Natural Number Arithmetic in the Theory of Finite Sets

TL;DR

This paper develops a finite set-based theory of natural numbers, exploring diverse natural number systems with varying lengths and closure properties, and constructing models for weak arithmetic systems.

Contribution

It introduces a finite set theory framework for natural numbers, defining hierarchies and methods to construct systems with specific closure properties, extending prior infinitary theories.

Findings

Natural number systems vary in length and closure properties.

Hierarchies of natural number systems are constructed with increasing power.

A method for constructing natural number systems with desired closure properties is described.

Abstract

We describe a theory of finite sets, and investigate the analogue of Dedekind's theory of natural number systems (simply infinite systems) in this theory. Unlike the infinitary case, in our theory, natural number systems come in differing lengths and with different closure properties. We give examples of natural number systems incomparable in length; we define hierarchies of natural number systems closed under increasingly powerful functions; and we describe a method by which to construct natural number systems with given closure properties. These natural number systems form natural models for various systems of weak arithmetic.