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Large Cardinals, Inner Models, and Determinacy: An Introductory Overview
2015
Notre Dame Journal of Formal Logic
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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]
2011
arXiv
pre-print
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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
2006
Bulletin of Symbolic Logic
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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
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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]
2017
arXiv
pre-print
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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
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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
2011
Archive for Mathematical Logic
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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]
2022
arXiv
pre-print
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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
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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
2007
Logic Journal of the IGPL
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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
2008
Journal of Symbolic Logic (JSL)
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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]
2021
arXiv
pre-print
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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
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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
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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
2012
Annals of Pure and Applied Logic
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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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