It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ε {1,…,ω}, ∑3 + n1 absoluteness implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑3 + n1 absoluteness is exactly that of n strong cardinals so that in particular (...) projective absoluteness is equiconsistent with the existence of infinitely many strong cardinals. It is also argued how this theorem is to be construed as the first step in the long range program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle positively a conjecture of Woodin and thereby solve the last Delfino problem. (shrink)

We extend the construction of Mitchell and Steel to produce iterable fine structure models which may contain Woodin limits of Woodin cardinals, and more. The precise level reached is that of a cardinal which is both a Woodin cardinal and a limit of cardinals strong past it.

Let κ R be the least ordinal κ such that L κ (R) is admissible. Let $A = \{x \in \mathbb{R} \mid (\exists\alpha such that x is ordinal definable in L α (R)}. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC - Replacement + "There exists ω Woodin cardinals which are cofinal in the ordinals." T has consistency strength weaker than that of the theory ZFC + (...) "There exists ω Woodin cardinals", but stronger than that of the theory ZFC + "There exists n Woodin Cardinals", for each n ∈ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = R ∩ M. Since M is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every Σ * n real is in A. (shrink)

We present a characterization of supercompactness measures for ω1 in L(R), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of $\delta_3^1 = \omega_2$ with $ZFC+AD^{L(R)}$ , and the second proving the uniqueness of the supercompactness measure over ${\cal P}_{\omega_1} (\lambda)$ in L(R) for $\lambda > \delta_1^2$.

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L.The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and (...) descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL ⇒ HODL ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals. (shrink)

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, (...) … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture. (shrink)

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core (...) models. (shrink)

We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n[g]$ for a Turing cone of reals x, where $M_n$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n[g]} = M_n,$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$, (...) $\delta _{\infty }$ is the least Woodin cardinal in $\mathcal {M}_{\infty }$, $\kappa _{\infty }$ is the least inaccessible cardinal in $\mathcal {M}_{\infty }$ above $\delta _{\infty }$, and $\Lambda $ is a partial iteration strategy for $\mathcal {M}_{\infty }$. It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n[g]} $ satisfies $\operatorname {GCH} $. (shrink)

We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...) the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy. (shrink)

We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...) the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy. (shrink)

We study Σ1-definable sets in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1-definable, the set of all stationary subsets of ω1 is not Σ1-definable and the complement of every Σ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1-definable well-ordering (...) of H and the existence of a Δ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1-definable subsets of κκ, in the case where κ itself has certain large cardinal properties. (shrink)

In this paper we explore a connection between descriptive set theory and inner model theory. From descriptive set theory, we will take a countable, definable set of reals, A. We will then show that , where is a canonical model from inner model theory. In technical terms, is a “mouse”. Consequently, we say that A is a mouse set. For a concrete example of the type of set A we are working with, let ODnω1 be the set of reals which (...) are ∑n definable over the model Lω1 , from an ordinal parameter. In this paper we will show that for all n 1, ODnω1 is a mouse set. Our work extends some similar results due to D.A. Martin, J.R. Steel, and H. Woodin. Several interesting questions in this area remain open. (shrink)

We introduce two new iteration games: the game , which is a strengthening of the weak iteration game, and the game , which is somewhat stronger than but weaker than the full iteration game of length ω1. For a countable M elementarily embeddable in some Vη, with two players I and II, we can show that II wins and that I does not win.

An inner model operator is a function M such that given a Turing degree d, M is a countable set of reals, d M, and M has certain closure properties. The notion was introduced by Steel. In the context of AD, we study inner model operators M such that for a.e. d, there is a wellorder of M in L). This is related to the study of mice which are below the minimal inner model with ω Woodin cardinals. As a (...) technical tool, we show that the alternative fine structure theory developed by Mitchell and Steel for mice with Woodin cardinals is equivalent to the traditional fine structure theory developed by Jensen for L. (shrink)

We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that , the Axiom of Determinacy, holds in the of a generic extension of : every uncountable cardinal is singular, and every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.

In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most (...) interesting “completion” of the ZFC-theory of L.ADL implies that L satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L. Nevertheless, there is a natural inner model of L, namely HODL, which satisfies ZFC.. The superscript “L” indicates, here and below, that the notion in question is to be interpreted in L.) HODL is reasonably close to the full L, in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL: what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (shrink)

We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated by every dominating real, but that this fails to hold in the tree Hechler extension. We prove a representation theorem for dominating reals in the standard Hechler extension: every dominating real eventually dominates a sandwich composition of the Hechler real with two (...) ground model reals that monotonically converge to infinity. We apply our results to negatively settle a conjecture of Brendle and Löwe (Conjecture 15 of [4]). We also answer a question due to Laflamme. (shrink)

Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals: and (2) o(κ) = κ⁺⁺. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play.

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found (...) one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L. One of the nice things about AD is that the theory ZF + AD + V = L appears as a choiceless “completion” of ZF in that any interesting question seems to find an at least attractive answer in that theory. Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension. (shrink)

We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.