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UNIVERSISM AND EXTENSIONS OF V

Part of: Philosophical aspects of logic and foundations Set theory General and miscellaneous specific topics

Published online by Cambridge University Press:  21 July 2020

CAROLIN ANTOS ,
NEIL BARTON  and
SY-DAVID FRIEDMAN
CAROLIN ANTOS
Affiliation:
FACHBEREICH PHILOSOPHIE/ZUKUNFTSKOLLEG UNIVERSITY OF KONSTANZ KONSTANZ, GERMANYE-mail: carolin.antos-kuby@uni-konstanz.de
NEIL BARTON
Affiliation:
FACHBEREICH PHILOSOPHIE UNIVERSITY OF KONSTANZ KONSTANZ, GERMANYE-mail: neil.barton@uni-konstanz.de
SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER UNIVERSITY OF VIENNA, VIENNA, AUSTRIAE-mail: sdf@logic.univie.ac.at
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Abstract

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favor of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.

Keywords

extensionsmultiversismuniversismset theoryadmissible set

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A99: None of the above, but in this section 03E70: Nonclassical and second-order set theories
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 1 , March 2021 , pp. 112 - 154
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Copyright
© Association for Symbolic Logic, 2020

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References

BIBLIOGRAPHY

Antos, C. (2015). Foundations of Higher-Order Forcing. Ph.D. Thesis, Universität Wien.Google Scholar
Antos, C. (2018). Class Forcing in Class Theory. Cham: Springer International Publishing, pp. 1–16.Google Scholar
Antos, C., & Friedman, S.-D. (2017). Hyperclass forcing in Morse-Kelley class theory. The Journal of Symbolic Logic, 82(2), 549–575.CrossRefGoogle Scholar
Antos, C., Friedman, S.-D., Honzik, R., & Ternullo, C. (2015). Multiverse conceptions in set theory. Synthese, 192(8), 2463–2488.CrossRefGoogle Scholar
Arrigoni, T., & Friedman, S.-D. (2013). The hyperuniverse program. Bulletin of Symbolic Logic, 19, 77–96.CrossRefGoogle Scholar
Bagaria, J. (1997). A characterization of Martin's Axiom in terms of absoluteness. The Journal of Symbolic Logic, 62(2), 366–372.CrossRefGoogle Scholar
Bagaria, J. (2005). Natural axioms of set theory and the continuum problem. In Proceedings of the 12th International Congress of Logic, Methodology, and Philosophy of Science. London: King's College London Publications, pp. 43–64.Google Scholar
Bagaria, J. (2008). Set theory. In The Princeton Companion to Mathematics, Princeton, NJ: Princeton University Press, pp. 302–321.Google Scholar
Barton, N. (2020a). Forcing and the universe of sets: must we lose insight? Journal of Philosophical Logic, 49(4), 575–612.CrossRefGoogle Scholar
Barton, N. (2020b). Are Large Cardinal Axioms Restrictive? Manuscript under review. Available from: https://philpapers.org/archive/BARALC-5.pdf.Google Scholar
Barton, N., & Friedman, S.-D. (2017). Maximality and ontology: how axiom content varies across philosophical frameworks. Synthese, 197(2), 623–649.CrossRefGoogle ScholarPubMed
Barwise, J. (1975). Admissible Sets and Structures: An Approach to Definability Theory. New York: Springer.CrossRefGoogle Scholar
Boolos, G. (1984). To be is to be a value of a variable (or to be some values of some variables). The Journal of Philosophy, 81(8), 430–449.CrossRefGoogle Scholar
Bukovský, L. (1973). Characterization of generic extensions of models of set theory. Fundamenta Mathematicae, 83(1), 35–46.CrossRefGoogle Scholar
Caicedo, A. E., Cummings, J., Koellner, P., & Larson, P. B. (2017). Foundations of Mathematics: Logic at Harvard, Essays in Honor of W. Hugh Woodin's 60th Birthday. Contemporary Mathematics, Vol. 690. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Cohen, P. (1966). Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, Inc.Google Scholar
Cummings, J. (2010). Iterated Forcing and Elementary Embeddings. Dordrecht, Netherlands: Springer, pp. 775–883.Google Scholar
Detlefsen, M. (1986). Hilbert's Program. Dordrecht, Netherlands: Reidel.CrossRefGoogle Scholar
Detlefsen, M. (1990). On an alleged refutation of Hilbert's program using Gödel's first incompleteness theorem. Journal of Philosophical Logic, 19, 343–377.CrossRefGoogle Scholar
Detlefsen, M. (2001). What does Gödel's second theorem say? Philosophia Mathematica, 9, 37–71.CrossRefGoogle Scholar
