Here is my current concern.
Say that a set $E$ has the half-property when there exist sets $F$ and $G$ such that $E$ is the disjoint union of $F$ and $G$, $F$ and $G$ being isomorphic. Trivially, no finite set has the half property, whereas (trivially as well) any countable set or the set of real numbers has the half-property.
It seems to me that any infinite set has the half-property, but I feel that this could be related to some axiomatic system (involving the choice? I don't know). If $E$ is (up to isomorphism) a power set, then take $E'$ such that $E=\mathcal P(E')$, a in $E'$ and take $F$ as the subsets containing a and $G$ as the subsets without a; can this be helpful to solve the problem?
Thanks in advance for your thoughts!
