A construction of set theory

TL;DR

This paper develops a generalized, category-theoretic framework that encompasses a relaxed form of set theory, avoiding classical logic constraints and providing a maximal implementation of Zermillo-Fraenkel-Choice axioms.

Contribution

It introduces a new, category-theoretic approach to set theory that simplifies usage and embeds classical axioms in a maximal, universal structure.

Findings

Relaxed set theory is easier to use with full precision.

A maximal implementation of Zermillo-Fraenkel-Choice axioms is constructed.

The framework avoids classical logic constraints like the Law of Excluded Middle.

Abstract

We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside this we find "relaxed" set theory, which is much easier to use with full precision than traditional axiomatic theories. There is also an implementation of the Zermillo-Fraenkel-Choice axioms that is maximal in the sense that any other implementation uniquely embeds in it.