how does the laws of the universe being not axiomatizable
…I never said they are not.
relate to the brain not using uncomputable functions?
That is unspecific: Do you mean it is using external oracles? It cannot use use them because they cannot exist because they’re four-sided triangles. If you mean that it is considering uncomputable functions, then it can do so symbolically, but it cannot evaluate them, not in finite time that is: The brain can consider the notion of four-sided triangles, but it cannot calculate the lengths of those sides given, say, an area and an angle or such. What would that even mean.
…I never said they are not.
The incompleteness theorem says that a consistent axiomatic formal system satisfying some conditions cannot be complete, so the universe as a formal system (supposed consistent, complete, expressive enough, …) cannot be axiomatized.
external oracles
What do you mean external?
The possibility of using physical phenomena as oracles for solving classically uncomputable problems in the real world is an open question. If you think this is logically as impossible as a four sided triangle you should give sources for this claim, not just some vague statements involving the incompleteness theorem. Prove this logical impossibility or give sources, thats all im asking.
Who says you cant take a first order logic sentence, codify it as a particular arrangement of certain particles and determine if the sentence was valid by observing how the particles behave? Some undiscovered physical phenomenon might make this possible… who knows. It would make possible the making of a real world machine that surpasses the turing machine in computability, no? How is this like a four sided triangle? The four sided triangle is logically impossible, but a hypercomputer is logically possible. The question is whether it is also physically possible, which is an open question.
The incompleteness theorem says that a consistent axiomatic formal system satisfying some conditions cannot be complete, so the universe as a formal system (supposed consistent, complete, expressive enough, …) cannot be axiomatized.
It can also be axiomisable but inconsistent. In principle, that is, but as said you’d annoy a lot of physicists.
What do you mean external?
As in the previously mentioned summation of the results of theoretical hypercomputation: “If uncomputable inputs are permitted, then uncomputable outputs can be produced”. Those oracles would be the input.
The possibility of using physical phenomena as oracles for solving classically uncomputable problems in the real world is an open question.
If they exist, then they can be used. We do that all the time in the sense that we’re pretending they exist, it’s useful to e.g. prove that an algorithm is optimal: We compare an implementable algorithm it with one that can e.g. see the future, can magically make all the right choices, etc. But they don’t exist.
If you think this is logically as impossible as a four sided triangle you should give sources for this claim
I already pointed you to an easy-going explanation of the proof by diagonalization. I’m not going to sit here and walk you through your homework. In fact I have given up explaining it to you because you’re not putting in the work, hence why I resorted to an analogy, the four-sided triangle.
Some undiscovered physical phenomenon might make this possible… who knows.
Are all thinkable phenomena possible? Can there be four-sided triangles?
The four sided triangle is logically impossible, but a hypercomputer is logically possible.
That is an assertion without substantiation, and for what it’s worth you’re contradicting the lot of Computer Science. A hypercomputer is a more involved, not as intuitive, four-sided triangle.
If you think that it’s logically possible, go back to that proof I pointed you to. I will not do so again.
The diagonalization argument you pointed me to is about the uncomputability of the halting problem. I know about it, but it just proves that no turing machine can solve the halting problem. Hypercomputers are supposed to NOT be turing machines, so theres no proof of the impossibility of hypercomputers to be found there.
I know diagonalization proofs, they dont prove what you say they prove. Cite any computer science source stating that the existence of hypercomputers are logically impossible. If you keep saying it follows from some diagonalization argument without showing how or citing sources ill move on from this.
