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TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

Published online by Cambridge University Press:  29 March 2012

ZACH WEBER
ZACH WEBER*
Affiliation:
University of Otago; University of Melbourne
*
*DEPARTMENT OF PHILOSOPHY, PO BOX 56, UNIVERSITY OF OTAGO, DUNEDIN 9054, NEW ZEALAND. E-mail:[email protected]
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Abstract

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 5 , Issue 2 , June 2012 , pp. 269 - 293
Copyright
Copyright © Association for Symbolic Logic 2012

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References

BIBLIOGRAPHY

Barwise, J., editor. (1977). Handbook of Mathematical Logic. Amsterdam: North-Holland.Google Scholar
Beall, Jc., Brady, R. T., Hazen, A. P., Priest, G., & Restall, G. (2006). Relevant restricted quantification. Journal of Philosophical Logic, 35, 587–598.CrossRefGoogle Scholar
Beall, Jc., & Murzi, J. (201x). Two flavors of curry paradox. Journal of Philosophy. To appear.Google Scholar
Beall, Jc., Priest, G., & Weber, Z. (2011). Can u do that? Analysis, 71(2), 280–285.Google Scholar
Bell, J. L. (2005). Set Theory: Boolean-Valued Models and Independence Proofs. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Brady, R. T. (1989). The non-triviality of dialectical set theory. In Priest, G., Routley, R., and Norman, J., editors. Paraconsistent Logic: Essays on the Inconsistent. Munich, Germany: Philosophia Verlag, pp. 437–470.CrossRefGoogle Scholar
Brady, R. (2006). Universal Logic. Stanford, CA: CSLI.Google Scholar
Brady, R., & Rush, P. (2008). What is wrong with cantor’s diagonal argument? Logique et Analyse, 51 (202).Google Scholar
Cantor, G. (1892). Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker Vereinigung, 1: 75–78. English trans. in [Ewald 1996], vol. 2.Google Scholar
Cantor, G. (1895). Beiträge zur begründung der transfiniten mengenlehre (erster artikel). Mathematische Annalen, 46, 481–512.CrossRefGoogle Scholar
Cantor, G. (1899). Letter to Dedekind. In van Heijenoort, J. editor. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
Daynes, A. (2000). A strictly finitary non-triviality proof for a paraconsistent system of set theory deductively equivalent to classical zfc minus foundation. Archive for Mathematical Logic, 39(8), 581–598.Google Scholar
Dedekind, R. (1888). Essays on the Theory of Numbers. New York: Dover.Google Scholar
Drake, F. (1974). Set Theory: An Introduction to Large Cardinals. Amsterdam: North Holland.Google Scholar
Dunn, J. M. (1988). The impossibility of certain higher-order non-classical logics with extensionality. In Austin, D. F., editor, Philosophical Analysis. Dordrecht: Kluwer, pp. 261–280.Google Scholar
Ewald, W. B. (1996). From Kant to Hilbert: A source book in the foundations of mathematics, 2 vols., Oxford: Oxford University Press.Google Scholar
Gödel, K. (1964). What is Cantor’s continuum problem? In Benacerraf, P., & Putnam, H., editors. (1964). Philosophy of Mathematics. Englewood Cliffs, N.J: Prentice-Hall, pp. 258–273.Google Scholar
Hallett, M. (1984). Cantorian Set Theory and Limitation of Size. Oxford, UK: Clarendon Press.Google Scholar
Hausdorff, F. (1957). Set Theory (third edition). New York, NY: Chelsea Publishing Co. First edition 1914.Google Scholar
Hausdorff, F. (2005). Hausdorff on Ordered Sets. Providence, RI: American Mathematical Society. Edited, with notes, by Plotkin, J. M..Google Scholar
Jech, T., editor. (1974). Axiomatic Set Theory. Providence, RI: American Mathematical Society.Google Scholar
Kanamori, A. (1994). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Berlin: Springer Verlag.Google Scholar
Kunen, K. (1971). Elementary embeddings and infinitary combinatorics. Journal of Symbolic Logic, 36(3), 407–413.CrossRefGoogle Scholar
Levy, A. (1960). Axiom schemata of strong infinity in axiomatic set theory. Pacific Journal of Mathematics, 10, 223–238.CrossRefGoogle Scholar
Levy, A. (1979). Basic Set Theory. Berlin: Springer Verlag. Reprinted by Dover in 2002.CrossRefGoogle Scholar