Enderton, H. (1972). A Mathematical Introduction to Logic. San Diego, CA: Harcourt/Academic Press.Google Scholar
Ewald, W. B. (1996). From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Vol. I. Oxford: Oxford University Press.Google Scholar
Friedman, S.-D. (2000). Fine Structure and Class Forcing. de Gruyter Series in Logic and its Applications, Vol. 3. Berlin: de Gruyter.CrossRefGoogle Scholar
Friedman, S.-D. (2006). Internal consistency and the inner model hypothesis. Bulletin of Symbolic Logic, 12(4), 591–600.CrossRefGoogle Scholar
Friedman, S.-D. (2016). Evidence for set-theoretic truth and the hyperuniverse programme. Proof, Truth, Computation, 3(4), 517–555.Google Scholar
Friedman, S.-D., Fuchino, S., & Sakai, H. F. (2020). On the set-generic multiverse. Proceedings of the Singapore Programme on Sets and Computations. Forthcoming.Google Scholar
Friedman, S.-D., & Honzik, R. (2016a). Definability of satisfaction in outer models. Journal of Symbolic Logic, 81(3), 1047–1068.CrossRefGoogle Scholar
Friedman, S.-D., & Honzik, R. (2016b). On strong forms of reflection in set theory. Mathematical Logic Quarterly, 62(1–2), 52–58.CrossRefGoogle Scholar
Friedman, S.-D., Welch, P., & Woodin, W. H. (2008). On the consistency strength of the inner model hypothesis. The Journal of Symbolic Logic, 73(2), 391–400.CrossRefGoogle Scholar
Fuchs, G., Hamkins, J. D., & Reitz, J. (2015). Set-theoretic geology. Annals of Pure and Applied Logic, 166(4), 464–501.CrossRefGoogle Scholar
Fujimoto, K. (2012). Classes and truths in set theory. Annals of Pure and Applied Logic, 163, 1484–1423.CrossRefGoogle Scholar
Gitman, V., & Hamkins, J. D. (2010). A natural model of the multiverse axioms. Notre Dame Journal of Formal Logic, 51(4), 475–484.CrossRefGoogle Scholar
Gitman, V., & Hamkins, J. D. (2017). Open determinacy for class games. In Caicedo, A. E., Cummings, J., Koellner, P., and Larson, P., editors. Foundations of Mathematics. Contemporary Mathematics, Vol. 690. Providence, RI: American Mathematical Society, pp. 121–144.CrossRefGoogle Scholar
Hale, B. (2013). Necessary Beings: An Essay on Ontology, Modality, and the Relations Between Them. Oxford: Oxford University Press.CrossRefGoogle Scholar
Hamkins, J. D. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5(3), 416–449.CrossRefGoogle Scholar
Hellman, G. (1989). Mathematics Without Numbers. Oxford: Oxford University Press.Google Scholar
Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Noviak, Z., and Simonyi, A., editors. Truth, Reference, and Realism. Budapest: Central European University Press, pp. 1–75.Google Scholar
Koellner, P. (2013). Hamkins on the multiverse. Exploring the Frontiers of Incompleteness conference series, Harvard University.Google Scholar
Kunen, K. (2013). Set Theory. London: College Publications.Google Scholar
Laver, R. (2007). Certain very large cardinals are not created in small forcing extensions. Annals of Pure and Applied Logic, 149(1–3), 1–6.CrossRefGoogle Scholar
Linnebo, Ø. (2006). Sets, properties, and unrestricted quantification. In Rayo, A., and Uzquiano, G., editors. Absolute Generality. Oxford: Oxford University Press, pp. 149–178.Google Scholar
Meadows, T. (2015). Naive infinitism. Notre Dame Journal of Formal Logic, 56(1), 191–212.CrossRefGoogle Scholar
Rayo, A., & Uzquiano, G. (2006). Introduction to absolute generality. In Rayo, A., & Uzquiano, G., editors. Absolute Generality. Oxford: Oxford University Press, pp. 1–19.Google Scholar
Rumfitt, I. (2015). The Boundary Stones of Thought: An Essay in the Philosophy of Logic. Oxford: Oxford University Press.CrossRefGoogle Scholar
Stanley, M. C. (2008). Outer Model Satisfiability. Unpublished Manuscript.Google Scholar
Steel, J. (2014). Gödel's program. In Kennedy, J., editor. Interpreting Gödel. Cambridge: Cambridge University Press.Google Scholar
Usuba, T. (2017). The downward directed grounds hypothesis and very large cardinals. Journal of Mathematical Logic, 17(2), 1750009.CrossRefGoogle Scholar
Uzquiano, G. (2003). Plural quantification and classes. Philosophia Mathematica, 11(1), 67–81.CrossRefGoogle Scholar
Welch, P., & Horsten, L. (2016). Reflecting on absolute infinity. Journal of Philosophy, 113(2), 89–111.CrossRefGoogle Scholar
Zermelo, E. (1930). On Boundary Numbers and Domains of Sets, Vol. 2. Oxford: Oxford University Press, pp. 1208–1233.Google Scholar