On foundations for deductive mathematics

TL;DR

The paper critiques the adequacy of mainstream foundations, notably ZFC, to faithfully support everyday deductive mathematics and proposes a new foundational framework built below set theory, centered on object generators and a non-binary logic core. It introduces the coherent limit axiom as a key principle that, within a relaxed set theory, yields a maximal, consistent foundation that embeds ZFC-like theories while remaining aligned with current mathematical practice. The approach emphasizes primitive generation of objects, minimal explicit assumptions, and global consistency checks, aiming to simplify foundations without sacrificing deductive power. If successful, this framework could resolve long-standing set-theoretic questions and improve the reliability and usability of foundations for mainstream mathematics.

Abstract

This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what is, or isn't, a foundation; and get clues as to how a foundation can be optimized for effective human use. For this we turn to history and professional practice of the subject. We have no asperations to Philosophy. The first section gives a short abstract discussion, focusing on the significance of consistency. The next briefly describes foundations, explicit and implicit, at a few key periods in mathematical history. We see, for example, that at the primitive level human intuitions are essential, but can be problematic. We also see that traditional axiomatic set theories, Zermillo-Fraenkel-Choice (ZFC) in particular, are not quite consistent with mainstream practice. The final section sketches the proposed new foundation and gives the basic argument that it is uniquely qualified to be considered {the} foundation of mainstream deductive mathematics. The ``coherent limit axiom'' characterizes the new theory among ZFC-like theories. This axiom plays a role in recursion, but is implicitly assumed in mainstream work so does not provide new leverage there. In principle it should settle set-theory questions such as the continuum hypothesis.