Take the tour ×  Let AC denote the Axiom of Choice. Let PP
denote the so-called "Partition Principle" which states that "If
S is a non-empty set and T is a non-empty 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? Or-on the other hand-have any
models of ZF been constructed in which pp is true and some
consequence of AC -such as the existence of non-measurable sets
of real numbers-is 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 counter-intuitive picture of the
"Set-theoretical Universe".  share|improve this question PP+DC
implies the existence of a nonmeasurable set -
mathoverflow.net/questions/22927/… –  Franç