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A multiverse perspective on the axiom of constructiblity [article]

Joel David Hamkins
2012 arXiv   pre-print
The argument appears to lose its force, 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.  ... 
arXiv:1210.6541v1 fatcat:5j6ixypdabalngbnsah4fw364a

Superstrong and other large cardinals are never Laver indestructible [article]

Joan Bagaria and Joel David Hamkins and Konstantinos Tsaprounis and Toshimichi Usuba
2014 arXiv   pre-print
<\kappa-closed forcing Q 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.  ... 
arXiv:1307.3486v2 fatcat:nsol2cdsebbhll5naa65mphuo4

A New Hope for the Symbolic, for the Subject

Norma M. Hussey
2020 Filozofski Vestnik  
The argument presented is that the prevailing and sustained incoherence of the mathematical ontology (i.e. set theory) underscores 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  ... 
doi:10.3986/fv.41.2.14 fatcat:xwyyhcxfpncqneml2h7a4pucfu

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.  ... 

Large cardinals and definable well-orders on the universe

Andrew D. Brooke-Taylor
2009 Journal of Symbolic Logic (JSL)  
We use 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.  ... 
doi:10.2178/jsl/1243948331 fatcat:7v6m4y4djzhlhgyykdma7rnjme

Strong Axioms of Infinity and the Search for V

W. Hugh Woodin
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
Gödel's Axiom of Constructibility, 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).  ... 
doi:10.1142/9789814324359_0023 fatcat:fqtwho4535drfltf56g7ur2gxe

The Axiom of Infinity and Transformations j: V → V

Paul Corazza
2010 Bulletin of Symbolic Logic  
The resulting transformation VV is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations VV 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  ... 
doi:10.2178/bsl/1264433797 fatcat:kvawthvypffkjiuxsmftux3izu

An equiconsistency for universal indestructibility

Arthur W. Apter, Grigor Sargsyan
2010 Journal of Symbolic Logic (JSL)  
The equiconsistency is relative to 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.  ... 
doi:10.2178/jsl/1264433923 fatcat:q4ytle6ucjcrhpahtmmfk7iqo4

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.  ... 
arXiv:2205.01787v1 fatcat:eiezuj25afeqfggxalpauodkeu

Two remarks on Merimovich's model of the total failure of GCH [article]

Mohammad Golshani
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 AV κ ξ has size κ ξ , A contains all relevant information and Pǭ ∩ A ⋖ Pǭ.  ... 
arXiv:2102.00748v1 fatcat:bbmba6ccorgyva56yuv55kdgba

Internal Consistency and the Inner Model Hypothesis

Sy-David Friedman
2006 Bulletin of Symbolic Logic  
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.  ...  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.  ... 
doi:10.2178/bsl/1164056808 fatcat:vxxqnux3vvewtpl6a57fzsiowi

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.  ...  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  ... 
doi:10.1016/j.apal.2012.01.009 fatcat:yk2avi6w3rbtje5a2lco3ohl3y

Model Theoretic Characterizations of Large Cardinals Revisited [article]

Will Boney, Stamatis Dimopoulos, Victoria Gitman, Menachem Magidor
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.  ... 
arXiv:2202.00549v1 fatcat:igong7m4nrfkbmbmw2zwzhjhfy

Reinhardt cardinals in inner models [article]

Gabriel Goldberg
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.  ... 
arXiv:2107.13119v1 fatcat:sm7ada3ronfkbeo2eaibp7i7eq

Large Cardinals Beyond Choice

Joan Bagaria, Peter Koellner, W. Hugh Woodin
2019 Bulletin of Symbolic Logic  
The HOD Dichotomy Theorem states that if there is an 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.  ... 
doi:10.1017/bsl.2019.28 fatcat:3f7a4mqr2fe2vnetk3upm3b5cu
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