PP-2022-08: Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory

PP-2022-08: Khomskii, Yurii and Oddsson, Hrafn Valtýr (2022) Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory. [Pre-print] (Submitted)

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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.

Item Type: Pre-print
Report Nr: PP-2022-08
Series Name: Prepublication (PP) Series
Year: 2022
Uncontrolled Keywords: Non-classical set theory; paraconsistent set theory; paraconsistent logic; paraconsistent and paracomplete set theory
Subjects: Logic
Depositing User: Dr. Yurii Khomskii
Date Deposited: 11 Oct 2022 17:50
Last Modified: 11 Oct 2022 17:50
URI: https://eprints.illc.uva.nl/id/eprint/2224

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