Vop\v{e}nka's Alternative Set Theory in the Mathematical Canon of the 20th Century: Author's Translation from Czech

TL;DR

This paper explores Vopěnka's Alternative Set Theory, highlighting its origins, axiomatic foundations, and its role within 20th-century mathematics, emphasizing its philosophical and mathematical significance.

Contribution

It provides a comprehensive mapping of the mathematical context of Vopěnka's Alternative Set Theory and investigates its philosophical influences and development.

Findings

Links to nonstandard universes and Skolem's models

Axiomatization of Robinson's Nonstandard Analysis

Position within 20th-century mathematical logic

Abstract

Vop\v{e}nka's Alternative Set Theory can be viewed both as an evolution and as a revolution: it is based on his previous experience with nonstandard universes, inspired by Skolem's construction of a nonstandard model of arithmetic, and its inception has been explicitly mentioned as an attempt to axiomatize Robinson's Nonstandard Analysis. Vop\v{e}nka preferred working in an axiomatic theory to investigating its individual models; he also viewed other areas of nonclassical mathematics through this prism. This article is a contribution to the mapping of the mathematical neighbourhood of the Alternative Set Theory, and at the same time, it submits a challenge to analyze in more detail the genesis and structure of the philosophical links that eventually influenced the Alternative Set Theory.