A universal extender model without large cardinals in <i>V</i>

Mitchell, William J.; Schindler, Ralf

doi:10.2178/jsl/1082418531

Cited by 16 publications

(53 citation statements)

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“…The following simple lemma implies that N x is a universal weasel in the sense of N * x . Its proof is an adaptation of the proof by Mitchell and Schindler in [11] that K c is universal.…”

Section: The L[ E]-model N Xmentioning

confidence: 92%

Derived Models Associated to Mice

Steel

2008

Computational Prospects of Infinity

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We shall present some results on the the properties of derived models of mice, and on the existence of mice with large derived models. The basic plan of the paper is:Sections 1-5. Introduction. Preliminary definitions and background material. The iteration independence of the derived model associated to a mouse. The mouse set conjecture in derived models associated to mice.Sections 6-8 The Solovay sequence in derived models associated to mice. A method of "translating away" extenders overlapping Woodin cardinals in mice.Sections 9-14. Some partial results on capturing Σ 2 1 truths in AD + models by mice. (That is, partial results on the mouse set conjecture.)Section 15 A theorem of Woodin on the consistency strength of AD + + θ 0 < θ. Section 16The global mouse set conjecture implies its local version.Section 17 Capturing sets of reals in R-mice.

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“…The following simple lemma implies that N x is a universal weasel in the sense of N * x . Its proof is an adaptation of the proof by Mitchell and Schindler in [11] that K c is universal.…”

Section: The L[ E]-model N Xmentioning

confidence: 92%

Derived Models Associated to Mice

Steel

2008

Computational Prospects of Infinity

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“…Another is a use of the Woodin cardinal, to obtain a rich collection of coarse extenders into which we can embed. The latter relies on some ideas that trace back to Mitchell-Schindler [5], and has precursors in the proof of Lemma 11.1 of Steel [15] for universality of fully backgrounded constructions up to a Woodin cardinal, and in an argument of Steel (previously published only in the context of HOD-mice, in Sargsyan [9,Lemma 5.2]) obtaining high cofinality of the stack over a fully backgrounded construction up to a Woodin cardinal.…”

Section: δ Is Subcompact and Sbh δ Holdsmentioning

confidence: 99%

Equiconsistencies at subcompact cardinals

Neeman

Steel

2015

Arch. Math. Logic

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Abstract. We present equiconsistency results at the level of subcompact cardinals. Assuming SBH δ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both 2(δ) and 2 δ fail, then δ is subcompact in a class inner model. If in addition 2(δ + ) fails, we prove that δ is Π 2 1 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π 2 1 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ +(n) supercompact for all n < ω.We state some results at this level, and indicate how they are proved.MSC 2010: 03E45, 03E55. Keywords: subcompact cardinals, inner models, long extenders, coherent sequences, square.We dedicate this paper to Rich Laver, a brilliant mathematician and a kind and generous colleague. §1. Introduction. We present equiconsistency results at the level of subcompact cardinals. The methods we use extend further, to levels which are interlaced with the axioms κ is κ +(n) supercompact, for n < ω. The extensions will be carried out in a sequel to this paper, Neeman-Steel [7], but we indicate in this paper some of the main ideas involved.Our reversals assume iterability for countable substructures of V . By a strictly short extender we mean an extender F which maps its critical point strictly above its strength. Let SBH δ be the statement that for every countable H ≺ V δ , the good player has a winning strategy in the full iteration game of length ω 1 + 1 on the transitive collapse of H, with only strictly short extenders allowed, and with the iteration trees restricted to linear compositions of normal, non-overlapping, plus 2 trees. This is a special case of the Strategic Branches Hypothesis of .Recall that a sequencewhere Lim(C β ) is the set of limit points of C β . A thread through a coherent sequence is a club C ⊆ δ, so that for every α ∈ Lim(C) ∩ Z, C α = C ∩ α. The statement that there is a coherent sequence on δ that cannot be threaded is denoted P(δ). We will be

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“…Let T and U be the resulting trees on (K c ) Mz and N of length λ + 1 for some ordinal λ. It follows that (K c ) Mz wins the comparison by universality of the premouse (K c ) Mz inside M z (see Section 3 in [MSch04]). Therefore we have that there exists an iterate R = M T λ of (K c ) Mz and an iterate N * = M U λ of N such that N * R. Moreover there is no drop on the main branch on the N -side of the coiteration.…”

Section: Order Of Constructionmentioning

confidence: 95%

“…Therefore it follows that R * is not (2n − 1)-small and thus MT b is not (2n − 1)-small. By the iterability proof of Chapter 9 in [St96] (adapted as in [MSch04]) applied inside M z we can re-embed the model MT b into a model of the (K c ) Mzconstruction. This yields that (K c ) Mz is not (2n − 1)-small, contradicting our assumption that it is (2n − 1)-small.…”

Section: Order Of Constructionmentioning

confidence: 99%

“…This coiteration is successful using Lemma 2.11 (2) because we have N * |δ N * = R|δ N * and R and M # 2n−2 (N * |δ N * ) are both iterable above δ N * , using Subclaim 1 for the iterate R of (K c ) Mz . If there is no drop on the main branch in T , then R = M T λ wins the comparison by universality of (K c ) Mz inside M z (again see Section 3 in [MSch04]). If there is a drop on the main branch in T , then R also wins the comparison, because in this case we have that ρ ω (R) < δ N * and ρ ω (M # 2n−2 (N * |δ N * )) = δ N * .…”

Section: Order Of Constructionmentioning

confidence: 99%

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