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> if you reject the axiom of choice then the construction that Cantor does to construct a real isn't valid Are you talking about Cantor's argument that the rea
by OscarCunningham 7y ago
> if you reject the axiom of choice then the construction that Cantor does to construct a real isn't valid
Are you talking about Cantor's argument that the reals are uncountable? That doesn't need choice.
- thaumasiotes 7y agoElaborating, the hypothesis that Cantor disproves is "The real numbers are countable -- that is to say, the real numbers can be put into one-to-one correspondence with the natural numbers". You never have to use the axiom of choice, because the hypothesis tells you there is a one-to-one function between the reals and the naturals. You can then order the reals in the order suggested by their image in the naturals: f(0), f(1), f(2), ...