Division by three

• Published in 2006
• Added on 2014-11-17

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.

Links

• https://arxiv.org/abs/math/0605779v1
• https://arxiv.org/pdf/math/0605779v1.pdf

Other information

BibTeX entry

@article{Doyle2006,
	key = {Doyle2006},
	type = {article},
	title = {Division by three},
	author = {Doyle, Peter G. and Conway, John Horton},
	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 = {},
	date_added = {2014-11-17},
	date_published = {2006-05-01},
	urls = {https://arxiv.org/abs/math/0605779v1,https://arxiv.org/pdf/math/0605779v1.pdf},
	collections = {Fun maths facts},
	url = {https://arxiv.org/abs/math/0605779v1 https://arxiv.org/pdf/math/0605779v1.pdf},
	urldate = {2014-11-17},
	month = {may},
	year = 2006
}