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A multiverse perspective on the axiom of constructiblity
[article]

2012
*
arXiv
*
pre-print

The argument appears to lose its force,

arXiv:1210.6541v1
fatcat:5j6ixypdabalngbnsah4fw364a
*in*contrast, on an upwardly extensible concept of set,*in*light of the various facts showing that*models*of set theory generally have extensions to*models*of*V*= ... I shall argue that the commonly held*V*not equal L via maximize position, which rejects the axiom of constructibility*V*= L on the basis that it is restrictive, implicitly takes*a*stand*in*the pluralist ... the full*universe**V*can be end-*extended*to*a**universe**in*which*V*= L holds; and so on. ...##
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Superstrong and other large cardinals are never Laver indestructible
[article]

2014
*
arXiv
*
pre-print

<\kappa-closed forcing Q

arXiv:1307.3486v2
fatcat:nsol2cdsebbhll5naa65mphuo4
*in*V_\theta, the*cardinal*\kappa\ will exhibit none of the*large**cardinal*properties with target \theta\ or larger. ...*In*fact, all these*large**cardinal*properties are superdestructible: if \kappa\ exhibits any of them, with corresponding target \theta, then*in*any forcing extension arising from nontrivial strategically ... of ground*models*over which the*universe**V*was obtained by forcing. ...##
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A New Hope for the Symbolic, for the Subject

2020
*
Filozofski Vestnik
*

The argument presented is that the prevailing and sustained incoherence of the mathematical ontology (i.e. set theory) underscores

doi:10.3986/fv.41.2.14
fatcat:xwyyhcxfpncqneml2h7a4pucfu
*a*contemporary deficit of humanity's symbolic organization which,*in*turn ... Hugh Woodin (mathematician) and colleagues*in*the present moment,*in*the context of the mathematical ontology proposed and elaborated by Alain Badiou (philosopher). ... As such, the*extender**models*are constructed as refinements of*V*, which preserve enough*extenders*from*V*to witness that the*large**cardinal*axiom holds. 102 By the year 2000, the inner*model*program ...##
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Page 3253 of Mathematical Reviews Vol. , Issue 96f
[page]

1996
*
Mathematical Reviews
*

*Without*any hypothesis

*in*

*V*, Dodd and Jensen developed K as

*a*

*model*that has as much strength as possible

*without*actually containing

*a*measurable

*cardinal*, and

*extended*the famous Jensen covering theorem ... Jensen: starting with

*a*measurable

*cardinal*

*in*the real

*universe*

*V*, one can construct relative to it to get

*a*nice L-like

*model*L[U]

*in*which measurability is retained. ...

##
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Large cardinals and definable well-orders on the universe

2009
*
Journal of Symbolic Logic (JSL)
*

We use

doi:10.2178/jsl/1243948331
fatcat:7v6m4y4djzhlhgyykdma7rnjme
*a*reverse Easton forcing iteration to obtain*a**universe*with*a*definable well-order, while preserving the GCH and proper classes of*a*variety of very*large**cardinals*. ... By choosing the*cardinals*at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the*large**cardinal*properties*without*having to meet any non-trivial master conditions ... This research was conducted at the Kurt Gödel Research Center for Mathematical Logic, The*University*of Vienna, with support from the Austrian Science Fund (FWF) project P16790-N04. ...##
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Strong Axioms of Infinity and the Search for V

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

Gödel's Axiom of Constructibility,

doi:10.1142/9789814324359_0023
fatcat:fqtwho4535drfltf56g7ur2gxe
*V*= L, provides*a*conception of the*Universe*of Sets which is perfectly concise modulo only*large**cardinal*axioms which are strong axioms of infinity. ... However the axiom*V*= L limits the*large**cardinal*axioms which can hold and so the axiom is false. ... Then there is*a*borel function π : R → R such that*A*= π −1 [B]. (transitive)*model*of ZFC (of course one cannot prove such*a*set exists*without*appealing to*large**cardinal*axioms). ...##
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The Axiom of Infinity and Transformations j: V → V

2010
*
Bulletin of Symbolic Logic
*

The resulting transformation

doi:10.2178/bsl/1264433797
fatcat:kvawthvypffkjiuxsmftux3izu
*V*→*V*is strong enough to account for virtually all*large**cardinals*, but is at the same time*a*natural generalization of an assertion about transformations*V*→*V*known to be ... An axiom asserting the existence of*a**large**cardinal*can naturally be viewed as*a*strong Axiom of Infinity. ...*In**a*similar vein, from*a**model*of ZFC together with arbitrarily*large*weak Reinhardt*cardinals*, one can obtain*a**model*of ZFC*in*which the weak Reinhardt*cardinals*are bounded*in*the*universe*: Starting ...##
###
An equiconsistency for universal indestructibility

