# Division by three

• Published in 2006
In the collection
We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 cross A and 3 cross B then there is a one-to-one correspondence between A and B. The first such proof, due to Lindenbaum, was announced by Lindenbaum and Tarski in 1926, and subsequently lost'; Tarski published an alternative proof in 1949. We argue that the proof presented here follows Lindenbaum's original.

### BibTeX entry

@article{Doyle2006,
title = {Division by three},
author = {Doyle, Peter G. and Conway, John Horton},
url = {https://arxiv.org/abs/math/0605779v1 https://arxiv.org/pdf/math/0605779v1.pdf},
urldate = {2014-11-17},
abstract = {We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 cross A and 3 cross B then there is a one-to-one correspondence between A and B. The first such proof, due to Lindenbaum, was announced by Lindenbaum and Tarski in 1926, and subsequently lost'; Tarski published an alternative proof in 1949. We argue that the proof presented here follows Lindenbaum's original.},
comment = {},
month = {may},
year = 2006,
collections = {Fun maths facts}
}