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.