Turing machines can’t exist, either.
Oh no! You got me there!
Why do you need uncountable infinities for hypercomputers, though?. I see that Martin Davis criticism has to do with that approach, and I agree this approach seems silly. But, it doesnt seem to cover all potential fronts for hypercomputers. Im not talking about current approaches to quantum computing either. What if some yet unknown physical law makes arrangements of particles somehow solve the first order logic validity problem, which is also not in R? Doesnt involve uncountable infinity at all. Again, im not saying its possible, just that theres no purely logical proof of impossibility, thats all.
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.
