Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T17:59:24.334Z Has data issue: false hasContentIssue false
  • English
  • Français

THE SET-THEORETIC MULTIVERSE

Published online by Cambridge University Press:  09 August 2012

JOEL DAVID HAMKINS
JOEL DAVID HAMKINS*
Affiliation:
Department of Philosophy, New York University, Mathematics Program, The Graduate Center of The City University of New York, and Department of Mathematics, The College of Staten Island of CUNY
*
*DEPARTMENT OF PHILOSOPHY, NEW YORK UNIVERSITY, 5 WASHINGTON PLACE, NEW YORK, NY 10003, MATHEMATICS PROGRAM, THE GRADUATE CENTER OF THE CITY UNIVERSITY OF NEW YORK, 365 FIFTH AVENUE, NEW YORK, NY 10016, DEPARTMENT OF MATHEMATICS, THE COLLEGE OF STATEN ISLAND OF CUNY, STATEN ISLAND, NY 10314, E-mail: [email protected], http://jdh.hamkins.org
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 5 , Issue 3 , September 2012 , pp. 416 - 449
Copyright
Copyright © Association for Symbolic Logic 2012

References

BIBLIOGRAPHY

Blass, A. (1984). The interaction between category theory and set theory. In Mathematical Applications of Category Theory (Denver, Col., 1983), Vol. 30 of Contemp. Math.Providence, RI: Amer. Math. Soc., pp. 5–29.CrossRefGoogle Scholar
Cohen, P. J. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, 50, 1143–1148.CrossRefGoogle ScholarPubMed
Cohen, P. J. (1964). The independence of the continuum hypothesis II. Proceedings of the National Academy of Sciences of the United States of America, 51, 105–110.CrossRefGoogle ScholarPubMed
Cohen, P. J. (1966). Set theory and the continuum hypothesis. New York, NY: W. A. Benjamin, Inc.Google Scholar
Fuchs, G., Hamkins, J. D., & Reitz, J. (submitted). Set-theoretic geology. Submitted. Preprint available fromhttp://arxiv.org/abs/1107.4776.Google Scholar
Freiling, C. (1986). Axioms of symmetry: Throwing darts at the real number line. Journal of Symbolic Logic, 51(1), 190–200.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., Welch, P., & Woodin, W. H. (2008). On the consistency strength of the inner model hypothesis. Journal of Symbolic Logic, 73, 391–400.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
Hamkins, J. D. (in press). The set-theoretical multiverse: A natural context for set theory. Annals of the Japan Association for Philosophy of Science. To appear.Google Scholar
Hamkins, J. D. (2003a). Extensions with the approximation and cover properties have no new large cardinals. Fundamenta Mathematicae, 180(3), 257–277.CrossRefGoogle Scholar
Hamkins, J. D. (2003b). A simple maximality principle. Journal of Symbolic Logic, 68(2), 527–550.CrossRefGoogle Scholar
Hamkins, J. D. (2005). The ground axiom. Oberwolfach Report, 55, 3160–3162.Google Scholar
Hamkins, J. D. (2009). Some second order set theory. In Ramanujam, R., and Sarukkai, S., editors. ICLA 2009, volume 5378 of LNAI. Amsterdam: Springer–Verlag, pp. 36–50.Google Scholar
Hamkins, J. D. (2009a). What are some reasonable-sounding statements that are independent of ZFC? MathOverflow. Available fromhttp://mathoverflow.net/questions/6594 (version: 2009-11-25).Google Scholar
Hamkins, J. D., Linetsky, D., & Reitz, J. (submitted). Pointwise definable models of set theory. Submitted. Preprint available fromhttp://arxiv.org/abs/1105.4597.Google Scholar
Hamkins, J. D., & Löwe, B. (2008). The modal logic of forcing. Transactions of the American Mathematical Society, 360, 1793–1817.CrossRefGoogle Scholar
Hamkins, J. D., Reitz, J., & Woodin, W. H. (2008). The Ground Axiom is consistent with V ≠ HOD. Proceedings of the AMS, 136, 2943–2949.CrossRefGoogle Scholar
Hamkins, J. D., & Seabold, D. (in preparation). Boolean ultrapowers.Google Scholar
Krömer, R. (2001). Tarski’s axiom of inaccessibles and Grothendieck universes—historical and critical remarks on the foundations of category theory. In Logic, Vol. 21 (Polish), Vol. 2312 of Acta Univ. Wratislav. Wrocław, Poland: Wydawn. Uniw. Wrocław., pp. 45–57.Google Scholar
Lakatos, I., editor. (1967). Problems in the Philosophy of Mathematics. Amsterdam: North-Holland Publishing Co. Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, vol. 1.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
Luzin, N. N. (1935). Sur les ensembles analytiques nuls. Fundamenta Mathematicae, 25, 109–131, 1935.CrossRefGoogle Scholar
Maddy, P. (1998). V = L and Maximize. In Makowski, J. A., and Raave, E. V., editors. Logic Colloquium ’95 Proceedings of ASL, Haifa, Isreal 1995. Berlin: Springer. pp. 134–152.Google Scholar
Martin, D. A. (2001). Multiple universes of sets and indeterminate truth values. Topoi, 20(1), 5–16.CrossRefGoogle Scholar
Peano, G. (1889). The principles of arithmetic, presented by a new method. pages pp. 83–97 in van Heijenoort, 1967.Google Scholar
Reitz, J. (2006). The Ground Axiom. PhD Thesis, The Graduate Center of the City University of New York.Google Scholar
Reitz, J. (2007). The Ground Axiom. Journal of Symbolic Logic, 72(4), 1299–1317.CrossRefGoogle Scholar
Solovay, R. M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25, 287–304.CrossRefGoogle Scholar
Solovay, R. M. (2010). Personal communication, October 31, 2010.Google Scholar
Stavi, J., & Väänänen, J. (2003). Reflection principles for the continuum. In Zhang, Y., editor. Logic and Algebra, Vol. 302 of AMS Contemporary Mathematics Series. Providence, RI: American Mathematical Society.Google Scholar
Steel, J. R. (2004). Generic absoluteness and the continuum problem. Slides for a talk at the Laguna workshop on philosophy and the continuum problem (P. Maddy and D. Malament organizers) March, 2004. Available fromhttp://math.berkeley.edu/steel/talks/laguna.ps.Google Scholar
van Heijenoort, J. (1967). From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
Woodin, W. H. (2004). Recent development’s on Cantor’s Continuum Hypothesis. In Proceedings of the Continuum in Philosophy and Mathematics. Copenhagen, Denmark: Carlsberg Academy, November 2004.Google Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre. (German). Fundumenta Mathematicae, 16, 29–47.CrossRefGoogle Scholar