On foundations for deductive mathematics

TL;DR

This paper explores the relationships between mathematical primitives, foundations, and deductive practice, proposing a new foundation based on the coherent limit axiom that aims to better align with mainstream mathematical reasoning.

Contribution

It introduces a novel foundational framework for deductive mathematics, emphasizing the role of the coherent limit axiom and its potential to optimize human mathematical practice.

Findings

Traditional set theories like ZFC are not fully consistent with mainstream practice.

The proposed foundation uniquely qualifies as the basis for deductive mathematics.

The coherent limit axiom could resolve set-theoretic questions such as the continuum hypothesis.

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.