A Model Theory for the Potential Infinite

TL;DR

This paper develops a model theoretic framework for mathematics based on the potential infinite, where sets are indefinitely extensible finite entities, providing a sound and complete interpretation for classical first-order logic.

Contribution

It introduces a dynamic model theory for the potential infinite, interpreting the universal quantifier with an implicit reflection principle and establishing soundness and completeness.

Findings

Dynamic models are sound and complete for first-order logic.

Finite formulas can be interpreted within finite parts of the increasing model.

The potential infinite allows for a consistent development of mathematics without actual infinity.

Abstract

We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely extensible finite. The main adoption is the interpretation of the universal quantifier, which has an implicit reflection principle. Each universal quantification refers to an indefinitely large, but finite set. The quantified sets may increase, so after a reference by quantification, a further reference typically uses a larger, still finite set. We present the concepts for classical first-order logic and show that these dynamic models are sound and complete with respect to the usual inference rules. Moreover, a finite set of formulas requires a finite part of the increasing model for a correct interpretation.