Without any choice axioms, are there free ultrafilters on the natural numbers? If not, can we prove the existence of ANY free ultrafilters, on any set?
2026-04-05 02:19:27.1775355567
What is necessary for having a free ultrafilter?
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No. To both questions.
It is consistent that the axiom of choice fails and there are no free ultrafilters on the natural numbers; and it is also consistent that the axiom of choice fails and there are no free ultrafilters on any set.
I'm not sure who proved the first result. The second result is due to Andreas Blass.