School of Historical and Philosophical Studies - Research Publications
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ItemReal Analysis in Paraconsistent Logic(Springer Science and Business Media LLC, 2012-10-01)
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ItemTHE INADEQUACY OF A PROPOSED PARACONSISTENT SET THEORY REPLY(CAMBRIDGE UNIV PRESS, 2011-03)In Bjørdal (2010), Bjørdal presents a paraconsistent set theory in which ∀x(x _= x) is a theorem. The author rightly claims that, while not trivializing (in the sense of proving everything), results like this are to be avoided. The set theory presented in Bjørdal (2010) is based on that of Weber (2010b), but with an introduced definition of identity—which is used, in effect, as a new axiom. With this added notion of identity, the non-self-identity of every object does in fact obtain; and so the set theory presented by Bjørdal is inadequate.......
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ItemA Paraconsistent Model of Vagueness(OXFORD UNIV PRESS, 2010-10)
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ItemExtensionality and Restriction in Naive Set Theory(Springer Science and Business Media LLC, 2010-02)
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ItemTRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY(Cambridge University Press (CUP), 2010-03)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.