countable

The solution is either to distinguish between countable and enumerable and decidable, or to use intuitionistic logic.

The proof that the cardinality of the reals is non-countable is perfectly constructive, see [1] for a discussion of these and related issues.

This is actually a very intuitive argument for why the reals (oops I mean decimals) are not countable.

If one uses classical logic there are even many countable, even finite sets which are beyond computable distinction.

You will inevitably find that the definable real numbers, like the computable reals, appear to be countable.

This forces us to add extra axioms saying which sets are ok: countable choice, dependent choice etc.

So the fact that something is computable, per se, isn't exactly what makes the real numbers countable rather than uncountable.

So the fact that the computable reals and the definable reals appear countable, is, at least to me, much like the statement that there are countable models of your favorite formal system.

Using rational numbers or other countable sets in this position usually leads to undesirable biases and or circuity in the model.

On the other hand, you could prove a metatheoretic result, constructing a model which only contains sets which are forced to exist by some finite formula (and therefore the model is countable).

After all, Skolem's paradox says that, meta-theoretically, we could have countable models of the classical real numbers as well.

To show the finite version implies the infinite version, you can use the countable version of Zorn's lemma (which you prove from the countable theorem of choice -- note it's a theorem, not an axiom).

The natural numbers don't merely correspond to countable things - countable things are the original source of the much more abstract idea of "natural numbers".

"Flight" can be countable, but as a generalized state/phase, I think it wouldn't be.