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MODAL STRUCTURALISM AND REFLECTION

Published online by Cambridge University Press:  14 June 2018

SAM ROBERTS
SAM ROBERTS*
Affiliation:
Department of Philosophy, IFIKK, University of Oslo
*
*DEPARTMENT OF PHILOSOPHY IFIKK UNIVERSITY OF OSLO POSTBOKS 1020 BLINDERN 0315 OSLO NORWAY E-mail: sam.roberts@ifikk.uio.noURL: http://samrroberts.net
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Abstract

Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

Keywords

03A0503E5503E3097E2097E3000A3003B45structuralismmodal structuralismset theorymodal logicplural logiclarge cardinalsreflection principles
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 823 - 860
Copyright
Copyright © Association for Symbolic Logic 2018 

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References

BIBLIOGRAPY

Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70(19), 661–679.CrossRefGoogle Scholar
Bernays, P. (1976). On the problem of schemata of infinity in axiomatic set theory. In Müller, G. H., editor. Sets and Classes: On the Work by Paul Bernays. Studies in Logic and the Foundations of Mathematics, Vol. 84. Amsterdam: North-Holland, pp. 121–172.CrossRefGoogle Scholar
Boolos, G. (1998). The iterative conception of set. In Jeffrey, R., editor. Logic, Logic, and Logic. Cambridge, MA: Harvard University Press, pp. 13–29.Google Scholar
Boolos, G. (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press. Edited by Jeffrey, Richard.Google Scholar
Clarke-Doane, J. (2017). What is the benacerraf problem? In Pataut, F., editor. New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity. New York: Springer, pp. 93–125.Google Scholar
Cotnoir, A. J. & Baxter, D. L. M. (2014). Composition as Identity. Oxford: Oxford University Press.CrossRefGoogle Scholar
Devlin, K. J. (1984). Constructibility. Perspectives in Mathematical Logic. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Drake, F. (1974). Set Theory: An Introduction to Large Cardinals. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.Google Scholar
Feferman, S., Friedman, H. M., Maddy, P., & Steel, J. R. (2000). Does mathematics need new axioms? The Bulletin of Symbolic Logic, 6(4), 401–446.CrossRefGoogle Scholar
Fujimoto, K. (2012). Classes and truths in set theory. Annals of Pure and Applied Logic, 163(11), 1484–1523.CrossRefGoogle Scholar
Gödel, K. (1964). What is cantor’s continuum problem? In Benacerraf, P., and Putnam, H., editors. Philosophy of Mathematics. Upper Saddle River, NJ: Prentice Hall, pp. 470–785.Google Scholar
Hale, B. (1996). Structuralism’s unpaid epistemological debts. Philosophia Mathematica, 4(2), 124–147.CrossRefGoogle Scholar
Hellman, G. (1989). Mathematics without Numbers: Towards a Modal-Structural Interpretation. Oxford: Clarendon.Google Scholar
Hellman, G. (1996). Structuralism without structures. Philosophia Mathematica, 4(2), 100–123.Google Scholar
Hellman, G. (2002). Maximality vs. extendability: Reflections on structuralism and set theory. In Malament, D., editor. Reading Natural Philosophy. Chicago: Open Court, pp. 335–361.Google Scholar
Hellman, G. (2005). Structuralism. In Shapiro, S., editor. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press, pp. 536–562.CrossRefGoogle Scholar
Hellman, G. (2011). On the significance of the burali-forti paradox. Analysis, 71(4), 631–637.CrossRefGoogle Scholar
Hellman, G. (2015). Infinite possibilities and possibilities of infinity. In Auxier, R. E., Anderson, D. R., and Hahn, L. E., editors. The Philosophy of Hilary Putnam. La Salle, IL: Open Court, pp. 259–278.Google Scholar
Hewitt, S. (2012). Modalising plurals. Journal of Philosophical Logic, 41(5), 853–875.CrossRefGoogle Scholar
Hughes, G. & Cresswell, M. (1996). A New Introduction to Modal Logic. London: Routledge.CrossRefGoogle Scholar
Kanamori, A. (2003). The Higher Infinite (second edition). Berlin: Springer.Google Scholar
Koellner, P. (2006). On the question of absolute undecidability. Philosophia Mathematica, 14(2), 153–188.CrossRefGoogle Scholar
Koellner, P. (2009). On reflection principles. Annals of Pure and Applied Logic, 157(2–3), 206–219.CrossRefGoogle Scholar
Kunen, K. (2011). Set Theory. London: College Publications.Google Scholar
Lévy, A. (1965). A Hierarchy of Formulas in Set Theory. Memoirs of the American Mathematical Society, Vol. 57. Providence, RI: American Mathematical Society.Google Scholar
Lévy, A. & Vaught, R. (1961). Principles of partial reflection in the set theories of Zermelo and ackermann. Pacific Journal of Mathematics, 11, 1045–1062.CrossRefGoogle Scholar
Lewis, D. (1991). Parts of Classes. Oxford: Blackwell.Google Scholar
Linnebo, Ø. (2010). Pluralities and sets. Journal of Philosophy, 107(3), 144–164.CrossRefGoogle Scholar
Linnebo, Ø. (2013). The potential hierarchy of sets. The Review of Symbolic Logic, 6, 205–228.CrossRefGoogle Scholar
Linnebo, Ø. (2017). Plural quantification. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). Available at https://plato.stanford.edu/archives/sum2017/entries/plural-quant/.Google Scholar
Parsons, C. (1974). Sets and classes. Noûs, 8(1), 1–12.CrossRefGoogle Scholar
Paseau, A. (2007). Boolos on the justification of set theory. Philosophia Mathematica, 15(1), 30–53.CrossRefGoogle Scholar
Pettigrew, R. (2012). Indispensability arguments and instrumental nominalism. The Review of Symbolic Logic, 5(4), 687–709.CrossRefGoogle Scholar
Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, 64(1), 5–22.CrossRefGoogle Scholar
Rayo, A. & Yablo, S. (2001). Nominalism through de-nominalization. Noûs, 35(1), 74–92.CrossRefGoogle Scholar
Roberts, S. (2017). A strong reflection principle. The Review of Symbolic Logic, 10(4), 651–662.CrossRefGoogle Scholar
Tait, W. W. (1998). Zermelo’s conception of set theory and reflection principles. In Schirn, M., editor. Philosophy of Mathematics Today. New York: Oxford University Press, pp. 469–483.Google Scholar
Uzquiano, G. (1999). Models of second-order Zermelo set theory. Bulletin of Symbolic Logic, 5(3), 289–302.CrossRefGoogle Scholar
Uzquiano, G. (2003). Plural quantification and classes. Philosophia Mathematica, 11(3), 67–81.CrossRefGoogle Scholar
Uzquiano, G. (2011). Plural quantification and modality. Proceedings of the Aristotelian Society, 111(2pt2), 219–250.CrossRefGoogle Scholar
Uzquiano, G. (2014). Mereology and modality. In Kleinschmidt, S., editor. Mereology and Location. Oxford: Oxford University Press, pp. 33–56.CrossRefGoogle Scholar
Welch, P. (2017). Global reflection principles. In Sober, E., Niiniluoto, I., and Leitgeb, H., editors. Logic, Methodology and Philosophy of Science: Proceedings of the Fifteenth International Congress. London: College Publications, pp. 18–36.Google Scholar
Zermelo, E. (1996). On boundary numbers and domains of sets: New investigations in the foundations of set theory. In Ewald, W., editor. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 2. New York: Oxford University Press, pp. 1219–1233.Google Scholar