entails that there are uncomputable functions, which I take to mean that there are uncomputable oracles in the physical world.
It means that there are functions that are not computable. You cannot, for example, write a program that decides, in finite time, whether an arbitrary program halts on a particular input. If you doubt that, have an easy-going explanation of the proof by diagonalisation.
We take the set of laws governing the universe and turn it into a formal system. How?
Ask a physicist, that’s their department not mine. Also I’d argue that the universe itself is a formal system, and lots of physicists would agree they’re onto the whole computability and complexity theory train. They may or may not agree to the claim that computer science is more fundamental than physics, we’re still working on that one.
Does the resulting formal system really meet all conditions of the incompleteness theorem?
Easily, because it will have to express the natural numbers. Have a Veritasium video on the whole thing. The two results (completeness and in computability) are fundamentally linked.
he turing machine wouldnt be adequate to model the brain in the sense that the brain, in that hypothetical scenario, would be a hypercomputer,
If the brain is a hypercomputer then, as already said, you’re not talking physics any more, you’re in the realms of ex falso quodlibet.
Hypercomputers are just as impossible as a village barber who shaves everyone in the village who does not shave themselves: If the barber shaves himself, then he doesn’t shave himself. If he shaves himself, then he doesn’t shave himself. Try to imagine a universe in which that’s not a paradox, that’s the kind of universe you’re claiming we’re living in when you’re claiming that hypercomputers exist.
It means that there are functions that are not computable. You cannot, for example, write a program that decides, in finite time, whether an arbitrary program halts on a particular input. If you doubt that, have an easy-going explanation of the proof by diagonalisation.
Ask a physicist, that’s their department not mine. Also I’d argue that the universe itself is a formal system, and lots of physicists would agree they’re onto the whole computability and complexity theory train. They may or may not agree to the claim that computer science is more fundamental than physics, we’re still working on that one.
Easily, because it will have to express the natural numbers. Have a Veritasium video on the whole thing. The two results (completeness and in computability) are fundamentally linked.
If the brain is a hypercomputer then, as already said, you’re not talking physics any more, you’re in the realms of ex falso quodlibet.
Hypercomputers are just as impossible as a village barber who shaves everyone in the village who does not shave themselves: If the barber shaves himself, then he doesn’t shave himself. If he shaves himself, then he doesn’t shave himself. Try to imagine a universe in which that’s not a paradox, that’s the kind of universe you’re claiming we’re living in when you’re claiming that hypercomputers exist.