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

Text
Paraconsistent Paper.pdf - Submitted Version Available under License Creative Commons Attribution No Derivatives. Download (486kB) |

## 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 |

## Actions (login required)

View Item |