No, because proofs have to consist of a finite number of words. Thus there are only countably many proofs of anything. In particular, there are only countably many reals between 0 and 1 which can be expressed in a finite number of words.
Are there a finite number of words? Language seems to grow and adapt to new concepts as needed, perhaps there is an infinity of linguistic descriptions available to us.
It's not. It's a single proof about a set, a set that's assumed to be uncountable in standard ZF set theory.
The "axiom system" that (supposedly) contain a countable number of axioms. But these too are constructs of set theory. We still create proofs one by one of theories about axiom systems with infinite axiom - so we have a countable/enumerable set of such theories.
The proof systems to we can see or touch involve this enumerable properties. Perhaps you could change that with an analogue computer that a person could input "any" "quantity" into. But that's outside math as things stand.
Do you mean the proof that 0.25 >= 0 and the proof that 1/e >= 0 count as the same one, because there's a more general proof that a set of values including those is >= 0? But then where do you draw the line? When can you consider 2 proofs different enough to count as different ones?
I think you have a slightly stricter definition of "a proof" than me. I would consider a proof that all the numbers in (0,1) are positive to also be a proof that the number 0.5 is positive, as well as the number 1/e, and Champernowne's constant.
Since the original question was about uncountably many mathematical truths I would say we have one proof that proves uncountably many mathematical truths.
It is an abstract proof of a generator for concrete proofs of specific assignments to variables. The potential is uncountable, but only a countable subset will ever be invoked.
I don't know what it really means for a proof to be invoked, and I also don't really like the idea of separating proofs into concrete and abstract proofs. Either it proves something or it doesn't.
