A foundation for deductive mathematics

TL;DR

This paper proposes a new set-theoretic foundation for deductive mathematics that aligns with mainstream practice and addresses inconsistencies in traditional axiomatic set theories like ZFC.

Contribution

It introduces a set theory framework that corresponds to the maximal model used in standard mathematics, clarifies the role of axioms, and identifies the limitations of ZFC models.

Findings

Standard practice implicitly uses the proposed set theory.

Most of traditional axiomatic set theory is irrelevant to everyday mathematics.

The 'coherent limit axiom' holds in the maximal model and not in others.

Abstract

Set theory is widely believed to provide a secure foundation for deductive mathematics, but current set theories do not quite do this. The mainstream essentially uses na\"\i ve set theory. After Russell's paradox showed this to be inconsistent, the patch ``don't say `set of all sets' '' was added. The resulting methodology has been extremely successful, but still lacks a consistent foundation. The set theory community extracted properties of na\"\i ve set theory to use as axioms, culminating in the Zermillo-Fraenkel-Choice (ZFC) axioms. Unfortunately they missed an axiom, and ZFC as it stands is not consistent with standard methodology. This paper addresses these issues. The first dozen pages (Sections 1--5) gives primitives, defines sets in this context, and verifies that these have the properties used in standard practice. Sections 6--7 relates this to traditional axiomatic set theory. We show the sets here correspond to the sets in a maximal model for the ZFC axioms. Section 8 gives the ``coherent limit axiom'', considered obviously true in mainstream practice, and shows it holds in the maximal model and fails in all others. There are several qualitative conclusions. First, standard mainstream practice implicitly takes place in the set theory described here. This also shows there are no ``hidden axioms'': we already have the full toolkit. Second, most of the axiomatic set theory of the last hundred years is irrelevant to standard mathematical practice. The ZFC models produced by forcing, for example, are essentially never maximal, and therefore do not constrain or inform standard practice.