Applicable Mathematics in a Minimal Computational Theory of Sets

TL;DR

This paper enhances a static logical framework for set theories, demonstrating that a minimal predicatively acceptable theory can be computational and capable of developing significant portions of applied mathematics, including real numbers as a class.

Contribution

It introduces a method to extend the framework with definitions without losing its static nature and shows the minimal theory's computational power and applicability to real analysis.

Findings

The minimal theory's elements are denoted by closed terms in its transitive model.

The minimal theory suffices to develop large parts of applied mathematics.

Real numbers can be incorporated as a proper class within the theory.

Abstract

In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored. In this work we first improve that framework by enriching it with means for coherently extending by definitions its theories, without destroying its static nature or violating any of the principles on which it is based. Then we turn to investigate within the enriched framework the power of the minimal (predicatively acceptable) theory in it that proves the existence of infinite sets. We show that that theory is a computational theory, in the sense that every element of its minimal transitive model is denoted by some of its closed terms. (That model happens to be the second universe in Jensen's hierarchy.) Then we show that already this minimal theory suffices for developing very large portions (if not all) of scientifically applicable mathematics. This requires treating the collection of real numbers as a proper class, that is: a unary predicate which can be introduced in the theory by the static extension method described in the first part of the paper.