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See Andrej Bauer's proof that they are indeed uncountable: https://mathoverflow.net/questions/30643/are-real-numbers-countable-in-constructive-mathematics https
by ogogmad 6y ago
See Andrej Bauer's proof that they are indeed uncountable: https://mathoverflow.net/questions/30643/are-real-numbers-countable-in-constructive-mathematics https://mathoverflow.net/questions/30643/are-real-numbers-co...
- adamnemecek 6y agoI'm still unconvinced.
- fractionalhare 6y agoIs there a specific issue you have with this proof? With respect, I think you may be conflating a constructive proof of the uncountability of the reals with a proof that the set of all computable reals is countable. These are two different claims. A statement about the reals which is provable under constructive mathematics does not reduce to a statement about the computable reals and vice versa. A set can be constructively definable without all of its elements being constructively enumerable.
- ogogmad 6y agoGiven any explicit sequence of real numbers, it's always possible to generate a number not in that sequence. In fact, this can be done algorithmically using essentially Cantor's method.