Reference Request: Independence of the ultrafilter lemma from ZF - MathOverflow

Reference Request: Independence of the ultrafilter lemma from ZF

2011-03-22T09:31:24Z

I'm looking for references for the following facts concerning the ultrafilter lemma (~ "there exist non-principal ultrafilters"):

The ultrafilter lemma is independent of ZF.
ZF + the ultrafilter lemma does not imply the Axiom of Choice.

I would prefer an overview article / book that links to the original papers instead of the original papers themselves. That's because I don't rely on these facts; I only want to give some context for my readers.

Answer by Michael Greinecker for Reference Request: Independence of the ultrafilter lemma from ZF

2011-03-22T10:14:58Z

The axiom "There exists some nonprincipalultrafilter." is discussed in Horst Herrlich, "The Axiom of Choice" as WUF(?) and in Rubin and Rubin "Equivalents of the Axiom of Choice, II." under [206]. The latter is kind of the standard ressource for these kinds of questions.

Here are papers providing proofs:

For 1:

MR0480028 (58 #227) Pincus, David ; Solovay, Robert M. Definability of measures and ultrafilters. J. Symbolic Logic 42 (1977), no. 2, 179--190.

For 2:

MR0480027 (58 #226) Pincus, David . Adding dependent choice to the prime ideal theorem. Logic Colloquium 76 (Oxford, 1976), pp. 547--565. Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977.

Answer by arsmath for Reference Request: Independence of the ultrafilter lemma from ZF

2011-03-22T16:40:10Z

If you're just interested in context, rather than proofs, Eric Schechter's book Handbook of Analysis and Its Foundations talks about the relation between existence of ultrafilters, various weak notions of choice, and standard results in analysis.

Answer by Andreas Blass for Reference Request: Independence of the ultrafilter lemma from ZF

2011-03-22T20:44:20Z

In any of the formulations mentioned (so far) in the comments, the ultrafilter lemma is independent of ZF but weaker than AC. That the strongest form (all filters can be extended to ultrafilters) doesn't imply AC is a theorem of J.D. Halpern and A. Lévy ["The Boolean prime ideal theorem does not imply the axiom of choice" in Axiomatic Set Theory, Proc. Symp. Pure Math. XIII part 1, pp. 83-134]. That ZF doesn't prove even the weakest form (there exists a nonprincipal ultrafilter on some set) is a theorem of mine ["A model without ultrafilters," Bull. Acad. Polon. Sci. 25 (1977) pp. 329-331], building on S. Feferman's construction of a model with no non-principal ultrafilters on the set of natural numbers ["Some applications of the notions of forcing and generic sets," Fundamenta Mathematicae 55 (1965) pp. 325-345].