 4.4 yields the following claim: Claim 1.
Let M be an x -premouse and λ ∈ M be an ordinal such that M has θ Woodincardinals below λ . Suppose that M is of class S α above λ and no proper initial segmentof M is of class S α above λ , where α is a successor ordinal. Let δ denote the leastWoodin cardinal of M above λ . Then(1) If M | = φ ∅ I [ ˙ W ∅ I ] , then Player I has a winning strategy in G .(2) If M | = φ ∅ II [ ˙ W ∅ II ] , then Player II has a winning strategy in G .(3) M | = φ ∅ I [ ˙ W ∅ I ] ∨ φ ∅ II [ ˙ W ∅ II ] . The theorem is now immediate from the claim. (cid:3)
It seems very likely that the hypotheses of Theorem 5.1 are optimal. However, theproof in [Agu] does not seem to adapt easily to show this.5.2. σ -algebras. The proof of Theorem 4.4 adapts to prove the determinacy of various σ -algebras. As an example, we consider the smallest σ -algebra containing all projectivesets. We show that a sufficient condition for its determinacy is the existence of x -premiceof class S ω +1 for every x ∈ R . Theorem 5.4.
Suppose that for each x ∈ ω ω there is an x -premouse of class S ω +1 .Then, every set in the smallest σ -algebra on ω ω containing the projective sets is deter-mined.Proof Sketch. Let A belong to the σ -algebra in the statement. Let { Σ α ( Π ω ) : α < ω } denote the Borel hierarchy built starting from sets which are countable intersections ofprojective sets, i.e., Σ ( Π ω ) = Π ω consists of all countable intersections of projectivesets and Σ α ( Π ω ) consists of all countable unions of sets each of which is the complementof a set in Σ β ( Π ω ) for some β < α . Standard arguments show that these pointclassesconstitute a hierarchy of sets in the smallest σ -algebra containing the projective sets.It follows that there is α < ω and x ∈ ω ω such that A ∈ Σ α (Π ω )( x ), i.e., such that A belongs to Σ α ( Π ω ) and that A has a σ -projective code (in the sense of Section 3) whichis recursive in x .Let [ A ] be a σ -projective code for A which is recursive in x and in which no com-plements appear. To show that A is determined, it suffices to show that the decoding ONG GAMES AND σ -PROJECTIVE SETS 21 game for [ A ] is determined. Let us denote this game by G . It is a simple clopen game;it is not quite of rank ω + 1, so determinacy does not follow immediately from Theorem4.4, but it follows from the proof:Given a partial play p of the game and a model N , we define sets ˙ W p I ( N ) and ˙ W p II ( N ),formulae φ I [ ˙ W p I ( N )] and φ II [ ˙ W p II ( N )], and names ˙ B ˙ p,In and ˙ B ˙ p,IIn as in the proof ofTheorem 4.4. The cases where gr( G p ) ≤ ω are exactly as in the proof of Theorem4.4. The remaining cases are very slightly different—for this, let us define the notion of subrank . Recall that G has the following rules:(1) Players I and II begin by alternating ω many rounds to play a real number y ∈ ω ω .(2) Afterwards, they alternate a finite amount of turns as follows: letting [ A ] = [ A ]and supposing [ A n ] has been defined, [ A n ] is a σ -projective code for a union oran intersection of sets, say { B i : i ∈ ω } , with each B i of smaller (Borel) rank.By the rules of the decoding game, one of the players needs to play a naturalnumber i , thus selecting a σ -projective code [ B i ] for B i ; we let [ A n +1 ] = [ B i ].(3) Eventually, a stage n ∗ is reached in which [ A n ∗ ] is a code for a projective set; atthis point the players continue with the rules of the decoding game for [ A n ∗ ].Given a play p of G in which a player needs to play a natural number i and [ A n ] hasbeen defined as above, we say that the game subrank of G p is the least γ such that A n ∈ Σ γ (Π ω )( x ) or A n ∈ Π γ (Σ ω )( x ).We now continue the definition of the sets ˙ W p I ( N ) and ˙ W p II ( N ), formulae φ I [ ˙ W p I ( N )]and φ II [ ˙ W p II ( N )], and names ˙ B ˙ p,In and ˙ B ˙ p,IIn . Assume p is such that ω < gr( G p ), so thatthe sets have not been defined already; we proceed by induction on the subrank of G p : Case . The subrank of G p is defined and nonzero