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I suppose this is not an easy question unless we formally define what counts as a theorem. For the purpose of this question, let us conservatively suppose that we are taking ZFC as our foundation, and assume that every theorem is some implication of the axioms.

With that in mind, my immediate thought is that indeed the set of provable theorems must be countable, as the set of (finite) first order logic statements are countable.

However, such argument seems valid on the prior assumption that every theorem can be expressed with a finite amount of first order logic symbols, and I am not convinced that this is true.

Thus, are there provable theorems which cannot be expressed by a finite amount of symbols? In other words, do the axioms of ZFC imply the existence of some knowable tautology that is beyond that which first order logic could express with a finite amount of symbols?

Graviton
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    Regardless of whether or not there exist (in your words) "provable theorems which cannot be expressed by a finite amount of symbols", proofs themselves consist of a finite sequence of symbols drawn from a finite alphabet. So the number of proofs is necessarily countable. So even if there exist "theorems which cannot be expressed by a finite amount of symbols" (omitting the word 'provable'), the number of provable theorems (whether finitely expressible or not) must be countable simply because the number of proofs themselves is necessarily countable. – John Forkosh Nov 13 '21 at 09:39
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    @JohnForkosh I concur, but I cannot seem to escape the possibility of a truth which could only be proven with an infinite series of steps. Certainly, perhaps such a proof could be compacted by using a higher-order logic, but I don't think that implies that it is possible to write it out within an equivalent finite string of FOL symbols alone. – Graviton Nov 13 '21 at 09:43
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    Re "possibility of a truth which could only be proven with an infinite series of steps", I think "infinite proof" (whether countably infinite or uncountable) is more-or-less an oxymoron because proofs are human constructions, so necessarily finite by (any sensible) definition. Even humanly-conceivable truths are (I think) necessarily finite. although that wouldn't rule out the existence of humanly-inconceivable truths which aren't. But logic and proofs are linguistically-expressible human ideas, hence always finite. – John Forkosh Nov 13 '21 at 10:05
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    For further insight (where all of mine comes from, anyway) you might want to google "Curry-Howard isomorphism", e.g., https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence, which demonstrates the correspondence between proofs and programs. Then you just have to convince yourself that there ain't no infinite programs. – John Forkosh Nov 13 '21 at 10:16

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