McKubre-Jordens, M., & Weber, Z. (2012). Real analysis in paraconsistent logic. Journal of Philosophical Logic. To Appear.CrossRefGoogle Scholar
Moore, G. H. (1982). Zermelo’s Axiom of Choice. New York: Springer Verlag.Google Scholar
Moore, G. H. (1995). The origins of Russell’s paradox: Russell, Couturat, and the antinomy of infinite number. In Hintikka, J., editor. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics. Dordrecht: Kluwer, pp. 215–239.Google Scholar
Mortensen, C. (1995). Inconsistent Mathematics. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
Moschovakis, Y. (2006). Notes on Set Theory (second edition). New York: Springer.Google Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–241.Google Scholar
Priest, G. (1989). Reductio ad absurdum et modus tollendo ponens. In Priest, G., Routley, R., and Norman, J., editors. Paraconsistent Logic: Essays on the Inconsistent. Munich, Germany: Philosophia Verlag, pp. 613–626.Google Scholar
Priest, G. (2000). Inconsistent models of arithmetic, II: The general case. Journal of Symbolic Logic, 65, 1519–1529.CrossRefGoogle Scholar
Priest, G. (2006). In Contradiction: A Study of the Transconsistent (second edition). Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Priest, G., & Routley, R. (1989). Applications of paraconsistent logic. In Priest, G., Routley, R., and Norman, J., editors. Paraconsistent Logic: Essays on the Inconsistent. Munich, Germany: Philosophia Verlag, pp. 151–186.CrossRefGoogle Scholar
Priest, G., Routley, R., & Norman, J., editors. (1989). Paraconsistent Logic: Essays on the Inconsistent. Munich, Germany: Philosophia Verlag.Google Scholar
Restall, G. (1992). A note on naïve set theory in LP. Notre Dame Journal of Formal Logic, 33, 422–432.Google Scholar
Restall, G. (1993). How to be Really contraction free. Studia Logica, 52, 381–391.Google Scholar
Restall, G. (1994). On logics without contraction. PhD Thesis, The University of Queensland.Google Scholar
Routley, R. (1977). Ultralogic as universal? Relevance Logic Newsletter, 2, 51–89. Reprinted in Routley (1980).Google Scholar
Routley, R. (1980). Exploring Meinong’s Jungle and Beyond. Canberra, Australia: Philosophy Department, RSSS, Australian National University. 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.Google Scholar
Rubin, H., & Rubin, J. E. (1963). Equivalents of the Axiom of Choice. Amsterdam: North Holland.Google Scholar
Russell, B. (1905). On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society, 4, 29–53.Google Scholar
Russell, B. (1937). The Principles of Mathematics (second edition). London: George Allen & Unwin.Google Scholar
Slaney, J. K. (1990). A general logic. Australasian Journal of Philosophy, 68, 74–88.CrossRefGoogle Scholar
Tarski, A. (1962). Some problems and results relevant to the foundations of set theory. In Nagel, E., Suppes, P., and Tarski, A. editors Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress. Berkeley: Stanford University Press, pp. 125–135.Google Scholar
van Bendegem, J. P. (2003). Classical arithmetic is quite unnatural. Logic and Logical Philosophy, 11, 231–249.Google Scholar
van Dalen, D. (2004). Logic and Structure. Berlin: Springer.CrossRefGoogle Scholar
van Heijenoort, J., editor. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
von Neumann, J. (1925). Axioms for 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. 346–354.Google Scholar
Weber, Z. (2010a). A paraconsistent model of vagueness. Mind, 119(476), 1025–1045.CrossRefGoogle Scholar
Weber, Z. (2010b). Extensionality and restriction in naive set theory. Studia Logica, 94(1), 87–104.Google Scholar
Weber, Z. (2010c). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3(1), 71–92.Google Scholar
Zermelo, E., 1904. Neuer Beweis, dass jede Menge Wohlordnung werden kann (Aus einem an Herrn Hilbert gerichteten Briefe). Mathematische Annalen, 59, 514–516. Translated in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, pp. 139–141.CrossRefGoogle Scholar