Models for a paraconsistent set theory

Abstract

In this paper the existence of natural models for a paraconsistent version of naive set theory is discussed. These stand apart from the previous attempts due to the presence of some non-monotonic ingredients in the comprehension scheme they fulfill. Particularly, it is proved here that allowing the equality relation in formulae defining sets, within an extensional universe, compels the use of nonmonotonic operators. By reviewing the preceding attempts, we show how our models can naturally be obtained as fixed points of some functor acting on a suitable category (stressing the use of fixed-point arguments in obtaining such alternative semantics).

Paraconsistent Set Theory

“…In a more recent investigation (see [14]), the question of the existence of natural models for a paraconsistent version of naive set theory is reconsidered. There it is proved that allowing the equality relation in formulas defining sets (within an extensional universe) compels the use of non-monotonic operators, and that fixed points can be used to obtain certain kind of models.…”

Paraconsistent set theory by predicating on consistency

“…In a more recent investigation (cf. [23]), the question of the existence of natural models for a paraconsistent version of naive set theory is reconsidered. There it is proved that allowing the equality relation in formulae defining sets (within an extensional universe) compels the use of non-monotonic operators, and that fixed points can be used to obtain certain kind of models.…”

“…The symbol " " will be used for denoting membership in the metalanguage, while "∈" will denote the membership relation symbol in the object (firstorder) language. T. Libert considers in [23] This leads naturally to a 3-valued paraconsistent logic. It is noteworthy to stress the similarity between Libert's interpretation structures and the partial models underlying the theory of quasi-truth, introduced by I. Mikenberg, N. da Costa and R. Chuaqui in [25].…”

Extensionality and Restriction in Naive Set Theory

“…Weyl points out that most of the sets used in mathematics cannot be obtained or checked by listing. "No one can describe an infinite set other than by indicating properties which are characteristic of the elements of the set" [46] (23). Relying on quasi-constructive extensional metaphors when doing this type of work, he indicates, is "nonsensical."…”

“…See[23] for models giving separate consideration to the extensions and antiextensions of sets 10. With thanks to Ross Brady.…”