Thanks, Andrés (and similar belated congratulations), this is great. In retrospect I had seen $\mathbb E$ and the book you mention, but at least the random part is news to me.

@Peter: In that case this question is still open, because about $\Omega$ you can prove many nontrivial things, e.g. that it is ML-random. But is it possible to define "nontrivial"?

@Peter: I'm sure Voevodsky had in mind something more like my answer than Rodrigo Freire's answer. But if we really want to talk about all the facts about $S$ being undecided then Rodrigo's answer is just as correct as mine (and simpler). Because for any finitely many $n_i$ one could make another definition of my column of $\Omega$ where the facts about whether $n_i\in$ that column are built into the definition.

I guess the conclusion here is that Voevodsky's claim should have been reformulated, replacing "nothing" by "only finitely much".