David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Philosophical Studies 49 (1):37 - 61 (1986)
As the story goes, the source of the paradoxes of naive set theory lies in a conflation of two distinct conceptions of set: the so-called iterative, or mathematical, conception, and the Fregean, or logical, conception. While the latter conception is provably inconsistent, the former, as Godel notes, "has never led to any antinomy whatsoever". More important, the iterative conception explains the paradoxes by showing precisely where the Fregean conception goes wrong by enabling us to distinguish between sets and proper classes, collections that are "too big" to be sets. While I agree wholeheartedly with this distinction, in this paper I argue first that the iterative conception does not provide an explanation of all of the set theoretic paradoxes. I then argue that we need to reconsider the distinction between sets and proper classes rather more carefully. The result will be that ZFC does not capture the iterative conception in its full generality. I close by offering a more general theory that, arguably, does.
|Keywords||set theory proper classes iterative conception of set|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Geoffrey Hellman (1990). Toward a Modal-Structural Interpretation of Set Theory. Synthese 84 (3):409 - 443.
Similar books and articles
Pierluigi Miraglia (2000). Finite Mathematics and the Justification of the Axiom of Choicet. Philosophia Mathematica 8 (1):9-25.
Bradley E. Wilson (1991). Are Species Sets? Biology and Philosophy 6 (4):413-431.
George Boolos (1971). The Iterative Conception of Set. Journal of Philosophy 68 (8):215-231.
Adam Rieger (2011). Paradox, ZF and the Axiom of Foundation. In D. DeVidi, M. Hallet & P. Clark (eds.), Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell. Springer.
Gabriel Uzquiano (2002). Categoricity Theorems and Conceptions of Set. Journal of Philosophical Logic 31 (2):181-196.
Stephen Pollard (1985). Plural Quantification and the Iterative Concept of Set. Philosophy Research Archives 11:579-587.
Alexander Paseau (2007). Boolos on the Justification of Set Theory. Philosophia Mathematica 15 (1):30-53.
M. D. Potter (1993). Iterative Set Theory. Philosophical Quarterly 44 (171):178-193.
Mark F. Sharlow (1987). Proper Classes Via the Iterative Conception of Set. Journal of Symbolic Logic 52 (3):636-650.
Thomas Forster (2008). The Iterative Conception of Set. Review of Symbolic Logic 1 (1):97-110.
Added to index2009-01-28
Total downloads29 ( #64,368 of 1,101,944 )
Recent downloads (6 months)6 ( #52,474 of 1,101,944 )
How can I increase my downloads?