You said "acccording to classical mathematics", which is not constructive so I assumed you were working in ZFC or some similar set theory.
Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't satisfy your idea of "finite definition"
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.
Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't satisfy your idea of "finite definition"