Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory

[Joao Marcos]

Pode ser de interesse para alguns dos colegas.

JM

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arXiv:2210.00057
Date: Fri, 30 Sep 2022 19:39:37 GMT (37kb)

Title: Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory
Authors: Yurii Khomskii, Hrafn Valt\'yr Oddsson
Categories: math.LO
MSC-class: 03E70, 03B53, 03B60
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We present a novel treatment of set theory in a four-valued paracomplete and
paraconsistent logic, i.e., a logic in which propositions can be neither true
nor false, and can be both true and false. By prioritising a system with an
ontology of non-classical sets that is easy to understand and apply in
practice, our approach overcomes many of the obstacles encountered in previous
attempts at such a formalization. We propose an axiomatic system BZFC, obtained
by analysing the ZFC-axioms and translating them to a four-valued setting in a
careful manner. We introduce the anti-classicality axiom postulating the
existence of non-classical sets, and prove a surprising results stating that
the existence of a single non-classical set is sufficient to produce any other
type of non-classical set. We also look at bi-interpretability results between
BZFC and classical ZFC, and provide an application concerning Tarski semantics,
showing that the classical definition of the satisfaction relation yields a
logic precisely reflecting the non-classicality in the meta-theory.
\\ ( https://arxiv.org/abs/2210.00057 , 37kb)
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