The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.

The fact, I think, was about the undecidability (read: independence from ZFC or maybe ZF axioms) of a certain seemingly very "natural" sentence about the convergence of sequences of holomorphic functions in one variable.

So the question is:

What are some natural undecidable sentences about holomorphic functions? Where by "undecidable" I mean independent from the ZFC or the ZF axioms of set theory, and by "natural" I mean something that is not manifestly designed to be an independence result and possibly that arised quite autonomously from Logic.

Edit: I'm aware there are some independence results (I think by Kranz and Di Biase) related to the boundary behaviour of holomorphic functions. The "fact" I wanted to recall is not part of this theory, though independence examples related to this theory are well accepted in the answers.

uniqueanswer, perhaps none do, since one can often contribute ideas from different perspectives or additional gloss and so on. $\endgroup$ – Joel David Hamkins Feb 22 '13 at 0:15