English

On Dividing by Two in Constructive Mathematics

Logic 2018-04-13 v1 Combinatorics

Abstract

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

Keywords

Cite

@article{arxiv.1804.04490,
  title  = {On Dividing by Two in Constructive Mathematics},
  author = {Andrew Swan},
  journal= {arXiv preprint arXiv:1804.04490},
  year   = {2018}
}
R2 v1 2026-06-23T01:21:42.103Z