A foundation for deductive mathematics
Frank Quinn
TL;DR
The paper addresses foundational gaps in set theory by introducing relaxed sets built from object generators and a primitive, non-2-valued assertion logic, paired with the Coherent Limit Axiom. It constructs a universal well-founded system $\mathbb{W}$ and a universal almost well-ordered pairing that embeds all well-founded structures, showing these give a maximal ZFC-like model and align standard mathematics with this framework. It demonstrates that ZFC axioms are effectively captured within this maximal model and argues that forcing models are typically non-maximal and thus do not constrain ordinary practice. The work argues that mainstream mathematics already operates within this maximal, coherent framework, and provides detailed connections to classical set theory via recursion, Cantor-like arguments, and the Beth function, while clarifying the role and necessity of the Coherent Limit Axiom.
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\"ıve 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\"ıve 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.