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### Inner Models for Large Cardinals [chapter]

William J. Mitchell
2012 Handbook of the History of Logic
models for large cardinals during 6 the last part of the last century, beginning at the end of Kanamori's chapter [Kanamori, 2010b], which is to say at about 1965.  ...  The model L[A] can be viewed as the smallest model which contains the structure of the set A, and since this chapter is concerned with models including 15 large cardinal structure, essentially all models  ...  DEVELOPMENT OF INNER MODELS WITHOUT FINE STRUCTURE Large cardinals in L The study of L-like models for large cardinals naturally starts with L itself.  ...

### Inner models with large cardinal features usually obtained by forcing

Arthur W. Apter, Victoria Gitman, Joel David Hamkins
2011 Archive for Mathematical Logic
We construct a variety of inner models exhibiting large cardinal features usually obtained by forcing.  ...  For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal.  ...  Must there also be such inner models for the very large large cardinals? Test Question 6.  ...

### Inner Models for Large Cardinals [chapter]

Springer Monographs in Mathematics
models for large cardinals during 6 the last part of the last century, beginning at the end of Kanamori's chapter [Kanamori, 2010b], which is to say at about 1965.  ...  The model L[A] can be viewed as the smallest model which contains the structure of the set A, and since this chapter is concerned with models including 15 large cardinal structure, essentially all models  ...  DEVELOPMENT OF INNER MODELS WITHOUT FINE STRUCTURE Large cardinals in L The study of L-like models for large cardinals naturally starts with L itself.  ...

### Internal Consistency and the Inner Model Hypothesis

Sy-David Friedman
2006 Bulletin of Symbolic Logic
We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals.  ...  There is a forcing extension L[G] of L in which GCH fails at every regular cardinal. Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals.  ...  Thus a statement ϕ is internally consistent relative to large cardinals iff Icon(ZFC + ϕ) follows from Icon(ZFC + LC) for some large cardinal axiom LC.  ...

### Page 4409 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews
Part of the development of inner model theory for large cardinals has been to establish the converse in these models, that Jonsson cardinals are Ramsey.  ...  cardinal is Ramsey in situations at the level of in- ner models for Woodin cardinals.  ...

### Page 3919 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews
The inner model program is to capture stronger and stronger large cardinal hypotheses by finding canonical inner models for them, and producing core models is a way to get at more subtle consequences for  ...  An inner model for a large cardinal hypothesis is a “minimal” transitive model containing all the ordinals in which that hypothesis obtains, an L-like model exhibiting the minimal commitments of the hypothesis  ...

### Strong Compactness and the Ultrapower Axiom [article]

Gabriel Goldberg
2017 arXiv   pre-print
More formally, if there is a weak extender model for supercompactness that satisfies UA, then this model absorbs all large cardinals.  ...  The following is therefore a precise test question for the inner model program. Question 1. Is UA consistent with a supercompact cardinal?  ...

### The Landscape of Large Cardinals [article]

Rohan Srivastava
2022 arXiv   pre-print
By a large cardinal, we mean any cardinal κ whose existence is strong enough of an assumption to prove the consistency of ZFC.  ...  The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory.  ...  Any clarity this paper provides about large cardinals ought to be attributed to him, and any deficiencies attributed to my inability to properly distill his insights.  ...

### Inner models with large cardinal features usually obtained by forcing [article]

Arthur Apter and Victoria Gitman and Joel David Hamkins
2011 arXiv   pre-print
We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals.  ...  If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly  ...  Must there also be such inner models for the very large large cardinals? Test Question 6.  ...

### Page 7030 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews
The results appeal to techniques for preservation and destruction of large cardinals due to Joel Hamkins A.  ...  Loosely speaking, core models are extensions of L that are maximal inner models in that they exhibit regularity of structure in a maximal sense “just before” the existence of a large cardinal.  ...

### Large Cardinals and Topology: a Short Retrospective and Some New Results

S. G. Da Silva
2007 Logic Journal of the IGPL
models with measurable cardinals.  ...  The author's purpose is to present some applications of large cardinals in general topology, pointing out that there are several topological problems that cannot be settled without dealing with inaccessible  ...  Acknowledgements The author would like to thank the referee for his (or her) careful reading of the paper and for providing several useful comments and suggestions.  ...

### On the consistency strength of the inner model hypothesis

Sy-David Friedman, Philip Welch, W. Hugh Woodin
2008 Journal of Symbolic Logic (JSL)
The Inner Model Hypothesis (IMH) and the Strong Inner Model Hypothesis (SIMH) were introduced in [4]. In this article we establish some upper and lower bounds for their consistency strength.  ...  The Inner Model Hypothesis asserts that the universe has been maximised with respect to internal consistency: The Inner Model Hypothesis (IMH): If a statement φ without parameters holds in an inner model  ...  Theorem 2 The IMH implies that there is an inner model with measurable cardinals of arbitrarily large Mitchell order. Proof.  ...

### Strongly compact cardinals and ordinal definability [article]

Gabriel Goldberg
2021 arXiv   pre-print
We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal.  ...  We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega-Jonsson algebra.  ...  By completely different techniques, however, Woodin [6] showed that such a dichotomy holds for the (noncanonical) inner model HOD under large cardinal hypotheses. Theorem (Woodin's HOD dichotomy).  ...

### Page 27 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews
Since then, inner models of the form L[E], where E is a coherent sequence of extenders, have become the focus of concerted investigation as the canonical inner models of various large cardinals hypothe  ...  These models are, loosely speaking, minimal in the sense of having just enough structure either for affirming the existence of these large cardinals or (in the case of the “core models”) for having minimal  ...

### Page 7159 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews
This paper studies minimal large cardinal hypotheses for establish- ing characterizations of those B C @, constructible from a real, i.e.  ...  Woodin cardinals were much stronger than the cardinals previously encompassed by inner models, and for reasons having to do with the putative limits on definability imposed by them, they became the goal  ...
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