Stacking mice

Jensen, Ronald Björn; Schimmerling, Ernest; Schindler, Ralf; Steel, John R.

doi:10.2178/jsl/1231082314

Cited by 43 publications

(83 citation statements)

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“…If a maximal K c construction converges to a transitive inner model containing the ordinals then the resulting model has covering properties. For instance, the authors of [2] introduced a K c construction and showed that if it converges to a transitive inner model containing the ordinals and κ is an inaccessible cardinal then, assuming there is no inner model with a superstrong cardinal, cf(Ord ∩ S(K c |κ)) ≥ κ where S is the stack of all countably iterable mice extending K c |κ and projecting to κ.…”

Section: V [G] "P[t ] = (P[s]) C " a Set Of Reals A Is Called Univmentioning

confidence: 99%

Covering with universally Baire operators

Sargsyan

2015

Advances in Mathematics

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We introduce a covering conjecture and show that it holds below AD R + "Θ is regular". We then use it to show that in the presence of mild large cardinal axioms, P F A implies that there is a transitive model containing the reals and ordinals and satisfying AD R + "Θ is regular". The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of AD + + θ 0 < Θ.One of the central themes in set theory is to identify canonical inner models which compute successor cardinals correctly. A prototype of such results is Jensen's famous covering theorem which in particular implies that provided 0 # doesn't exist, for every cardinal κ ≥ ω 2 , cf((κ + ) L ) ≥ κ where L is the constructible universe.Clearly "canonical inner model" is open for interpretations. For an inner model theorist, the canonical objects of a set theoretic universe are the sets coded by a mixture of fine extender sequences and the universally Baire sets. Recall that a set of reals is universally Baire if its continuous preimages in all compact Hausdorff spaces have the property of Baire 1 . * 2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. † Keywords: Mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH. ‡ This material is partially based upon work supported by the NSF Grant No DMS-1201348. Part of this paper was written while the author was a Leibniz Fellow at the Mathematisches Forschungsinstitut Oberwolfach.1 A set of reals is said to have the property of Baire if it is different from an open set by a meager set. 1In more set theoretic terms, a set of reals A is κ-universally Baire (or simply κ-uB) if there are trees T and S on κ × ω such that p[T ] = A and for every partial ordering P of size < κ and for every generic g ⊆ P, In most cases, to show that a K c construction converges it is enough to show that the countable submodels of the models produced during the K c construction are countably iterable. This is the content of the iterability conjecture of [19] (see Conjecture 6.5 of [19]). The best partial result is that a K c construction converges provided there is no non-domestic mouse (see [1]). One of the modern new techniques in inner model theory is the core model 2 induction and the work in this paper started by asking how the core model induction can be used to prove instances of the iterability conjecture. We remark that from now on by "K c construction" we mean the K c construction intro-Naturally, an inner model theorist will conjecture that transitive inner models that are closed under all the universally Baire sets of the universe and also are closed under robust embeddings have covering properties, i.e., if M is this hypothetical universe then for many cardinals κ, cf ((κ + ) M ) ≥ κ. Taking this intuition seriously, when we try to prove the iterability conjecture via core model induction, instead of proving instances of iterability conjecture, we ...

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Section: V [G] "P[t ] = (P[s]) C " a Set Of Reals A Is Called Univmentioning

confidence: 99%

Covering with universally Baire operators

Sargsyan

2015

Advances in Mathematics

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“…Baumgartner showed that the proper forcing axiom PFA (see subsection 1.2) is consistent relative to the existence of supercompact cardinals, see Shelah [18,Chapter VII]; it is widely expected that, once fine structural inner model theory has been developed enough, it will be shown that PFA is in fact equiconsistent with a strong large cardinal axiom. The best available result to date is from Jensen-SchimmerlingSchindler-Steel [9], where it is shown that PFA implies the existence of nondomestic mice (see subsection 1.3), see also Andretta-Neeman-Steel [1].…”

Section: Large Cardinal Axioms See Kanamorimentioning

confidence: 99%

“…Many consequences of PFA can be considered natural features of the universe of sets (for example, the failure of the continuum hypothesis, the failure of square principles, the singular cardinal hypothesis, determinacy in L(R), and generic absoluteness of L(R); see for example Bekkali [3], Todorčević [20], Viale [22], and Jensen-Schimmerling-Schindler-Steel [9]) thus providing evidence for its acceptance as a natural extension of ZFC. However, even if one does not consider PFA to be "natural", it seems reasonable that some common features of forcing axioms and other similar strong reflection principles will eventually be considered as natural as large cardinal axioms.…”

Section: Forcing Axiomsmentioning

confidence: 99%

Cardinal preserving elementary embeddings

Caicedo¹

2010

Logic Colloquium 2007

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Abstract. Say that an elementary embedding j : N → M is cardinal preserving if CAR M = CAR N = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V . We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j : V → M . §1. Introduction. This paper is the first of a series attempting to investigate the structure of (not necessarily fine structural) inner models of the set theoretic universe under assumptions of two kinds:1. Forcing axioms, holding either in the universe V of all sets or in both V and the inner model under study, and 2. Agreement between (some of) the cardinals of V and the cardinals of the inner model. I try to be as self-contained as is reasonably possible, given the technical nature of the problems under consideration. The notation is standard, as in Jech [8]. I assume familiarity with inner model theory; for fine structural background and notation, the reader is urged to consult Steel [19] and Mitchell [15].In the remainder of this introduction, I include some general observations on large cardinal theory, forcing axioms, and fine structure, and state the main results of the paper.Consider set theory with the axiom of choice as formalized by the Gödel-Bernays axioms GBC, so we can freely treat proper classes. An inner model (or simply, a model) is a transitive class model M of the Zermelo-Fraenkel ZFC axioms containing all the ordinals. If M is a model and ϕ is a statement, ϕ M is the assertion that ϕ holds in M . If is a definable term, M indicates the interpretation of inside M . Denote by ORD the class of ordinals and by CAR the class of cardinals. The cofinality of an ordinal α is denoted cf(α). All our embeddings are elementary and non-trivial, and the classes involved are inner models; the critical point of such an embedding j is denoted cp(j).

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“…13 See [Fuc08] and [JSSS07] for a further discussion. The strength of a non-domestic mouse exceeds what is known as the AD R hypothesis.…”

Section: Definition 213 ([Men74]mentioning

confidence: 99%

Combined Maximality Principles up to large cardinals

Fuchs

2009

J. symb. log.

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The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high. As a by-product, assuming the consistency of a supercompact cardinal, I show that it is consistent that the least weakly compact cardinal is indestructible.

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Stacking mice

Cited by 43 publications

References 12 publications

Covering with universally Baire operators

Covering with universally Baire operators

Cardinal preserving elementary embeddings

Combined Maximality Principles up to large cardinals

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