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