I've heard that reals is the least (well, I'll say it in intentionally vague way, so it will precisely reflect my current understanding) number-like set that is closed under an operation of taking limit points of its subsets.
Also, it's known that there are real numbers that can't be computed (almost all, by cardinality considerations).
Is there some properly defined subset of reals which has only the ones that can be defined in some suitable formalism and that by construction would also have all its limit points included (because they could have been defined in the same formalism)?
The idea is that such subset would have been countable but still would fit the definition of reals. I'm asking to shed some light on where such an idea is flawed (which is implied, again, by a diagonal argument of uncountability of reals).
