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What are these non-classical versions of ZFC defined by realizability?
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 ...
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Complexity of constructive arithmetical truth vs second order arithmetic
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) ...