Hmm, that's a good question. My first thought is that, if a number has some representation, then we can always define a Turing machine which outputs that number

Hmm, that's a good question. My first thought is that, if a number has some representation, then we can always define a Turing machine which outputs that number; by doing the equivalent of a hard-coded 'print "..."' statement. However, such machines may require an infinite transition table; so the computable numbers are those which are finitely representable; which limits us to those whose digits follow some sort of algorithm. Relating that to "definable" numbers, it seems I've probably missed some subtlety with Chaitin's constant: since (a) defining extra digits via axioms seems to be adding (algorithmic) information to our system, so defining all of its digits seems to give infinite information; yet (b) it's trivial to do this in a finite way, like "all subsequent digits are zero". Maybe I'll think about this more, but it seems like my intuition is a bit off (this area of mathematics is notoriously nuanced).