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  1. Determinacy in strong cardinal models.P. D. Welch - 2011 - Journal of Symbolic Logic 76 (2):719 - 728.
    We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. $\mathrm{D}\mathrm{e}\mathrm{t}\text{\hspace{0.17em}}({\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D})$ ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-1 mice and hence inner models with proper classes of strong cardinals. where ${\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D}\phantom{\rule{0ex}{0ex}}$ (AQI) is the pointclass of boldface ${\mathrm{\Pi }}_{1}^{1}$ -inductive (respectively arithmetically quasi-inductive) sets of reals.
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  • Iterates of the Core Model.Ralf Schindler - 2006 - Journal of Symbolic Logic 71 (1):241 - 251.
    Let N be a transitive model of ZFC such that ωN ⊂ N and P(R) ⊂ N. Assume that both V and N satisfy "the core model K exists." Then KN is an iterate of K. i.e., there exists an iteration tree J on K such that J has successor length and $\mathit{M}_{\infty}^{\mathit{J}}=K^{N}$. Moreover, if there exists an elementary embedding π: V → N then the iteration map associated to the main branch of J equals π ↾ K. (This answers (...)
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  • Covering at limit cardinals of K.William J. Mitchell & Ernest Schimmerling - 2023 - Journal of Mathematical Logic 24 (1).
    Assume that there is no transitive class model of [Formula: see text] with a Woodin cardinal. Let [Formula: see text] be a singular ordinal such that [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] is a regular cardinal in K. Then [Formula: see text] is a measurable cardinal in K. Moreover, if [Formula: see text], then [Formula: see text].
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  • $K$ without the measurable.Ronald Jensen & John Steel - 2013 - Journal of Symbolic Logic 78 (3):708-734.
    We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definablecore modelthat is close toVin various ways.
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  • Universally baire sets and definable well-orderings of the reals.Sy D. Friedman & Ralf Schindler - 2003 - Journal of Symbolic Logic 68 (4):1065-1081.
    Let n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n - 2 strong cardinals) that every $\Sigma_n^1-set$ of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses "David's trick" in the presence of inner models with strong cardinals.
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  • Universally Baire sets and definable well-orderings of the reals.S. Y. D. Friedman & Ralf Schindler - 2003 - Journal of Symbolic Logic 68 (4):1065-1081.
    Let n ≥ 3 be an integer. We show that it is consistent that every σ1n-set of reals is universally Baire yet there is a projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.
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  • Supercomplete extenders and type 1 mice: Part I.Q. Feng & R. Jensen - 2004 - Annals of Pure and Applied Logic 128 (1-3):1-73.
    We study type 1 premice equipped with supercomplete extenders. In this paper, we show that such premice are normally iterable and all normal iteration trees of type 1 premice has a unique cofinal branch. We give a construction of an KC type model using supercomplete type 1 extenders.
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  • Projective Well-orderings of the Reals.Andrés Eduardo Caicedo & Ralf Schindler - 2006 - Archive for Mathematical Logic 45 (7):783-793.
    If there is no inner model with ω many strong cardinals, then there is a set forcing extension of the universe with a projective well-ordering of the reals.
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  • Making all cardinals almost Ramsey.Arthur W. Apter & Peter Koepke - 2008 - Archive for Mathematical Logic 47 (7-8):769-783.
    We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...)
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  • A Universal Extender Model Without Large Cardinals In V.William Mitchell & Ralf Schindler - 2004 - Journal of Symbolic Logic 69 (2):371-386.
    We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model Kc which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and is universal with respect to set sized premice.
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