Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T15:12:47.805Z Has data issue: false hasContentIssue false
  • English
  • Français

THE POTENTIAL HIERARCHY OF SETS

Published online by Cambridge University Press:  14 March 2013

ØYSTEIN LINNEBO
ØYSTEIN LINNEBO*
Affiliation:
Birkbeck University of London and University of Oslo
*
*DEPARTMENT OF PHILOSOPHY, BIRKBECK UNIVERSITY OF LONDON, MALET STREET, LONDON WC1 7HX, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 2 , June 2013 , pp. 205 - 228
Copyright
Copyright © Association for Symbolic Logic 2013 

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

Benacerraf, P., & Putnam, H., editors. (1983). Philosophy of Mathematics: Selected Readings (second edition). Cambridge: Cambridge University Press.Google Scholar
Bernays, P. (1935). On Platonism in Mathematics. Reprinted in Benacerraf & Putnam(1983).Google Scholar
Boolos, G. (1984). To be is to be a value of a variable (or to be some values of some variables). Journal of Philosophy, 81, 430–449. Reprinted in Boolos (1998).Google Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 5–21. Reprinted in Boolos (1998).CrossRefGoogle Scholar
Boolos, G. (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press.Google Scholar
Burgess, J. P. (2004). E pluribus unum: Plural logic and set theory. Philosophia Mathematica, 12, 193–221.Google Scholar
Burgess, J. P., & Rosen, G. (1997). A Subject with No Object. Oxford: Oxford University Press.Google Scholar
Cantor, G. (1883). Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Teubner, B. G., Leipzig. Translated in (Ewald, 1996).Google Scholar
Drake, F. (1974). Set Theory: An Introduction to Large Cardinals. Amsterdam, the Netherlands: North-Holland.Google Scholar
Dummett, M. (1993). What Is Mathematics About? In His Seas of Language. Oxford: Clarendon, pp. 429–445.Google Scholar
Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathemat ics, Vol. 2. Oxford: Oxford University Press.Google Scholar
Ferreirós, J. (2007). Labyrinth of Thought (second edition). Basel, Switzerland: Birkhaüser.Google Scholar
Fine, K. (1981). First-order modal theories i-sets. Nouˆs, 15, 177–205.Google Scholar
Fine, K. (2005). Our knowledge of mathematical objects. In Gendler, T. S., and Hawthorne, J., editors, Oxford Studies in Epistemology, Vol. 1. Oxford: Oxford University Press, pp. 89–109.Google Scholar
Gödel, K. (1933). The present situation in the foundations of mathematics. In Gödel (1995).Google Scholar
Gödel, K. (1995). Collected Works, Vol. III. Oxford: Oxford University Press.Google Scholar
Hellman, G. (1989). Mathematics without Numbers. Oxford: Clarendon.Google Scholar
Hewitt, S. T. (2012). Modalising plurals. Journal of Philosophical Logic, 41, 853–875.Google Scholar
Hodes, H. (1984). On modal logics which enrich first-order S. Journal of Philosophical Logic, 13, 423–454.CrossRefGoogle Scholar
Jané, I. (2010). Idealist and realist elements in Cantor’s approach to set theory. Philosophia Mathematica, 18, 193–226.CrossRefGoogle Scholar
Lévy, A. (1960). Axiom schemata of strong infinity in axiomatic set theory. Pacific Journal of Mathematics, 10, 223–238.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.Google Scholar
Linnebo, Ø. (2009). Bad company tamed. Synthese, 170, 371–391.CrossRefGoogle Scholar
Linnebo, Ø. (2010). Pluralities and sets. Journal of Philosophy, 107, 144–164.Google Scholar
Linnebo, Ø. (2012). Plural quantification. In Stanford Encyclopedia of Philosophy. Available fromhttp://plato.stanford.edu/archives/fall2012/entries/plural-quant/.Google Scholar
Parsons, C. (1977). What is the iterative conception of set? In Butts, R., and Hintikka, J., editors. Logic, Foundations of Mathematics, and Computability Theory. Dordrecht, the Netherlands: Reidel, pp. 335–367. Reprinted in Benacerraf & Putnam (1983) and Parsons (1983a).CrossRefGoogle Scholar
Parsons, C. (1983a). Mathematics in Philosophy. Ithaca, NY: Cornell University Press.Google Scholar
Parsons, C. (1983b). Sets and Modality. In Mathematics in Philosophy. Cornell, NY: Cornell University Press, pp. 298–341.Google Scholar
Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, LXIV, 5–22. Reprinted in Benacerraf & Putnam (1983).Google Scholar
Rumfitt, I. (2005). Plural terms: Another variety of reference. In Bermudez, J. L., editor, Thought, Reference and Experience. Oxford: Clarendon, pp. 84–123.Google Scholar
Shapiro, S. and Wright, C. (2006). All things indefinitely extensible. In Rayo, A., and Uzquiano, G., editors. Absolute Generality. Oxford: Oxford University Press, pp. 255– 304.CrossRefGoogle Scholar
Studd, J. (forthcoming). The iterative conception of sets: A (bi-)modal explication. Journal of Philosophical Logic. DOI: 10.1007/s10992-012-9245-3.Google Scholar
Uzquiano, G. (2011). Plural quantification and modality. Proceedings of the Aristotelian Society, 111, 219–250.Google Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche. Fundamenta Mathematicae, 16, 29–47. Translated in Ewald (1996).Google Scholar