Suppose we add the axiom to ZF that for any set $S$, and any partition $P$ of $S$, the cardinality of $P$ is less than or equal to the cardinality of $S$. It is known that the axiom of choice implies that axiom. My question is, is the converse true? Or are there models of ZF where that axiom holds but the axiom of choice doesn't?
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The converse appears to be essentially the partition principle, and whether it implies the axiom of choice is an open problem. – Brian M. Scott Dec 15 '20 at 01:09
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2I mean, I close this as a duplicate about once a month. It is literally the most recent question under the [axiom-of-choice] tag at the time you are writing this question! – Asaf Karagila Dec 15 '20 at 01:26