A question about the Axiom of Choice

Question

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".

Answer 1

To my best knowledge, the Banaschewski-Moore paper from some twenty years ago is pretty much the last recorded progress on the topic.

The two main papers on the subject are the Banaschewski-Moore paper and a paper by Higasikawa, both are from more or less twenty years ago.

1. Bernhard Banaschewski, Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame J. Formal Logic 31 (3), (1990), 375–381.

2. Masasi Higasikawa, Partition principles and infinite sums of cardinal numbers. Notre Dame J. Formal Logic 36 (1995), no. 3, 425–434.

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 non-measurable 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 non-measurable set. But in fact even much weaker versions of $\sf PP$ imply the existence of non-measurable, 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 non-measurable 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 non-measurable set.

Sierpinski, W. "L’axiome de M. Zermelo et son rôle dans la théorie des ensembles et l’analyse." Bulletin de l’Académie des Sciences de Cracovie, Classe des Sciences Math., Sér. A (1918), 97-152.