Results in Theoretical CS independent of ZFC I'm going to ask a quite vague question, since the borderline between theoretical computer science and math is not always easy to distinguish.

QUESTION: Are you aware of any interesting result in CS which 
is either independent of ZFC (i.e. standard set theory),
or that was originally proven in ZFC (+ some other axiom) and only later proven in ZFC alorne?

I'm asking because I'm close to finish my PhD thesis, and my main result (the determinacy of a class of games which are used to give "game semantics" to a probabilistic modal $\mu$-calculus) is at the moment proven in ZFC extended with other Axioms (namely the negation of the Continuum hypothesis $\neg CH $ and, Martin's Axiom $MA$).

So the setting is clearly Computer Science (the modal $\mu$-calculus is a temporal logic, and I'm extending it to work with probabilistic systems).

I would like to cite in my thesis other examples (if you are aware of any) of this kind.

Thank you in advance,

bye

Matteo
While I am not aware of any such results, other than your own, I think you could broaden the scope somewhat and ask: what results in TCS have been proved using (any kind of) non-standard axioms. By non-standard here I mean something other than classical logic with ZF (or ZFC).

A beautiful example of the kind of work I have in mind are Alex Simpson's results on properties of programming languages using synthetic domain theory. He uses intuitionistic set theory with axioms that contradict classical logic.

Also, Alex and I used intuitionistic axioms with anti-classical continuity principles to show results about Banach-Mazur computabilty.

However, none of the mentioned examples have an "open" status, like your proofs, because we know that the non-standard axioms we used can be understood simply as working inside a model of intuitionistic mathematics, where the model can be shown to exist in ZFC. So the non-standard setup is really a way of getting things done more elegantly, and in principle they could be done in straight ZFC (although I am afraid to think about how exactly that would go).