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.