Are there non-intuitive consequences of the ultrafilter lemma?

Question

The ultrafilter lemma states that, for every set $X$, every filter on $X$ is contained in an ultrafilter on $X$.

We know this gives us a weak form of choice... and we know the non-intuitive consequences of the Axiom of Choice, too.

Are there good "paradoxical" results that follow from the ultrafilter lemma alone? (I'm willing to accept the existence of non-measurable sets as reasonable, so I'm looking for something a bit less intuitive than that.)

Answer 1

The Banach–Tarski paradox can be proved using nothing more than the Hahn–Banach theorem, which itself is a consequence of the Ultrafilter Lemma. So as far as "paradoxical results" go, this is the most famous one.

But the Ultrafilter Lemma cannot prove the following things:

1. Every infinite set has a countably infinite subset. In particular, it is possible that such set is a dense set of reals.

2. The real numbers are hereditarily Lindelöf, or that $\Bbb N$ is Lindelöf.

3. Continuity of $f\colon\Bbb R\to\Bbb R$ at a point $x$ is equivalent to sequential continuity at $x4 (or generally between metric spaces).

4. Given $X,Y\subseteq\Bbb R$ such that both $X$ and $Y$ are strictly smaller than $\Bbb R$ in cardinal, then $|X\cup Y|<2^{\aleph_0}$.

And many many other weird consistency results without choice are consistent with the Ultrafilter Lemma.