Inner models in the region of a Woodin limit of Woodin cardinals

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that(1) every level of is a sound, tame mouse, and(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

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Projective well-orderings and bounded forcing axioms

Journal of Symbolic Logic ◽

10.2178/jsl/1120224728 ◽

2005 ◽

Vol 70 (2) ◽

pp. 557-572

Author(s):

Andrés Eduardo Caicedo

Keyword(s):

Large Cardinal ◽

Forcing Axioms ◽

Woodin Cardinals ◽

Inner Models

AbstractIn the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

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Inner models from extended logics: Part 1

Journal of Mathematical Logic ◽

10.1142/s0219061321500124 ◽

2020 ◽

pp. 2150012

Author(s):

Juliette Kennedy ◽

Menachem Magidor ◽

Jouko Väänänen

Keyword(s):

Continuum Hypothesis ◽

Second Order ◽

Order Logic ◽

First Order Logic ◽

First Order ◽

Woodin Cardinals ◽

Inner Models ◽

Inner Model ◽

Second Order Logic ◽

The Continuum

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see text] of [Formula: see text] are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model.

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The Jensen Covering Property

Journal of Symbolic Logic ◽

10.2307/2694959 ◽

2001 ◽

Vol 66 (4) ◽

pp. 1505-1523 ◽

Cited By ~ 5

Author(s):

E. Schimmerling ◽

W. H. Woodin

Keyword(s):

Large Cardinal ◽

Core Model ◽

Covering Property ◽

Prikry Forcing ◽

Covering Lemma ◽

Woodin Cardinals ◽

Inner Models ◽

Coherent Sequence ◽

Measurable Cardinals

The Jensen covering lemma says that either L has a club class of indiscernibles, or else, for every uncountable set A of ordinals, there is a set B ∈ L with A ⊆ B and card (B) = card(A). One might hope to extend Jensen's covering lemma to richer core models, which for us will mean to inner models of the form L[] where is a coherent sequence of extenders of the kind studied in Mitchell-Steel [8], The papers [8], [12], [10] and [1] show how to construct core models with Woodin cardinals and more. But, as Prikry forcing shows, one cannot expect too direct a generalization of Jensen's covering lemma to core models with measurable cardinals.Recall from [8] that if L[] is a core model and α is an ordinal, then either Eα = ∅, or else Eα is an extender over As in [8], we assume here that if Eα is an extender, then Eα is below superstrong type in the sense that the set of generators of Eα is bounded in (crit(Eα)). Let us say that L[] is a lower-part core model iff for every ordinal α, Eα is not a total extender over L[]. In other words, if L[] is a lower-part core model, then no cardinal in L[] is measurable as witnessed by an extender on . Other than the “below superstrong” hypothesis, we impose no bounds on the large cardinal axioms true in the levels of a lower-part core model.

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Scales in K(ℝ) at the end of a weak gap

Journal of Symbolic Logic ◽

10.2178/jsl/1208359049 ◽

2008 ◽

Vol 73 (2) ◽

pp. 369-390 ◽

Cited By ~ 8

Author(s):

J. R. Steel

Keyword(s):

Induction Hypothesis ◽

The Other ◽

Core Model ◽

The Core ◽

Woodin Cardinals ◽

Other Hand

In this note we shall proveTheorem 0.1. Letbe a countably ω-iterable-mouse which satisfies AD, and [α, β] a weak gap of. Supposeis captured by mice with iteration strategies in ∣α. Let n be least such that ; then we have that believes that has the Scale Property.This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated.Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.

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THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

Journal of Symbolic Logic ◽

10.1017/jsl.2019.78 ◽

2019 ◽

Vol 85 (1) ◽

pp. 338-366 ◽

Cited By ~ 1

Author(s):

JUAN P. AGUILERA ◽

SANDRA MÜLLER

Keyword(s):

Set Theory ◽

Consistency Strength ◽

Axiom Of Determinacy ◽

Woodin Cardinals ◽

Inner Model ◽

Image Position ◽

Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

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INNER MODEL THEORETIC GEOLOGY

Journal of Symbolic Logic ◽

10.1017/jsl.2015.64 ◽

2016 ◽

Vol 81 (3) ◽

pp. 972-996 ◽

Cited By ~ 1

Author(s):

GUNTER FUCHS ◽

RALF SCHINDLER

Keyword(s):

Set Theory ◽

Main Theme ◽

Solid Core ◽

Strong Cardinal ◽

The Third ◽

Inner Models ◽

Inner Model ◽

Basic Concepts ◽

Woodin Cardinal ◽

The Universe

AbstractOne of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

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OPTIMAL PROOFS OF DETERMINACY II

Journal of Mathematical Logic ◽

10.1142/s0219061302000175 ◽

2002 ◽

Vol 02 (02) ◽

pp. 227-258 ◽

Cited By ~ 6

Author(s):

ITAY NEEMAN

Keyword(s):

General Lemma ◽

Woodin Cardinals ◽

Base Theory

We present a general lemma which allows proving determinacy from Woodin cardinals. The lemma can be used in many different settings. As a particular application we prove the determinacy of sets in [Formula: see text], n ≥ 1. The assumption we use to prove [Formula: see text] determinacy is optimal in the base theory of [Formula: see text] determinacy.

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