Is every ultrafilter generated by a small set?

Question

If $|X| = \kappa$, can it be the case that every ultrafilter $\mathcal{F}$ on $X$ is generated by a set $\mathcal{F}_0 \subseteq PX$ of size $<2^\kappa$? Here I say that $\mathcal{F}$ is generated by $\mathcal{F}_0$ if $\mathcal{F}$ is the smallest filter containing $\mathcal{F}_0$. Let's say that $\mathcal{F}$ is small-generated if there exists $\mathcal{F}_0$ which generates $\mathcal{F}$ and has size $|\mathcal{F}_0| <2^\kappa$. So the question is whether every ultrafilter is small-generated. I'd like to show that the answer is no. If it helps to assume that $\kappa$ is regular or something, that would be fine.

Since there are $2^{2^\kappa}$-many ultrafilters on $X$, an easy counting argument shows that not every ultrafilter is generated by a set of size $\kappa$ (there can be at most $2^\kappa$ of these). Nor can every ultrafilter be generated by a set of size $\lambda$ if $2^\lambda < 2^{2^{\kappa}}$. So under GCH, the answer is in fact no. But without GCH, we could have $\lambda < 2^\kappa$ but $2^\lambda = 2^{2^\kappa}$. So counting only gets us so far.

It's also easy to show that if $\mathcal{F}$ is not small-generated, then for any $A \subseteq X$, either the filter generated by $\mathcal{F} \cup \{A\}$ is not small-generated, or else the filter generated by $\mathcal{F} \cup \{\neg A\}$ is not small generated, where $\neg A$ is the complement of $A$. This suggests trying to build a non-small-generated ultrafilter by starting with a non-small-generated filter and expanding it inductively. But I think the union of a small set of non-small-generated filters can be small-generated, so I don't know what to do at limit steps. And anyway, I don't even know of a base case to use.

Answer 1

In fact it is a more tricky problem to get a model with a "small-generated" non-principal uf over $\omega$. See Kunen p 289 (old book) p 345 (new book), where an uf base of cardinality $\aleph_1$ in a model with $\mathfrak c >\aleph_1$ is generically defined. Or Baumgartner/Laver in Annals of Mathematical Logic 17 (1979) 271-288 where it is shown that some ufs in $L$ remain uf bases in $\omega_2$-long Sacks-forcing iterated extensions of $L$.

Answer 2

Let $F \subseteq [\kappa]^{\kappa}$ be an independent family (every finite boolean combination has size $\kappa$) of size $2^{\kappa}$. Put $E = \{\kappa \setminus \bigcap A: A \in [F]^{\aleph_0}\}$. Let $U$ be an ultrafilter containing $F \cup E$. Then $U$ is not generated by fewer than $2^{\kappa}$ sets. You can easily generalize this construction to get $2^{2^{\kappa}}$ such ultrafilters.

It is, however, unknown if it is consistent to have a uniform ultrafilter on $\omega_1$ which can be generated by fewer than $2^{\omega_1}$ sets.