2010
*
Journal of Symbolic Logic (JSL)
*

The equiconsistency is relative to

doi:10.2178/jsl/1264433923
fatcat:q4ytle6ucjcrhpahtmmfk7iqo4
*a**cardinal*weaker*in*consistency strength than*a*Woodin*cardinal*, Stewart Baldwin's notion of hyperstrong*cardinal*. ... We obtain an equiconsistency for*a*weak form of*universal*indestructibility for strongness. ... Thus, for any λ which is sufficiently*large*, there is an elementary embedding witnessing the λ strongness of κ*in*K generated by*a*(κ, λ)-*extender*such that*in*the target*model*, κ is*a*strong*cardinal*. ...##
###
The Landscape of Large Cardinals
[article]

2022
*
arXiv
*
pre-print

By

arXiv:2205.01787v1
fatcat:eiezuj25afeqfggxalpauodkeu
*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. ...##
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Two remarks on Merimovich's model of the total failure of GCH
[article]

2021
*
arXiv
*
pre-print

*model*N of M such that M and N have the same bounded subsets of λ, λ is

*a*singular

*cardinal*

*in*N, (λ^+i)^N=(λ^+i)^M, for i=1,2,3, and N 2^λ=λ^+. ... Let M denote the Merimovich's

*model*

*in*which for each infinite

*cardinal*λ, 2^λ=λ^+3. ... Now, working

*in*

*V*[G κ ξ +1 ], by [1] , we can pick some elementary submodel

*A*of the

*large*part of the

*universe*, so that

*A*⊇

*V*κ ξ has size κ ξ ,

*A*contains all relevant information and Pǭ ∩

*A*⋖ Pǭ. ...

##
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Internal Consistency and the Inner Model Hypothesis

2006
*
Bulletin of Symbolic Logic
*

There is

doi:10.2178/bsl/1164056808
fatcat:vxxqnux3vvewtpl6a57fzsiowi
*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*. ... 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*. ... It is conjectured that core*model*theory can be*extended*from strong*cardinals*to Woodin*cardinals*,*without*any*large**cardinal*assumptions. ...##
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Foundational implications of the Inner Model Hypothesis

2012
*
Annals of Pure and Applied Logic
*

The Inner

doi:10.1016/j.apal.2012.01.009
fatcat:yk2avi6w3rbtje5a2lco3ohl3y
*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. ... If*a*statement φ*without*parameters holds*in*an inner*universe*of some outer*universe*of*V*(i.e.,*in*some*universe*compatible with*V*), then it already holds*in*some inner*universe*of*V*. 17 See [15 ...##
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Model Theoretic Characterizations of Large Cardinals Revisited
[article]

2022
*
arXiv
*
pre-print

*In*[Bon20],

*model*theoretic characterizations of several established

*large*

*cardinal*notions were given. ... We continue this work, by establishing such characterizations for Woodin

*cardinals*(and variants), various virtual

*large*

*cardinals*, and subtle

*cardinals*. ... We can also assume that our

*universe*is the

*V*κ of

*a*much larger ZFC-

*model*

*in*which κ is subtle and the classes

*in*this case are the

*V*κ+1 of this

*model*. ...

##
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Reinhardt cardinals in inner models
[article]

2021
*
arXiv
*
pre-print

*A*

*cardinal*is weakly Reinhardt if it is the critical point of an elementary embedding from the

*universe*of sets into

*a*

*model*that contains the double powerset of every ordinal. ... This note establishes the equiconsistency of

*a*proper class of weakly Reinhardt

*cardinals*with

*a*proper class of Reinhardt

*cardinals*

*in*the context of second-order set theory

*without*the Axiom of Choice ... Even though we work

*without*AC, for us

*a*

*cardinal*is an ordinal number that is not

*in*bijection with any smaller ordinal. ...

##
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Large Cardinals Beyond Choice

2019
*
Bulletin of Symbolic Logic
*

The HOD Dichotomy Theorem states that if there is an

doi:10.1017/bsl.2019.28
fatcat:3f7a4mqr2fe2vnetk3upm3b5cu
*extendible**cardinal*, , then either HOD is "close" to*V*(*in*the sense that it correctly computes successors of singular*cardinals*greater than ) or HOD ... This conjecture implies that (assuming the existence of an*extendible**cardinal*) that the first alternative holds-HOD is "close" to*V*. This is the future*in*which pattern prevails. ... Theorem 7. 8 ( 8*Universality*). Suppose that N is*a*weak*extender**model*of the supercompactness of κ. ...
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