Transfinite cardinals in paraconsistent set theory (Q2890698)

This paper establishes a theory of cardinal numbers, supported by a naive set theory, using a paraconsistent logic to ensure nontriviality. The scope of the paper is very wide: from the Cantor-Schröder-Bernstein theorem through to an introductory account of inaccessible and measurable cardinals and, importantly, reflection theorems. The paper highlights difficulties in the concept of cardinality in such a theory, and suggests methods for dealing with such difficulties. NEWLINENEWLINENEWLINE The early parts of the paper are an exposition of familiar and elementary properties of sets in naive set theory, with some careful attention to detail when contradictions may be involved. Some surprises include Routley reducts, or sets that have been `relativized' by a contradiction that has been proven in the theory, and paradoxical results concerning ordinals (Burali-Forti is true, for example). NEWLINENEWLINENEWLINE It is often the case in nonclassical set theories that the axiom of choice plays a critical role, and this paper is no exception. However, usually choice is used as a kind of boundary which separates the finite from the infinite; in the theory presented in the paper, the notions of finitude and infinity are sufficiently rich to prove the axiom of (global) choice and the well-ordering theorem. Perhaps more importantly to the working mathematician, it introduces the (aptly named) concept of incompactness; a kind of finitude associated with all sets. Its role in the theory is similar to that of the usual compactness of a closed, bounded interval of the reals. Moreover, incompactness is a key ingredient in the naive version of the well-ordering theorem. A further key observation from the sections of the paper dealing with transfinite cardinals (the paper contains a proof of transfinite induction) is that the hierarchy determined by Cantor's theorem is essentially a structure which, under the view of naive set theory, expresses inconsistencies. NEWLINENEWLINENEWLINE On the whole, the paper clearly shows that it is not only possible to develop a rich, nontrivial naive set theory using a paraconsisent logic, but that there are many fundamental issues that may be brought into view only in such a theory. It will likely become a key reference for those working in approaches to nonclassical mathematics alternative to the constructivist.