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The only examples of ultranets/ultrafilters described in Bourbaki and Willard are the trivial ones (generated by a single point). I know that their existence relies in general on the axiom of choice or the ultrafilter lemma.

I have two questions:

has any non-trivial ultranet been constructed?

if not:

is there any strong reason why no non-trivial ultranet has/will ever be constructed?

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    $\begingroup$ What do you mean by "constructed"? Many ultranets have been "constructed" using the Axiom of Choice. Presumably those are not "constructions" in your sense. There are models of ZF without any nonprincipal ultrafilters - see ams.org/mathscinet-getitem?mr=476510 $\endgroup$
    – François G. Dorais
    Commented Dec 22, 2010 at 16:13
  • $\begingroup$ @Dorais. By "constructed" I mean (although I don't how to express it completely formal) a way that gives more insight than just the knowledge of its existence. The same as in Willard: "Explicit constructions of free ultrafilters have never been acomplished" Correct me if I'm wrong: according to that paper there is no hope on finding such a construction only with ZF, as it is consistent that no non-trivial ultrafilter exists at all. $\endgroup$
    – Gabriel Furstenheim
    Commented Dec 22, 2010 at 16:42
  • $\begingroup$ @François: it seems to me that the paper you cite in your above comment answers the OP's question. Upon reading the question, I myself searched briefly for such a result but didn't find it. Perhaps you should leave this as an actual answer? $\endgroup$
    – Pete L. Clark
    Commented Dec 22, 2010 at 17:16

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It is not possible to prove in ZF alone that there is a nonprincipal ultrafilter. This was established by Andreas Blass in 1977. [A model without ultrafilters, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 329–331, MR0476510]

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