Ultrafilter Richness: Explore Options Beyond AC

[BDV]

Hello,

With the axiom of choice, we are left with two options for ultrafilters:
a) principal ultrafilters, built from a singleton {x}.
b) nonpricipal filters of which all contain the cofinite filter, ergo complements of finite sets subalgebras.

Isn't this kind of flimsy? To get to more exotic/exciting objects does one:
give up AC
or
gives up the ultra in ultafilter (the A or X\A is in F condition)?

 

[nomadreid]

Why flimsy? For example, using an ultrafilter construction and without abandoning AC, you end up with measurable cardinals, which can do all sorts of interesting things. What sort of "exotic" objects did you have in mind?

 

[BDV]

I meant more exotic than the principal ultrafilters and complements of finite set subalgebras.

I realized last evening that non-principal ultrafilters may also contain objects with complements greater than finite sets. So there may be some richness right there.

I just wanted to get a good feel of the conceptual reach/limits of ultrafilters before I sink a significant amount of work in them. Ars longa, vita brevis.