Extending the extensional level of the Minimalist Foundation to axiomatic set theories

TL;DR

This paper extends the extensional level of the Minimalist Foundation with new rules, making it equivalent to both constructive and classical axiomatic set theories, thus broadening its foundational applicability.

Contribution

It introduces rule-based extensions to the Minimalist Foundation's extensional level, establishing equivalence with key axiomatic set theories.

Findings

Extensions are equivalent to constructive set theories

Extensions are equivalent to classical set theories

Framework broadens foundational options for mathematics

Abstract

We introduce extensions by rules of the extensional level of the Minimalist Foundation which turn out to be equivalent to constructive and classical axiomatic set theories.