TITLE: How to decompose an infinite set into two isomorphic ones without choice?
QUESTION [10 upvotes]: Is it provable in ZF that each infinite set $A$ can be partitioned into two sets $B$ and $C$ which are isomorphic to each other, i.e., there is a bijective map from $B$ to $C$?

REPLY [9 votes]: To answer the question in Todd's answer, the principle $2|A|=|A|$ for all infinite $A$ is strictly weaker than AC.  This difficult result was proved by Halpern and Howard for ZFA (allowing atoms) and by Sageev for ZF.

REPLY [3 votes]: The question seems interesting to me. There's an old result of Tarski that the axiom of choice is a consequence of ZF + the assertion "every infinite set $u$ is in bijection with $u^2$". A proof of this fact is given in Stoll's Set Theory and Logic; see page 126 (in the Dover edition). 
However, exercise 10.5 on that page reads, "It is not known whether the formula $2u = u$, which follows from the axiom of choice (Theorem 9.5), implies the axiom of choice. Attempt a proof of this." Stoll's book was written some time ago, but I suspect this implication is still not known.