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I have a question about Cantor's first uncountability proof.

What definition of $\mathbb{R}$ did Cantor use in his proof? Is that, for example, build on the axiom about bounded sets or build on the axiom formulated using interval nests?

I tried to find it in the original proof, but I can't find it. Who can help me with a good source?

Maybe anyone can help me, thanks in advance!

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