Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF

TL;DR

The paper proves that a recently introduced nonclassical set theory T, which extends ZF and avoids countable models, is relatively consistent with ZF, marking progress in mathematical foundations.

Contribution

It demonstrates the relative consistency of the nonclassical theory T with ZF, supporting its potential as an advancement in foundational mathematics.

Findings

T is relatively consistent with ZF.

T is finitely axiomatized.

T does not have a countable model if it has any.

Abstract

Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF, that is finitely axiomatized, and that does not have a countable model (if it has a model at all, that is). Here we prove that T is relatively consistent with ZF. We conclude that this is an important step towards showing that T is an advancement in the foundations of mathematics.