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  1. A finite family weak square principle.Ernest Schimmerling - 1999 - Journal of Symbolic Logic 64 (3):1087-1110.
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  • Jónsson cardinals, erdös cardinals, and the core model.W. J. Mitchell - 1999 - Journal of Symbolic Logic 64 (3):1065-1086.
    We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ-Erdös in K. In the absence of the Steel core model K we prove the same conclusion for any model L[E] such that either V = L[E] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is (...)
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  • Square In Core Models, By, Pages 305 -- 314.Ernest Schimmerling & Martin Zeman - 2001 - Bulletin of Symbolic Logic 7 (3):305-314.
    We prove that in all Mitchell-Steel core models, □k holds for all k. From this we obtain new consistency strength lower bounds for the failure of □k if k is either singular and countably closed, weakly compact, or measurable. Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □k holds iff k is not subcompact.
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  • Two Upper Bounds on Consistency Strength of $negsquare{aleph_{omega}}$ and Stationary Set Reflection at Two Successive $aleph{n}$.Martin Zeman - 2017 - Notre Dame Journal of Formal Logic 58 (3):409-432.
    We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...)
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  • Dodd parameters and λ-indexing of extenders.Martin Zeman - 2004 - Journal of Mathematical Logic 4 (01):73-108.
    We study generalizations of Dodd parameters and establish their fine structural properties in Jensen extender models with λ-indexing. These properties are one of the key tools in various combinatorial constructions, such as constructions of square sequences and morasses.
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  • ◇ at Mahlo cardinals.Martin Zeman - 2000 - Journal of Symbolic Logic 65 (4):1813-1822.
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  • When cardinals determine the power set: inner models and Härtig quantifier logic.Jouko Väänänen & Philip D. Welch - forthcoming - Mathematical Logic Quarterly.
    We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has (...)
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  • Core models with more Woodin cardinals.J. R. Steel - 2002 - Journal of Symbolic Logic 67 (3):1197-1226.
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  • The self-iterability of L[E].Ralf Schindler & John Steel - 2009 - Journal of Symbolic Logic 74 (3):751-779.
    Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal K which is not a limit of Woodin cardinals there is some cutpoint t K > a>ω1 are cardinals, then ◊$_{K.\lambda }^* $ holds true, and if in addition λ is regular, then ◊$_{K.\lambda }^* $ holds true.
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  • The Jensen covering property.E. Schimmerling & W. H. Woodin - 2001 - Journal of Symbolic Logic 66 (4):1505-1523.
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  • The definability of E in self-iterable mice.Farmer Schlutzenberg - 2023 - Annals of Pure and Applied Logic 174 (2):103208.
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  • The core model for almost linear iterations.Ralf-Dieter Schindler - 2002 - Annals of Pure and Applied Logic 116 (1-3):205-272.
    We introduce 0• as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “ does not exist”. Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for determining the exact consistency strength of the assumption in the statement of the 12th Delfino problem pp. 221–224)).
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  • Square in core models.Ernest Schimmerling & Martin Zeman - 2001 - Bulletin of Symbolic Logic 7 (3):305-314.
    We prove that in all Mitchell-Steel core models, □ κ holds for all κ. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of □ κ if κ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □ κ holds (...)
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  • On a Chang Conjecture. II.Ralf-Dieter Schindler - 1998 - Archive for Mathematical Logic 37 (4):215-220.
    Continuing [7], we here prove that the Chang Conjecture $(\aleph_3,\aleph_2) \Rightarrow (\aleph_2,\aleph_1)$ together with the Continuum Hypothesis, $2^{\aleph_0} = \aleph_1$ , implies that there is an inner model in which the Mitchell ordering is $\geq \kappa^{+\omega}$ for some ordinal $\kappa$.
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  • Characterization of □κin core models.Ernest Schimmerling & Martin Zeman - 2004 - Journal of Mathematical Logic 4 (01):1-72.
    We present a general construction of a □κ-sequence in Jensen's fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.
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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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  • Mice with finitely many Woodin cardinals from optimal determinacy hypotheses.Sandra Müller, Ralf Schindler & W. Hugh Woodin - 2020 - Journal of Mathematical Logic 20 (Supp01):1950013.
    We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...)
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  • Mice with finitely many Woodin cardinals from optimal determinacy hypotheses.Sandra Müller, Ralf Schindler & W. Hugh Woodin - 2020 - Journal of Mathematical Logic 20 (Supp01):1950013.
    We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...)
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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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  • Projective uniformization revisited.Kai Hauser & Ralf-Dieter Schindler - 2000 - Annals of Pure and Applied Logic 103 (1-3):109-153.
    We give an optimal lower bound in terms of large cardinal axioms for the logical strength of projective uniformization in conjuction with other regularity properties of projective sets of real numbers, namely Lebesgue measurability and its dual in the sense of category . Our proof uses a projective computation of the real numbers which code inital segments of a core model and answers a question in Hauser.
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  • Silver type theorems for collapses.Moti Gitik - 2020 - Annals of Pure and Applied Logic 171 (9):102825.
    Let κ be a cardinal of cofinality \omega_1 witnessed by a club of cardinals (κ_\alpha | \alpha < \omega_1) . We study Silver's type effects of collapsing of κ^+_\alphas 's on κ^+ . A model in which κ^+_\alphas 's (and also κ^+) are collapsed on a stationary co-stationary set is constructed.
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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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  • The strength of choiceless patterns of singular and weakly compact cardinals.Daniel Busche & Ralf Schindler - 2009 - Annals of Pure and Applied Logic 159 (1-2):198-248.
    We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that , the Axiom of Determinacy, holds in the of a generic extension of : every uncountable cardinal is singular, and every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.
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  • On the Consistency Strength of Two Choiceless Cardinal Patterns.Arthur W. Apter - 1999 - Notre Dame Journal of Formal Logic 40 (3):341-345.
    Using work of Devlin and Schindler in conjunction with work on Prikry forcing in a choiceless context done by the author, we show that two choiceless cardinal patterns have consistency strength of at least one Woodin cardinal.
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  • Approachable free subsets and fine structure derived scales.Dominik Adolf & Omer Ben-Neria - 2024 - Annals of Pure and Applied Logic 175 (7):103428.
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