Finite Sets and Counting

TL;DR

This paper develops a theory of finite sets inspired by Principia Mathematica, avoiding natural numbers, and applies it to analyze structures satisfying induction but not all Peano axioms.

Contribution

It introduces a novel finite set theory that does not rely on natural numbers and explores its implications for structures satisfying induction.

Findings

Finite set theory developed without natural numbers

Structures satisfying induction but not all Peano axioms analyzed

New results on the properties of such structures obtained

Abstract

We start by presenting a theory of finite sets using the approach which is essentially that taken by Whitehead and Russell in Principia Mathematica}, and which does not involve the natural numbers (or any other infinite set). This theory is then applied to prove results about structures which, like the natural numbers, satisfy the principle of mathematical induction, but do not necessarily satisfy the remaining Peano axioms.