When I said "according to classical mathematics" I did mean according to the normal orthodoxy, which does indeed mean ZFC. By "finite definition" I mean a finite sequence of statements in first or second order logic which can be proven to define a unique real number. (Even if, as with Chaitin's Constant, we can't figure out what that number is.)
The fact that this may not be a definition from my point of view is irrelevant - I was making a statement about what classical mathematics implies, and from the point of view of classical mathematics this is a perfectly reasonable definition.
Since the number of such statements is countable, and only some of them define real numbers, the set of such real numbers is countable. Being countable it is a set of measure 0, and therefore the "almost all" that I stated follows immediately.
The fact that this may not be a definition from my point of view is irrelevant - I was making a statement about what classical mathematics implies, and from the point of view of classical mathematics this is a perfectly reasonable definition.
Since the number of such statements is countable, and only some of them define real numbers, the set of such real numbers is countable. Being countable it is a set of measure 0, and therefore the "almost all" that I stated follows immediately.