Twist-Valued Models for Three-valued Paraconsistent Set Theory

Por • 2 dic, 2019 • Sección: Opinion

Walter Carnielli, Marcelo E. Coniglio

Boolean-valued models of set theory were introduced by Scott and Solovay in 1965 (and independently by Vopěnka in the same year), offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific three-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation, thus obtaining a paraconsistent model of ZF. We argue that that the implication operator of LPT0 defined in this paper is, in a sense, more suitable for a paraconsistent set theory than the implication of PS3: indeed, our implication allows for genuinely inconsistent sets (in a precise sense, [(w = w)] = 1/2 for some w). It is to be remarked that our implication does not fall under the definition of the so-called ‘reasonable implication algebras’ of Löwe and Tarafder. This suggests that ‘reasonable implication algebras’ are just one way to define a paraconsistent set theory, perhaps not the most appropriate. Our twist-valued models for LPT0 can be easily adapted to provide twist-valued models for PS3; in this way twist-valued models generalize Löwe and Tarafder’s three-valued ZF model, showing that all of them (including PS3) are models of ZFC. This offers more options for investigating independence results in paraconsistent set theory.

arXiv:1911.11833v1 [math.LO]

Logic (math.LO)

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