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Inner Models for Large Cardinals
[chapter]

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

2011
*
Archive for Mathematical Logic
*

We construct a variety of

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

2006
*
Bulletin of Symbolic Logic
*

We say that a statement is internally consistent iff it holds in some

doi:10.2178/bsl/1164056808
fatcat:vxxqnux3vvewtpl6a57fzsiowi
*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]

2017
*
arXiv
*
pre-print

More formally, if there is a weak extender

arXiv:1710.03586v1
fatcat:ur3e2ommeba3plidejq7lpy6cm
*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]

2022
*
arXiv
*
pre-print

By a

arXiv:2205.01787v1
fatcat:eiezuj25afeqfggxalpauodkeu
*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]

2011
*
arXiv
*
pre-print

We construct a variety of

arXiv:1111.0856v1
fatcat:tpoacnv27ndgbi6sqigtxsmlci
*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

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

2008
*
Journal of Symbolic Logic (JSL)
*

The

doi:10.2178/jsl/1208359050
fatcat:4aetlhenw5cc7ov5bcrz4uklla
*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]

2021
*
arXiv
*
pre-print

We improve the

arXiv:2107.00513v1
fatcat:dg2mbfbfgjhgjcsanzu2b7r6wm
*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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