On Dividing by Two in Constructive Mathematics

TL;DR

This paper explores the limitations of dividing by two in constructive mathematics, demonstrating that a classical isomorphism result does not hold in certain toposes without the axiom of choice.

Contribution

It shows that Bernstein's classical result fails in constructive mathematics by providing examples of toposes where the isomorphism does not imply the original sets are isomorphic.

Findings

Bernstein's result does not hold constructively

Examples of toposes where division by two fails

Constructive mathematics differs from classical in set isomorphism properties

Abstract

A classic result due to Bernstein states that in set theory with classical logic, but without the axiom of choice, for all sets $X$ and $Y$, if $X \times 2 \cong Y \times 2$ then also $X \cong Y$. We show that this cannot be done in constructive mathematics by giving some examples of toposes where it fails.