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A Natural Model of the Multiverse Axioms

Victoria Gitman, Joel David Hamkins
2010 Notre Dame Journal of Formal Logic  
If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of [Hama].  ...  The research of the first author has been supported by grants from the CUNY Research Foundation.  ...  The research of the second author has been supported in part by grants from the CUNY Research Foundation and the National Science Foundation.  ... 
doi:10.1215/00294527-2010-030 fatcat:tlqqmpijibeu5cgwyci2rizwii

A natural model of the multiverse axioms [article]

Victoria Gitman, Joel David Hamkins
2011 arXiv   pre-print
If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms introduced by Hamkins.  ...  The research of the first author has been supported by grants from the CUNY Research Foundation.  ...  The research of the second author has been supported in part by grants from the CUNY Research Foundation and the National Science Foundation.  ... 
arXiv:1104.4450v1 fatcat:5w5dg5hlczdpzoqwk3gizd74ce

Multiverse conceptions in set theory

Carolin Antos, Sy-David Friedman, Radek Honzik, Claudio Ternullo
2015 Synthese  
We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.  ...  Finally, in Section 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a 'vertical' multiverse, and give  ...  of the axioms and of the corresponding models.  ... 
doi:10.1007/s11229-015-0819-9 fatcat:skz656fdcfgstcnfotf2si5roq

The Copernican Multiverse of Sets [article]

Paul K. Gorbow, Graham E. Leigh
2021 arXiv   pre-print
the diversity of the set-theoretic multiverse.  ...  We develop an untyped framework for the multiverse of set theory.  ...  The authors are grateful for the comments and suggestions of the anonymous referee, and to Anton Broberg for permitting us to incorporate his argument for Lemma 5.1.  ... 
arXiv:2005.11087v7 fatcat:aqjzpsl6xncv5oy7ic55gvbqka

THE SET-THEORETIC MULTIVERSE

JOEL DAVID HAMKINS
2012 The Review of Symbolic Logic  
The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view.  ...  The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe.  ...  the natural numbers of the models.  ... 
doi:10.1017/s1755020311000359 fatcat:kjuotik6vffprhdaetkthyzdjm

Some Second Order Set Theory [chapter]

Joel David Hamkins
2008 Lecture Notes in Computer Science  
A ground model has access, via names and the forcing relation, to the objects and truths of the forcing extension. So there is a natural Kripke model lurking here.  ...  We build new models of set theory from existing models, via forcing and ultrapowers. These other models offer us glimpses of alternative universes and alternative truths. The Multiverse View.  ...  The Mantle, directedness, generic Mantle, generic HOD Downward set-directedness Definition  ... 
doi:10.1007/978-3-540-92701-3_3 fatcat:vw3nwx7b55d2rhnjt26os2g23y

Steel's Programme: Evidential Framework, the Core and Ultimate-L [article]

Joan Bagaria, Claudio Ternullo
2021 arXiv   pre-print
We address Steel's Programme to identify a 'preferred' universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the 'core hypothesis'.  ...  In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals).  ...  Then the MV T axioms imply that the multiverse has a core, which is the mantle (and a ground) of every world in the multiverse. 43 Proof.  ... 
arXiv:2011.14724v2 fatcat:ugqlnfda5vhm5fdqycqp4sdmbq

A RECONSTRUCTION OF STEEL'S MULTIVERSE PROJECT

Penelope Maddy, Toby Meadows
2020 Bulletin of Symbolic Logic  
In particular, we reconstruct his notion of a 'natural' theory, describe his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis.  ...  , the target is an array of universes, a multiverse.  ...  , Steel's axioms successfully axiomatize his natural toy models.  ... 
doi:10.1017/bsl.2020.5 fatcat:qeovhbt65ba4bcleuucjjb45cm

Maximality and ontology: how axiom content varies across philosophical frameworks

