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TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

Published online by Cambridge University Press:  14 January 2010

ZACH WEBER
ZACH WEBER*
Affiliation:
School of Philosophy and Historical Inquiry, University of Sydney
*
*SCHOOL OF PHILOSOPHY AND HISTORICAL INQUIRY, UNIVERSITY OF SYDNEY, NSW 2006 AUSTRALIA. E-mail:[email protected]
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Abstract

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 3 , Issue 1 , March 2010 , pp. 71 - 92
Copyright
Copyright © Association for Symbolic Logic 2010

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References

BIBLIOGRAPHY

Asmus, C. (2009). Restricted Arrow. Journal of Philosophical Logic, 38, 405–431.CrossRefGoogle Scholar
Batens, D., Mortensen, C., Priest, G., & van Bendegem, J.-P., editors. (2000). Frontiers of Paraconsistent Logic. Baldock, Hertfordshire, England and Philadelphia, PA: Research Studies Press.Google Scholar
Beall, J. C., Brady, R. T., Hazen, A. P., Priest, G., & Restall, G. (2006). Relevant restricted quantification. Journal of Philosophical Logic, 35, 587–598.CrossRefGoogle Scholar
Brady, R. (1971). The consistency of the axioms of the axioms of abstraction and extensionality in a three valued logic. Notre Dame Journal of Formal Logic, 12, 447–453.CrossRefGoogle Scholar
Brady, R., editor. (2003). Relevant Logics and Their Rivals, Volume II: A Continuation of the Work of Richard Sylvan, Robert Meyer, Val Plumwood and Ross Brady. With contributions by: Martin, Bunder, Andre, Fuhrmann, Andrea, Loparic, Edwin, Mares, Chris, Mortensen, and Alasdair, Urquhart. Aldershot, Hampshire, UK: Ashgate.Google Scholar
Brady, R. (2006). Universal Logic. Stanford, California: CSLI.Google Scholar
Brady, R. T., & Routley, R. (1989) The non-triviality of extensional dialectical set theory. In Priest, G., Routley, R., and Norman, J., editors. Paraconsistent Logic: Essays on the Inconsistent. Munich: Philosophia Verlag, pp. 415–436.CrossRefGoogle Scholar
da Costa, N. (2000). Paraconsistent mathematics. In Batens, D., Mortensen, G., Priest, G., and van Bendegem, J.-P., editors. Frontiers of Paraconsistent Logic. Baldock, Hertfordshire, England and Philadelphia, PA: Research Studies Press, pp. 165–180.Google Scholar
Drake, F. (1974). Set Theory: An Introduction to Large Cardinals. Amsterdam: North Holland Publishing Co.Google Scholar
Hallett, M. (1984). Cantorian Set Theory and Limitation of Size. Oxford Logic Guides. Oxford [Oxfordshire]: Clarendon Press, 1984.Google Scholar
Kunen, K. (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North Holland Publishing Co.Google Scholar
Levy, A. (1979) Basic Set Theory. Berlin, Heidelberg and New York: Springer Verlag. Reprinted by Dover, 2002.CrossRefGoogle Scholar
Libert, T. (2005). Models for paraconsistent set theory. Journal of Applied Logic, 3, 15–41.CrossRefGoogle Scholar
Mares, E. (2004). Relevant Logic. Cambridge, UK; New York: Cambridge University Press.CrossRefGoogle Scholar
Meyer, R. K., Routley, R., & Michael Dunn, J. (1978). Curry’s paradox. Analysis, 39, 124–128. Rumored to have been written only by Meyer.CrossRefGoogle Scholar
Petersen, U. (2000). Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica, 64, 365–403.CrossRefGoogle Scholar
Priest, G. (2006). In Contradiction: A Study of the Transconsistent. Oxford, UK: Oxford University Press. Second expanded edition of Priest (1987).CrossRefGoogle Scholar
Priest, G., Routley, R., & Norman, J., editors. (1989). Paraconsistent Logic: Essays on the Inconsistent. Munich: Philosophia Verlag.Google Scholar
Restall, G. (1992). A note on naïve set theory in LP. Notre Dame Journal of Formal Logic, 33, 422–432.CrossRefGoogle Scholar
Routley, R. (1980). Exploring Meinong’s Jungle and Beyond. Canberra: Philosophy Department, RSSS, Australian National University. Interim Edition, Departmental Monograph number 3.Google Scholar
Routley, R., & Meyer, R. K. (1976). Dialectical logic, classical logic and the consistency of the world. Studies in Soviet Thought, 16, 1–25.CrossRefGoogle Scholar
Rubin, H., & Rubin, J. E. (1985) [1963]. Equivalents of the Axiom of Choice. Amsterdam, North Holland Publishing Co.Google Scholar
Weber, Z. (forthcoming-a). Extensionality and restriction in naive set theory. Studia Logica.Google Scholar
Weber, Z. (forthcoming-b). Notes on inconsistent set theory. In Tanaka, K., Berto, F., Paoli, F., and Mares, E., editors. World Congress of Paraconsistency 4.Google Scholar
Zermelo, E. (1967). Investigations in the foundations of set theory. In van Heijenoort, J., editor. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press, pp. 200–215.Google Scholar