The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.
> The same word can have infinitely many meanings.
But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.
What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.
The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions.
Definitional recursion has to bottom out somewhere. (OK, it can also be circular, but I'm guessing you would not find that satisfactory.) Whatever words I use to define "function" you can always turn around and insist that I define those words. It's a never-ending game. It ultimately boils down to the definitions of words like "true" and "false, "same" and "different", whose meanings can only be communicated by way of examples: X and X are the same, X and Y are different.
But none of this has anything to do with the matter at hand. There are a finite number of atoms in the universe. Those atoms can only arrange themselves into a finite number of sentient creatures (or computers), each of which has only a finite brain in which can reside only a finite number of thoughts. So no matter how you slice it, the number of realized ideas in this universe is going to be not only countable but actually finite because there is only a finite amount of time before heat death.
