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Division by three

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Division by threePeter G. DoyleJohn Horton Conway Version dated 1994GNU FDL AbstractWe prove without appeal to the Axiom of Choice that for any setsAandB, if there is a one-to-one correspondence between 3 Aand3 Bthen there is a one-to-one correspondence first such proof, due to Lindenbaum, was announced by Linden-baum and Tarski in 1926, and subsequently lost ; Tarski publishedan alternative proof in 1949. We argue that the proof presented herefollows Lindenbaum s Classification numbers03E10 (Primary); 03E25 (Sec-ondary).1 IntroductionIn this paper we show that it is possible to divide by three . Specifically,we prove that for any setsAandB, if there is a one-to-one correspondencebetween 3 Aand 3 Bthen there is a one-to-one correspondence between John Conway collaborated on the research reported here, and has been listed as anauthor of this work since it was first distributed in 1994.

Division by three Peter G. Doyle John Horton Conway Version dated 1994 GNU FDLy 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 A and 3 B then there is a one-to-one correspondence between A and B.

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