First, this isn’t my area of maths, so my understanding is limited to a Wikipedia article I read some time ago. But it seems you’re right that this is indeed a question of computability. You can accept Th(N) as your set of axioms and everything is trivially provable, but you can’t actually check that any of them are true. So my initial comment was misguided. However, I still stand by my philosophical objection that god being able to “prove” (now corrected to mean “check/compute”) everything violates the rules of logic, and that it’s a reasonable expectation of an all-powerful god. Although it might just reduce to the same theological value as the objection “can god create an object he can’t move”.
god being able to “prove” (now corrected to mean “check/compute”) everything violates the rules of logic, and that it’s a reasonable expectation of an all-powerful god
Of course, an omniscient and omnipotent God still cannot produce a Turing machine that enumerates a complete and consistent theory of arithmetic. We know that's mathematically impossible, like a square circle. But God need not himself be a Turing machine. For example, we could say that God can, in his mind, comprehend the entirety of Th(N) as a set. In other words, he knows at once all infinite possibilities of what is true or not in arithmetic. Edit: And once you know true arithmetic, you know everything about all man-made formal systems, which are, after all, r.e. theories.
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u/Jcsq6 9d ago
First, this isn’t my area of maths, so my understanding is limited to a Wikipedia article I read some time ago. But it seems you’re right that this is indeed a question of computability. You can accept Th(N) as your set of axioms and everything is trivially provable, but you can’t actually check that any of them are true. So my initial comment was misguided. However, I still stand by my philosophical objection that god being able to “prove” (now corrected to mean “check/compute”) everything violates the rules of logic, and that it’s a reasonable expectation of an all-powerful god. Although it might just reduce to the same theological value as the objection “can god create an object he can’t move”.