There's nothing special about the definition. Use any construction you like, but do it over a countable model of set theory[1]. Now you have a countable set of real numbers, although there is no correspondence between it and the natural numbers within the model. The statement that 'there is an uncountable set' is also provable, although the set is countable. It's one of those Goedelian tricks, like the statement that is proven true but is unprovable.

That said, I prefer to say that only real numbers that can be emitted by computer programs exist.

[1] Use the Downward Lowenheim-Skolem theorem to get a countable set theory, and choose something like ZFC as your set theory.