They prove division by 3 preserves bijection between arbitrary sets without using the axiom of choice. From the intorduction: "For any sets A and B, if there i

They prove division by 3 preserves bijection between arbitrary sets without using the axiom of choice. From the intorduction: "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. Here 3 denotes the standard three-element set {0, 1, 2} and × denotes Cartesian product." There are a few related links and explanations on the arxiv trackbacks page - http://arxiv.org/tb/math/0605779 http://arxiv.org/tb/math/0605779