I define a non-empty set $S$ to be divisible by $2$ if it can be partitioned into two sets $\{A,B\}$ such that $A$ and $B$ have the same cardinality. I also define a non-empty set $S$ to be pair-partitionable if it can be partitioned into a partition that consist exclusively of unordered pairs. I have some questions regarding these definitions.
- Without the axiom of choice, can any infinite set be divided by $2$?
- Without the axiom of choice, is every infinite set pair-partitionable?
- Now, my main question. Without the axiom of choice, for every infinite set $S$, is being divisible by $2$ an equivalent property of being pair-partitionable?
