Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-w6k7h Total loading time: 0.26 Render date: 2022-06-27T05:24:06.588Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

THE POTENTIAL HIERARCHY OF SETS

Published online by Cambridge University Press:  14 March 2013

ØYSTEIN LINNEBO*
Affiliation:
Birkbeck University of London and University of Oslo
*
*DEPARTMENT OF PHILOSOPHY, BIRKBECK UNIVERSITY OF LONDON, MALET STREET, LONDON WC1 7HX, UK

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
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

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, 430449. Reprinted in Boolos (1998).CrossRefGoogle Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 521. 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, 193221.CrossRefGoogle 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. 429445.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, 177205.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. 89109.Google Scholar
Gödel, K. (1933). The present situation in the foundations of mathematics. In Gödel (1995).
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, 853875.CrossRefGoogle Scholar
Hodes, H. (1984). On modal logics which enrich first-order S. Journal of Philosophical Logic, 13, 423454.CrossRefGoogle Scholar
Jané, I. (2010). Idealist and realist elements in Cantor’s approach to set theory. Philosophia Mathematica, 18, 193226.CrossRefGoogle Scholar
Lévy, A. (1960). Axiom schemata of strong infinity in axiomatic set theory. Pacific Journal of Mathematics, 10, 223238.CrossRefGoogle Scholar
Lévy, A., & Vaught, R. (1961). Principles of partial reflection in the set theories of Zermelo and Ackermann. Pacific Journal of Mathematics, 11, 10451062.CrossRefGoogle Scholar
Linnebo, Ø. (2009). Bad company tamed. Synthese, 170, 371391.CrossRefGoogle Scholar
Linnebo, Ø. (2010). Pluralities and sets. Journal of Philosophy, 107, 144164.CrossRefGoogle 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. 335367. 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. 298341.Google Scholar
Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, LXIV, 522. Reprinted in Benacerraf & Putnam (1983).CrossRefGoogle Scholar
Rumfitt, I. (2005). Plural terms: Another variety of reference. In Bermudez, J. L., editor, Thought, Reference and Experience. Oxford: Clarendon, pp. 84123.CrossRefGoogle 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. 255304.Google Scholar
Studd, J. (forthcoming). The iterative conception of sets: A (bi-)modal explication. Journal of Philosophical Logic. DOI: 10.1007/s10992-012-9245-3.CrossRef
Uzquiano, G. (2011). Plural quantification and modality. Proceedings of the Aristotelian Society, 111, 219250.CrossRefGoogle Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche. Fundamenta Mathematicae, 16, 2947. Translated in Ewald (1996).CrossRefGoogle Scholar
58
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

THE POTENTIAL HIERARCHY OF SETS
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

THE POTENTIAL HIERARCHY OF SETS
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

THE POTENTIAL HIERARCHY OF SETS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *