Let AC denote the Axiom of Choice. Let PP denote the socalled "Partition Principle" which states that "If S is a nonempty set and T is a nonempty set of pairwise disjoint subsets of S, then S can be mapped onto T". It is well known that "AC implies PP" is provable in ZF, but the question of whether "PP implies AC" is provable in ZF has long been an open problem. What is the present status of this problem? Has any progress been made on it? Oron the other handhave any models of ZF been constructed in which pp is true and some consequence of AC such as the existence of nonmeasurable sets of real numbersis false? If "pp implies AC" could be proved in ZF, this would seem to provide a powerful philosophical argument for accepting AC. In my opinion, any set theory in which pp can be disproved yields a really counterintuitive picture of the "Settheoretical Universe".

The two main papers on the subject are the BanaschewskiMoore paper and a paper by Higasikawa, both are from more or less twenty years ago.
You can find a nice diagram of implications in Gregory Moore's book "Zermelo's Axiom of Choice" (which, oddly enough, is the second time I refer to on this site today). I have some master plan on how to prove its independence from the axiom of choice, but it's a wild and vague dream at the moment which doesn't worth much mentioning except for the fact that I believe, at the moment, that PP does not imply the axiom of choice. For whatever that is worth. One interesting fact on $\sf PP$ is that it implies the existence of nonmeasurable sets of real numbers all by itself. $\sf PP$ implies $\sf DC$, as well $\aleph_1\leq2^{\aleph_0}$ and therefore implies the existence of a nonmeasurable set. But in fact even much weaker versions of $\sf PP$ imply the existence of nonmeasurable, for example $\sf WPP$ which asserts $A\leq^\ast B\rightarrow B\nless A$, or in other words: if $B$ can be mapped onto $A$ then it cannot have a strictly smaller cardinality. The reason that $\sf WPP$ implies the existence of a nonmeasurable set is that $\Bbb R$ can always be mapped onto $[\Bbb R]^\omega$, the set of countably infinite sets of real numbers, and of course can be mapped into that set injectively. Since in $\sf ZF$ we have $\Bbb R\leq[\Bbb R]^\omega\leq^\ast\Bbb R$, in $\sf ZF+WPP$ we have that $\Bbb R$ is equipotent with $[\Bbb R]^\omega$. Sierpinski proved from this assumption that there exists a nonmeasurable set.


