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See Kleene realizability in Peano arithmetic for a similar question, but about PA instead of ZFC. (In particular, an answer as specific as Emil Jeřábek's answer would be great!) In the context of ...

Let us say that an arithmetic statement is constructively true iff it is realized by a computable function under Kleene's function realizability. Does the set of constructively true (first order) ...

Everything I know on this subject comes from Sacks book : "Higher recursion theory" Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$. We should have the result that $A \...

I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO users....