To be clear, and with all due respect, I think your answer, the statement you make, is wrong in itself: but I understand we'd have to discuss this in another place. I feel pretty confident about what the OP is after just as in "been there, done that", literally: I too have a philosophy and linguistics background, although meanwhile I was also a professional programmer and a logician by vocation... but of course I am partly guessing. Never mind, maybe the OP will speak up, anyway thanks for your kind feedback, Andrej, appreciated.

No worries, I have loads of those. :) I am more puzzled that you keep getting upvotes (but not I downvotes, except yours I guess) despite your answer is fundamentally wrong, as I have argued. But maybe this is not the place to have any actual confrontation.

As for "they are looking for the enumeration of provable statements", the OP goes directly into "sets", while Wolfram talks of "physics": which is where I have felt compelled to mention the distinction syntax/semantics and, subordinately, theoremhood vs provability.

On "enumerating all theorems", I might be wrong, why dishonest? The crucial distinction I am making there is between theoremhood and provability: and I do take it that the unprovable statement in the incompleteness theorem is true (of an arithmetic so and so). And there we have a theorem of arithmetic that cannot be enumerated (in the mathematical sense of enumeration).

I am too new to comment on your answer, but "enumerating all theorems" is impossible, not just "useless", so your answer is rather wrong (sure, model theory: not enough space here...). That said, I have not even mentioned Wolfram because indeed I find that effort essentially misguided, but the questioner here is a beginner, so I didn't think appropriate reading and answering his question to the letter, or saying "forget about Wolfram", I have rather tried to give basic coordinates: but I think you do have a point, I should have been more explicit.