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THE ITERATIVE CONCEPTION OF SET

Published online by Cambridge University Press:  01 June 2008

THOMAS FORSTER
THOMAS FORSTER*
Affiliation:
Centre for Mathematical Sciences
*
*CAMBRIDGE UNIVERSITY, CENTRE FOR MATHEMATICAL SCIENCES, CAMBRIDGE CB3 OWB, UK. E-mail: [email protected]
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Abstract

The two expressions ‘The cumulative hierarchy’ and ‘The iterative conception of sets’ are usually taken to be synonymous. However, the second is more general than the first, in that there are recursive procedures that generate some ill-founded sets in addition to well-founded sets. The interesting question is whether or not the arguments in favour of the more restrictive version – the cumulative hierarchy – were all along arguments for the more general version.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 1 , Issue 1 , June 2008 , pp. 97 - 110
Copyright
Copyright © Association for Symbolic Logic 2008

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References

BIBLIOGRAPHY

Barwise, J., & Moss, L. (1996). Vicious Circles. Stanford, CA:CSLI publications.Google Scholar
Boolos, G. (1971). The iterative conception of set. The Journal of Philosophy, 68, 215–231.CrossRefGoogle Scholar
Church, A. (1974). Set theory with a universal set. In Henkin, L., editor, Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics XXV. Providence, RI: AMS, pp. 297–308. Also in International Logic Review, 15, 11–23.Google Scholar
Conway, J. H. (2001). On Numbers and Games, 2nd edition. Peters, A. K., Natick, MA: Academic Press.Google Scholar
Esser, O. (1999). On the consistency of a positive theory. Mathematical Logic Quarterly, 45, 105–116.CrossRefGoogle Scholar
Esser, O. (2004). Une théorie positive des ensembles. Cahiers du Centre de Logique, vol. 13. Louvain-la-Neuve, Belgium: Academia-Bruylant. ISBN 2-8729-687-6.Google Scholar
Forster, T. E. (1983). Axiomatising set theory with a universal set. In Crabbé, M., ed. “La Theorie des ensembles de Quine" Cahiers du Centre de Logique, vol. 4., pp. 61–76. Louvain-La-Neuve, Belgium: CABAY. Available online from: www.dpmms.cam.ac.uk/~tf/old_nf.ps.Google Scholar
Forster, T. E. (2001). Church-Oswald models for set theory. In Logic, Meaning and Computation: Essays in Memory of Alonzo Church, Anderson, C. A. A.and Zeleny, M., editors, Studies in Epistemology, Logic, and Philosophy of Science. vol. 305. Dordrecht, Boston, and London: Kluwer.Google Scholar
Forster, T. E. (2003). Logic, Induction and Sets, LMS Undergraduate Texts in Mathematics, vol. 56. Cambridge University Press.CrossRefGoogle Scholar
Forti, M., & Honsell, F. (1983). Set theory with free construction principles. Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche, 10, 493–522.Google Scholar
Quine, W. V. (1968). Speaking of objects. In Ontological Relativity and Other Essays.Google Scholar
Quine, W. V. (1969). Set Theory and Logic, 2nd edition. Cambridge, MA: Belknap Press Harvard.Google Scholar