Problem #5

Does the Partition Principle imply the Axiom of Choice?

Open!!!

The Partition Principle ( $\mathsf{PP}$ ) states that for any sets $A$ and $B$ , if there is a surjection from $A$ to $B$ then there is an injection from $B$ to $A$ . This question asks whether the Partition Principle implies the Axiom of Choice over $\mathsf{ZF}$ .

Set Theory

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Known Partial Results

Pincus proved that $\mathsf{PP}$ implies that every well-ordered family of nonempty sets has a choice function.
By a result of Jensen, the above theorem of Pincus shows that $\mathsf{PP}$ also implies $\mathsf{DC}$ , the Axiom of Dependent Choice.

Notes

This question was first posed in the early 1900s and is one of the oldest open problems in set theory.

Reference for the problem statement

[BM90]Bernhard Banaschewski and Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame Journal of Formal Logic, 1990 [doi]

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