Skip to main content
    • Aa
    • Aa


  • ZACH WEBER (a1)

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.

Corresponding author
Hide All
Asmus C. (2009). Restricted Arrow. Journal of Philosophical Logic, 38, 405431.
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.
Beall J. C., Brady R. T., Hazen A. P., Priest G., & Restall G. (2006). Relevant restricted quantification. Journal of Philosophical Logic, 35, 587598.
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, 447453.
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.
Brady R. (2006). Universal Logic. Stanford, California: CSLI.
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. 415436.
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. 165180.
Drake F. (1974). Set Theory: An Introduction to Large Cardinals. Amsterdam: North Holland Publishing Co.
Hallett M. (1984). Cantorian Set Theory and Limitation of Size. Oxford Logic Guides. Oxford [Oxfordshire]: Clarendon Press, 1984.
Kunen K. (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North Holland Publishing Co.
Levy A. (1979) Basic Set Theory. Berlin, Heidelberg and New York: Springer Verlag. Reprinted by Dover, 2002.
Libert T. (2005). Models for paraconsistent set theory. Journal of Applied Logic, 3, 1541.
Mares E. (2004). Relevant Logic. Cambridge, UK; New York: Cambridge University Press.
Meyer R. K., Routley R., & Michael Dunn J. (1978). Curry’s paradox. Analysis, 39, 124128. Rumored to have been written only by Meyer.
Petersen U. (2000). Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica, 64, 365403.
Priest G. (2006). In Contradiction: A Study of the Transconsistent. Oxford, UK: Oxford University Press. Second expanded edition of Priest (1987).
Priest G., Routley R., & Norman J., editors. (1989). Paraconsistent Logic: Essays on the Inconsistent. Munich: Philosophia Verlag.
Restall G. (1992). A note on naïve set theory in LP. Notre Dame Journal of Formal Logic, 33, 422432.
Routley R. (1980). Exploring Meinong’s Jungle and Beyond. Canberra: Philosophy Department, RSSS, Australian National University. Interim Edition, Departmental Monograph number 3.
Routley R., & Meyer R. K. (1976). Dialectical logic, classical logic and the consistency of the world. Studies in Soviet Thought, 16, 125.
Rubin H., & Rubin J. E. (1985) [1963]. Equivalents of the Axiom of Choice. Amsterdam, North Holland Publishing Co.
Weber Z. (forthcoming-a). Extensionality and restriction in naive set theory. Studia Logica.
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.
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. 200215.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 37 *
Loading metrics...

Abstract views

Total abstract views: 340 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.