The concept of a set

TL;DR

This paper argues that the metaphysical basis of set theory is flawed and proposes a foundational system based on mathematical conceptualism, aligning better with actual mathematical practice than traditional Zermelo-Fraenkel set theory.

Contribution

It introduces a new foundational framework that distinguishes three types of collections and logic, improving the philosophical and practical understanding of sets in mathematics.

Findings

Sets are better understood as three types of collections with distinct logical foundations.

A new foundational system based on mathematical conceptualism is proposed.

This system aligns more closely with actual mathematical practice than ZF set theory.

Abstract

Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three different kinds of logic (classical, intuitionistic, and minimal). A foundational system incorporating this analysis and based on the principles of mathematical conceptualism accords better with actual mathematical practice than Zermelo-Fraenkel set theory does.