Hod mice and the Mouse Set Conjecture

Sargsyan, Grigor

doi:10.1090/memo/1111

Cited by 47 publications

(257 citation statements)

References 35 publications

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“…If N = P (M(U)), N ∈ F and there is some T with U T ∈ U such that N = P (M(T )), then by [20], there is a unique normal such T with U T ∈ U. We may then define π s N,N as the unique map as in (10) for any cofinal branch b ∈ (M sw ) Col(ω,max(s)) through T which "fixes s" as in (8) and (9).…”

Section: (Lemma 22)mentioning

confidence: 99%

“…To this end, let us work in M sw . Let K c be as defined in [5, Definition 2.7], 9 but with the following additivity adjustment: the critical point of an extender added (i.e., crit(G) for G as in [5, Definition 2.7 (a)]) is supposed to be above κ +Msw . In the light of Lemma 2.9 (a), the paper [11] shows that K c is fully iterable (inside M sw ).…”

Section: (Lemma 22)mentioning

confidence: 99%

“…Σ satisfies branch condensation, cf. [9,Definition 2.14], and (3) Σ is positional, cf. [9,Definition 2.35(4)].…”

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confidence: 99%

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Varsovian Models I

Sargsyan¹,

Schindler²

2018

J. symb. log.

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Let Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and Msw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of Msw via its iteration strategy Σ.We here show that Msw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence Msw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}},{\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it.These results generalize to models of the form Ms (x) for a cone of reals x, where Ms (x) denotes the least iterable inner model with a strong cardinal containing x. In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K,{\rm{\bar{\Sigma }}}]$.

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Section: (Lemma 22)mentioning

confidence: 99%

Section: (Lemma 22)mentioning

confidence: 99%

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Varsovian Models I

Sargsyan¹,

Schindler²

2018

J. symb. log.

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“…In unpublished work, Woodin strengthened the conclusion of Theorem 5.1 to get AD R + "Θ is regular" in the model L(uB , R) V [g] , making a positive answer to 5.4(1) plausible. However, 5.4(2) might be a more reasonable target for current inner-model-theoretic techniques such as those developed by Sargsyan [12]. One might also hope to do without the hypothesis that is measurable.…”

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confidence: 99%

Universally Baire Sets and Generic Absoluteness

Wilson¹

2017

J. symb. log.

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We prove several equivalences and relative consistency results involving notions of generic absoluteness beyond Woodin's (Σ 2 1 ) uB λ generic absoluteness for a limit of Woodin cardinals λ. In particular, we prove that two-step ∃ R (Π 2 1 ) uB λ generic absoluteness below a measurable cardinal that is a limit of Woodin cardinals has high consistency strength, and that it is equivalent with the existence of trees for (Π 2 1 ) uB λ formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many "failures of covering" for the models L(T, Vα) below a measurable cardinal.

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“…This leads to the core model induction, cf. [SchSt∞] and [Sa15], and specifically to Strong Mouse Set Capturing, cf. [Sa15, p. 8].…”

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confidence: 99%