And what is this idea that a set can be “defined” but not “constructively defined”? The language of set theory and first-order logic is *identical* in classical and constructive mathematics.

I think I understand the proof of Theorem 1.3 using classical reasoning, but am wondering where the proof of Theorem 1.3 fails to establish LEM in constructive mathematics.

In particular the proof of Theorem 1.3 ( that AC=>LEM) seems to use only a very weak, (and constructively accepted?) form of AC, namely that if we have precisely one or two distinct inhabited subsets of {0,1}, then we may pick an element from each one.

Are the sets A and B `constructively well defined’?

Is the problem that we `have’ two sets A and B but don’t `know’ if A=B?

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