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.)

  • $\begingroup$ Well, you should start by telling us what are some non-intuitive concepts which follow from the axiom of choice. Because most people don't know that, but without choice the set theoretic universe becomes even less intuitive. So adding back some choice, you can ask what starts to make sense again and what doesn't have to. But these are "negative results", and quite the opposite of your question. $\endgroup$ – Asaf Karagila Sep 24 '17 at 3:43
  • $\begingroup$ The Banach­–Tarski paradox, for example, follows from the Hahn–Banach theorem, which itself follows from the ultrafilter lemma. Is this what you're asking for? $\endgroup$ – Asaf Karagila Sep 24 '17 at 3:47
  • $\begingroup$ Look up "infinite hat guessing". $\endgroup$ – Nate Eldredge Sep 24 '17 at 5:53
  • $\begingroup$ Ah. I didn't think one would get all the way to the Banach-Tarski paradox with so little choice - but yes, that's exactly the sort of thing I had in mind. @AsafKaragila - would you mind posting that as an answer, even if it trivializes the question? And I'm aware that set theory gets weird regardless of whether or not you like choice... but I was asked this during a conversation with a non-mathematician. The strange consequences of choice seem to have a better publicist than the anti-choice oddities. $\endgroup$ – Eric Astor Sep 24 '17 at 11:59

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.


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