and the rules of G dictate that, after p , it is Player I’s turn.Let y ≥ T x and let M be any y -premouse which is a model of ZFC and of class S ω .Let p ∈ M . Then let ˙ W p I = ˙ W p I ( M ) be the set of all k ∈ ω such that M | = φ p ⌢ k I [ ˙ W p ⌢ k I ( M )] . This makes sense because for all k ∈ ω , the subrank of G p ⌢ k is smaller than the subrankof G p , so the set ˙ W p ⌢ k I ( M ) has been defined. Moreover, ˙ W p I belongs to M , since [ A ]is recursive in x and the formulae φ p ⌢ kI ( ˙ W p ⌢ k I [ M ]), as well as the sets ˙ W p ⌢ k I ( M ), aredefinable uniformly in p , as can be shown inductively by following this construction andthe proof of Theorem 4.5 (cf. the proof of [Nee04, Theorem 1E.1]).We let φ p I [ ˙ W p I ( M )] be the formula asserting that ˙ W p I is non-empty. Similarly, let˙ W p II = ˙ W p II ( M ) be the set of all k ∈ ω such that M | = φ p ⌢ k II [ ˙ W p ⌢ k II ( M )] . We let φ p II [ ˙ W p II ( M )] be the formula asserting that ˙ W p II is equal to ω . As before, the set˙ W p II belongs to M .The good P n -names ˙ B ˙ p,In and ˙ B ˙ p,IIn are defined as in the other cases. Case . The subrank of G p is defined and nonzero and the rules of G dictate that, after p , it is Player II’s turn.This case is similar to the preceding one. Case . p is the empty play.Let M be any x -premouse which is a model of ZFC and of class S ω +1 and let δ bethe smallest Woodin cardinal of M . In this case, ˙ W ∅ I is the canonical Col( ω, δ )-namefor all reals p such that φ pI ( ˙ W p I [ ˙ B ]) holds, where ˙ B ∈ M is a good Col( ω, δ )-name with respect to ˙ p , so that whenever G is Col( ω, δ )-generic over M , ˙ B [ G ] = ˙ W p I ( M [ G ]), where p = ˙ p [ G ].The name ˙ W ∅ II is defined analogously. The formulae φ ∅ I ( ˙ W ∅ I ) and φ ∅ II ( ˙ W ∅ II ) areobtained by applying Neeman’s theorem to ˙ W ∅ I and ˙ W ∅ II .This completes the definition of the names and formulae. After this, an argument asin the proof of Theorem 4.4 yields the following claim: Claim 1.
Let M be an x -premouse which is of class S ω +1 but has no proper initialsegment of class S ω +1 . Let δ denote the least Woodin cardinal in M . Then(1) If M | = φ ∅ I [ ˙ W ∅ I ] , then Player I has a winning strategy in G .(2) If M | = φ ∅ II [ ˙ W ∅ II ] , then Player II has a winning strategy in G .(3) M | = φ ∅ I [ ˙ W ∅ I ] ∨ φ ∅ II [ ˙ W ∅ II ] . The theorem is now immediate from the claim. (cid:3)
The following remains an interesting open problem:
Question 5.5.
What is the consistency strength of determinacy for sets in the smallest σ -algebra containing the projective sets? Acknowledgements
The first-listed author was partially suported by FWO grant number 3E017319 andby FWF grant numbers I4427 and I4513-N. The second-listed author, formerly knownas Sandra Uhlenbrock, was partially supported by FWF grant number P 28157 and inaddition gratefully acknowledges funding from L’OR´EAL Austria, in collaboration withthe Austrian UNESCO Commission and in cooperation with the Austrian Academyof Sciences - Fellowship
Determinacy and Large Cardinals . This project has receivedfunding from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk lodowska-Curie grant agreement No 794020 of the third-listed author(Project
IMIC: Inner models and infinite computations ). The third-listed author waspartially supported by FWF grant number I4039.
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Juan P. Aguilera, Department of Mathematics, Ghent University. Krijgslaan 281-S8, 9000Ghent, Belgium.Institut f¨ur diskrete Mathematik und Geometrie, Technische Universit¨at Wien. WiednerHauptstrasse 8-10, 1040 Wien, Austria.
Email address : [email protected] Sandra M¨uller, Institut f¨ur Mathematik, Universit¨at Wien. Kolingasse 14-16, 1090 Wien,Austria.
Email address : [email protected] Philipp Schlicht, Institut f¨ur Mathematik, Universit¨at Wien. Kolingasse 14-16, 1090Wien, AustriaSchool of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol,BS8 1UG, UK
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