Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
There are countably many proofs, but a proof does not entail a single truth. Many truths are entailed by a single proof, in fact an uncountable number of truths can be entailed by a single proof.
That does not mean that all truths can be written down or enumerated, but strictly speaking it is not sufficient to conclude that due to the fact that the set of all proofs are countable, that the set of all truths entailed by the proofs must also be countable.
None of this should be taken to violate Godel's incompleteness theorems.
Finally, it's worth mentioning that not all formal systems are limited to finite proofs. There are formal systems where theorems as well as proofs can be countably infinite in length and where there are uncountably many proofs. These systems, known as infinitary logic, are often reduceable to second order logic and hence are incomplete.
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.
