first order logic validity problem
Validity is RE (semidecidable), Satisfiability is undecidable.
How do we figure out that your fancy new law is actually the oracle you claim it is? It could be lying to us, to establish the thing as an oracle we’d have to be able to either inspect it or unit-test it over the whole infinite range.
Right, validity is semidecidable, just like the halting problem.
We might never know for certain that any natural law is true, we might never be certain that that oracle actually solves validity. But that doesnt prevent the oracle from working. My point is that its existence is possible, not that we will ever be able to trust it.
Besides, we dont know that the physical laws we work with today are true, but we nevetheless use them for practical purpuses all the time.
I mean if the point is that we know that we know nothing then I’ll agree.
Not my point… and you know it. My point is that even if we consider that proven theorems are known facts, we still dont know if hypercomputers are infeasible. We know, however, that i’ll never write python code that decides Validity because it is not (classically) decidable. But we have no theorems on the impossibility of hypercomputation.
