[nardi]

The parent says, and I quote: "...then you get that the reals are uncountable only 'because' of transcendental numbers."

But this is untrue, is it not? Because even if you removed all of the real transcendentals, the real numbers would still be uncountable because of the irrationals.

(Edit: Or are the non-transcendental irrationals countable? I can see how that might be true?)

[carstimon]

Your edit is correct. Non-transcendentals are algebraic by definition, and the algebraic numbers are countable.