Does there exist any rubric where provably transcendental real numbers emerge, in a meaningful way, as rare among all the transcendental numbers?
Here are some of the things I'm worried about:
1) To talk about provably transcendental numbers, it seems only fair to consider them as a subset of some sort of set of definable real numbers (relative to some appropriate language). If the language is countable, that means comparing two countable sets, so measure-theoretic language doesn't seem to help.
2) Some transcendentality proofs naturally apply to all the numbers in a definable uncountable set (which of course contains many undefinable numbers). Small variations of Liouville's famous original construction yield uncountable sets of transcendentals. So a countable language that can only encode a countable number of proofs can still establish the transcendentality of more than countably many numbers.
Perhaps something like this: relative to a fixed language one can define a complexity for definable transcendental reals by the length of their shortest defining formula. Among those of a given complexity, some fraction admit transcendentality proofs. Perhaps this fraction must go to 0 with the complexity for any reasonable theory? (This seems to me a meaningful question despite that attendant undecibilities concerning whether a formula defines a numbers, whether two formulas define the same number, etc.)