# Boolean Prime Ideal Theorem and non-principal ultrafilters

Somewhat related to my other question Existence of non-principal ultrafilters on sets, is it known whether it is consistent with ZF that every infinite set has a free (non-principal) ultrafilter, but not every filter on a Boolean algebra can be extended to an ultrafilter? (Again, use you favorite interpretation of "infinite" here. Large cardinals may be used.)

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In Jech's book "The Axiom of Choice" it is stated that the Prime Ideal Theorem may fail while every infinite set has a non-trivial ultrafilter. The proof is the exercise 8.5 (I don't have time to sketch it right now, hence the comment). –  Apostolos Sep 27 '11 at 14:02
Thank you. I will have a look at the exercise. –  Stefan Geschke Sep 27 '11 at 16:28