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Large Cardinals, Inner Models, and Determinacy: An Introductory Overview

P. D. Welch
2015 Notre Dame Journal of Formal Logic  
is an inner model with a Large Cardinal Γ".  ...  This programmatic approach to filling out V with wider and thicker inner models depending on the strength of the large cardinals existing in inner models of V , has become known as the "Inner Model program  ... 
doi:10.1215/00294527-2835083 fatcat:su5ti4hikbfgrdzzl3adeh23zq

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  ...  The third author's research has been additionally supported by research grants from the National Science Foundation and from the Simons Foundation.  ... 
arXiv:1111.0856v1 fatcat:tpoacnv27ndgbi6sqigtxsmlci

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.  ...  Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L [G] and V ?  ... 
doi:10.2178/bsl/1164056808 fatcat:vxxqnux3vvewtpl6a57fzsiowi

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.  ...  Jénsson cardinals and Ramsey cardinals are well-known large car- dinals that each imply the existence of 0*.  ... 

Strong Compactness and the Ultrapower Axiom [article]

Gabriel Goldberg
2017 arXiv   pre-print
used to construct and analyze canonical inner models.  ...  More formally, if there is a weak extender model for supercompactness that satisfies UA, then this model absorbs all large cardinals.  ... 
arXiv:1710.03586v1 fatcat:ur3e2ommeba3plidejq7lpy6cm

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  ... 

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.  ...  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.  ... 
doi:10.1007/s00153-011-0264-5 fatcat:ufdzoyy47ve37c5ka4uwfzlzzq

The Landscape of Large Cardinals [article]

Rohan Srivastava
2022 arXiv   pre-print
We assume basic familiarity with set theory, model theory, and the ZFC axioms, though certain concepts will be reviewed as necessary.  ...  We also discuss connections with the Continuum Hypothesis and provide some philosophical reflections on belief in the consistency of large cardinals.  ...  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.  ... 
arXiv:2205.01787v1 fatcat:eiezuj25afeqfggxalpauodkeu

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

2002 Mathematical Reviews  
Ele- mentary embeddings j: V < M from V into inner models M arise naturally in the theory of large cardinals, as such embeddings have formulations in ZFC via ultrafilters and, more generally, exten- ders  ...  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.  ... 
doi:10.1093/jigpal/jzm052 fatcat:6edqvzc53fbzjmgz6kig7qn5qe

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.  ...  Then V* is an outer model of V (V is an inner model of V*) iff the sets of V* include the sets of V and the classes of V* include the classes of V.  ...  Theorem 2 The IMH implies that there is an inner model with measurable cardinals of arbitrarily large Mitchell order. Proof.  ... 
doi:10.2178/jsl/1208359050 fatcat:4aetlhenw5cc7ov5bcrz4uklla

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.  ...  The HOD dichotomy Suppose M is an inner model, λ is a cardinal, and λ ′ is an M -cardinal.  ... 
arXiv:2107.00513v1 fatcat:dg2mbfbfgjhgjcsanzu2b7r6wm

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

2003 Mathematical Reviews  
Strong, Woodin, and superstrong cardinals, the large car- dinals that have been the focus of the recent inner model theory, are formulated in terms of extenders.  ...  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  ... 

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

1995 Mathematical Reviews  
problem”, and applied this in particular to show: If there is is an w-saturated ideal over a and a measurable cardinal, then there is an inner model with a Woodin cardinal.  ...  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  ... 

Foundational implications of the Inner Model Hypothesis

Tatiana Arrigoni, Sy-David Friedman
2012 Annals of Pure and Applied Logic  
The Inner Model Hypothesis (IMH) is a new axiomatic approach in set theory formulated by Sy-D. Friedman.  ...  The purpose of this paper is to illustrate the hypothesis, and discuss it with respect to the current debate on the consequences of independence results in set theory.  ...  axioms, see [10] and [1] . 6 Inner and core models for large cardinals can be regarded as generalizations of the universe L of constructible sets.  ... 
doi:10.1016/j.apal.2012.01.009 fatcat:yk2avi6w3rbtje5a2lco3ohl3y
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