Division by four

TL;DR

This paper presents a simple proof that for finite sets, multiple injections imply a single injection, and from this derives that multiple bijections imply a single bijection, clarifying a long-standing mathematical history.

Contribution

It provides a straightforward proof connecting finite multiple injections and bijections, resolving a complex historical debate in set theory.

Findings

nA<=nB implies A<=B for finite sets

nA==nB implies A==B for finite sets

Clarifies the relationship between multiple injections and bijections in finite sets

Abstract

Write A<=B if there is an injection from A to B, and A==B if there is a bijection. We give a simple proof that for finite n, nA<=nB implies A<=B. From the Cantor-Bernstein theorem it then follows that nA==nB implies A==B. These results have a long and tangled history, of which this paper is meant to be the culmination.