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  • 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.

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C. Asmus (2009). Restricted Arrow. Journal of Philosophical Logic, 38, 405431.

J. C. Beall , R. T. Brady , A. P. Hazen , G. Priest , & G. Restall (2006). Relevant restricted quantification. Journal of Philosophical Logic, 35, 587598.

R. Brady (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.

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T. Libert (2005). Models for paraconsistent set theory. Journal of Applied Logic, 3, 1541.

R. K. Meyer , R. Routley , & J. Michael Dunn (1978). Curry’s paradox. Analysis, 39, 124128. Rumored to have been written only by Meyer.

U. Petersen (2000). Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica, 64, 365403.

G. Priest (2006). In Contradiction: A Study of the Transconsistent. Oxford, UK: Oxford University Press. Second expanded edition of Priest (1987).

G. Restall (1992). A note on naïve set theory in LP. Notre Dame Journal of Formal Logic, 33, 422432.

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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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