Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:00:01.311Z Has data issue: false hasContentIssue false
  • English
  • Français

TWO ARGUMENTS AGAINST THE GENERIC MULTIVERSE

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

Published online by Cambridge University Press:  02 December 2020

TOBY MEADOWS
TOBY MEADOWS*
Affiliation:
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE UNIVERSITY OF CALIFORNIA, IRVINE
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper critically examines two arguments against the generic multiverse, both of which are due to W. Hugh Woodin. Versions of the first argument have appeared a number of times in print, while the second argument is relatively novel. We shall investigate these arguments through the lens of two different attitudes one may take toward the methodology and metaphysics of set theory; and we shall observe that the impact of these arguments depends significantly on which of these attitudes is upheld. Our examination of the second argument involves the development of a new (inner) model for Steel’s multiverse theory, which is delivered in the Appendix.

Keywords

philosophy of set theoryphilosophy of mathematicsset theoryforcing

MSC classification

Primary: 03A05: Philosophical and critical
Secondary: 00A30: Philosophy of mathematics 03E40: Other aspects of forcing and Boolean-valued models
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 2 , June 2021 , pp. 347 - 379
Check for updates
Copyright
© Association for Symbolic Logic, 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Avigad, J. (2003). Number theory and elementary arithmetic. Philosophia Mathematica, 11(3), 257–284.CrossRefGoogle Scholar
Bagaria, J., Castells, N., & Larson, P.. (2006). An Ω-Logic Primer. Basel: Birkhäuser Basel, pp. 1–28. https://doi.org/10.1007/3-7643-7692-91 Google Scholar
Bell, J. L. Set Theory: Boolean-Valued Models and Independence Proofs. Oxford, UK: Clarendon Press, 2005.CrossRefGoogle Scholar
Cohen, P. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, 50(6), 1143–1148.CrossRefGoogle ScholarPubMed
Cohen, P.. (1971). Comments on the foundations of set theory. In Scott, D., editor. Axiomatic Set Theory. Vol. XIII. Providence, RI: American Mathematical Society, pp. 9–16.CrossRefGoogle Scholar
Cohen, P.. (1974). Automorphisms of set theory. In Henkin, L., editor. Proceeding of the Tarski Symposium. Providence, RI: American Mathematical Society, pp. 325–330.CrossRefGoogle Scholar
Devlin, K. J. (1984). Constructibility. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Eniyat, A. (2010). Bi-interpretability vs mutual interpretability. Available from: https://cs.nyu.edu/pipermail/fom/2010-January/014325.html Google Scholar
Gödel, K. (1986). On formally undecidable propositions of principia mathematica and related systems. In Feferman, S., editor, Collected Works, Vol. 1. New York: OUP.Google Scholar
Hamkins, J. D. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5, 416–449.CrossRefGoogle Scholar
Hamkins, J. D. & Seabold, D. E.. (2012). Well-founded boolean ultrapowers as large cardinal embeddings (arxiv:1206.6075).Google Scholar
Jech, T. (2003). Set Theory. Heidelberg: Springer.Google Scholar
Kalmar, L. (1967). Response to mostowski’s paper. In Lakatos, I., editor, Problems in the Philosophy of Mathematics. Amsterdam: North Holland Publishing Company.Google Scholar
Kanamori, A. (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Heidelberg: Springer.Google Scholar
Koellner, P. (2009). Truth in Mathematics: The Question of Pluralism. London, UK: Palgrave Macmillan, pp. 80–116.Google Scholar
Kreisel, G. (1967). Response to Mostowski’s paper. In Lakatos, I., editor, Problems in the Philosophy of Mathematics. Amsterdam: North Holland Publishing Company.Google Scholar
Kunen, K. (2011). Set Theory (second edition). London, UK: College Publications.Google Scholar
Maddy, P. (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Mansfield, R. (1971). The theory of boolean ultrapowers. Annals of Mathematical Logic, 2(3), 297–323.CrossRefGoogle Scholar
Meadows, T. & Maddy, P.. (n.d.). A philosopher’s guide to Steel’s multiverse. In preparation.Google Scholar
Mostowski, A. (1967). Recent results in set theory. In Lakatos, I., editor, Problems in the Philosophy of Mathematics. Amsterdam: North Holland Publishing Company.Google Scholar
Reitz, J. (2007). The ground axiom. Journal of Symbolic Logic, 72(4), 1299–1317.CrossRefGoogle Scholar
Steel, J. R. (2014). Gödel’s program. In Kennedy, J., editor, Interpreting Gödel: Critical Essays. Cambridge, UK: Cambridge University Press.Google Scholar
Steel, J. R.. (2004) Generic absoluteness and the continuum problem. Unpublished handout.Google Scholar
Usuba, T. (2017). The downward directed grounds hypothesis and very large cardinals. ArXiv e-prints.CrossRefGoogle Scholar
Visser, A. (2004). Categories of theories and interpretations. Utrecht Logic Group Preprint Series, 228, 284–341.Google Scholar
Visser, A. & Friedman, H. M. (2014). When bi-interpretability implies synonymy. Logic Group preprint series, 320, 1–19.Google Scholar
Woodin, H. W. (2004). Set theory after Russell; the journey back to Eden. In Link, G., editor, 100 Years of Russell’s Paradox. Berlin, Germany: De Gruyter.Google Scholar
Woodin, H. W.. (2011). The realm of the infinite. In Heller, M. & Woodin, H. W., editors, Infinity: New Research Frontiers. Cambridge, UK: Cambridge University Press.Google Scholar
Woodin, H. W.. (2012). The Continuum Hypothesis, the Generic Multiverse of Sets, and the Ω Conjecture. Cambridge, UK: Cambridge University Press.Google Scholar
Woodin, H. W.. (2017). In search of Ultimate-L the 19th Midrasha mathematicae lectures. The Bulletin of Symbolic Logic, 23(1), 1–109.CrossRefGoogle Scholar