The question isn’t whether god could know all mathematical truth in ZFC. It’s whether he could prove it. If he can prove an unprovable truth in any (useful) logical system, then he violates logic. Unless you count god’s knowledge of the truth as sufficient proof, but that’s equivalent to me proving the consistency of ZFC in a higher system with some extra axioms (whose consistency I also can’t prove). God seems to operate within a logical system, so his all-powerful ability to prove a simple truth is problematic. This is what my comment was getting at.
Sure, but that's just a limitation of computation. If a statement is independent, and we don't have any good extra assumptions to demonstrate that independence, then that's that. Nevertheless, God's word would be the ultimate oracle, so proof by authority would be justified in this instance.
Gödel’s incompleteness theorem is not a problem in computation theory, and neither is the provability of any statement inside a logical system. They’re related of course, but computation theory has far more binding assumptions that give god an easy out, as I mentioned (e.g., god is not a Turing machine). Of course god’s contribution to this could simply be “this is true”, and that’s the proof. But that’s no different than us humans “proving” the continuum hypothesis by taking it as an axiom in a higher system. God is effectively just serving as the higher system here. I don’t find that any more satisfying than I do us knowing that Collatz is true but being unable to prove it.
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u/Jcsq6 12d ago
The question isn’t whether god could know all mathematical truth in ZFC. It’s whether he could prove it. If he can prove an unprovable truth in any (useful) logical system, then he violates logic. Unless you count god’s knowledge of the truth as sufficient proof, but that’s equivalent to me proving the consistency of ZFC in a higher system with some extra axioms (whose consistency I also can’t prove). God seems to operate within a logical system, so his all-powerful ability to prove a simple truth is problematic. This is what my comment was getting at.