Neil Barton, Sy-David Friedman
2017 Synthese  
An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology  ...  In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s11229-017-1336-9 pmid:32214510 pmcid:PMC7061924 fatcat:3zousqdz7bcw5b37hnoqx5ibbq

The Non-unique Universe

Gordon McCabe
2010 Foundations of physics  
to have only one possible model, and to thereby negate the possible existence of a multiverse, that theory must be such that it admits only a finite model.  ...  The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical  ...  In this precise sense, the axioms which define a mathematical structure are the axioms of a theory, and each member of a species of structure is a model of that axiomatic theory. 3 If our physical universe  ... 
doi:10.1007/s10701-010-9425-3 fatcat:s6pj4glakjhxbjpmeuxzhjckty

Some remarks on the mathematical structure of the multiverse [article]

Alan McKenzie
2016 arXiv   pre-print
The benefits of MWI may be retained, however, by postulating a multiverse of discrete, parallel, block universes which are identical to each other up to certain points in the MWI trunk before they diverge  ...  The mathematical hierarchy required by the enigmatic footnote 48a in the paper by Goedel leads to an explanation for a unitary evolution of deterministic quantum rules across the multiverse while accounting  ...  At a higher level still, further axioms describe a multitude of such multiverses, accounting for different constants of nature.  ... 
arXiv:1602.04247v1 fatcat:p3shjtstznaj5hvjam7udrlrrm

Topos-theoretic Model of the Deutsch multiverse [article]

Alexander K. Guts
2002 arXiv   pre-print
The Deutsch multiverse is collection of parallel universes. In this article a formal theory and a topos-theoretic model of the Deutsch multiverse are given.  ...  For this the Lawvere-Kock Synthetic Differential Geometry and topos models for smooth infinitesimal analysis are used.  ...  In this article a formal theory and a topos-theoretic model of the Deutsch multiverse are given.  ... 
arXiv:physics/0203071v1 fatcat:6kwij45qwncxnkxlxydazxw56q

From geometry to geology: An invitation to mathematical pluralism through the phenomenon of independence [article]

Jonas Reitz
2016 arXiv   pre-print
This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory.  ...  Direct study of this multiverse presents technical challenges due to its second-order nature.  ...  Acknowledgements Thanks to Mihir Chakraborty and Michele Friend for their generous and inclusive approach to mathematics and philosophy, truly the embodiment of pluralism, and to Joel David Hamkins for  ... 
arXiv:1609.00093v1 fatcat:hdmchiubvnernoxpuy5hznk244

The Search for New Axioms in the Hyperuniverse Programme [chapter]

Sy-David Friedman, Claudio Ternullo
2016 Boston Studies in the Philosophy of Science  
The satisfaction of these statements (H-axioms) in countable transitive models, the collection of which constitutes the 'hyperuniverse' (H), has remarkable first-order consequences, some of which we review  ...  Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the 'maximal iterative concept', and the programme identifies higher-order statements motivated  ...  (Zermelo (1930) , in Ewald (1996) , p. 1233) Zermelo's sequence of natural models can also be viewed as a tower-like multiverse, a 'vertical' multiverse, a collection of universes linearly ordered by  ... 
doi:10.1007/978-3-319-31644-4_10 fatcat:upzo5jef65godc3cs6ccxujeia

The Set-theoretic Multiverse : A Natural Context for Set Theory(Mathematical Logic and Its Applications)

Joel DAVID HAMKINS
2011 Annals of the Japan Association for Philosophy of Science  
Set・ theorists often take their subject as coiistitut・ing a foundation for thc rcst of mathematics, in the sense that other abstract mathematical objects can  ...  generic for set forcing), which is a model of the Ground Axiom.  ...  Although the multiverse position is fundamentally a philosephical position, an ontologieal claim about the nature of mathematical existenee, and the analysis of the multiverse view will  ... 
doi:10.4288/jafpos.19.0_37 fatcat:c7otfm7qxfav3e6zguudzwbwx4
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