To the point that you say things like the universe already is a formal system to which we can apply the incompleteness theorem.
And that is contentious, why? If the laws of the universe are formalisable, then the universe is isomorphic to that formalisation and as such also a formal system. We’re not talking being and immanence, here, we’re talking transcendent properties.
When I asked this, you only mentioned being able to express natural numbers. But can the formal system express them in the specific sense that we need here to use incompleteness?
How do you express them in ways that do not trigger incompleteness? Hint: You can’t. It’s a sufficient condition, there’s equivalent ones, if I’m not mistaken an infinite number of them, but that doesn’t matter because they’re all equivalent.
These are all giant holes you skipped
These are all things you would understand if I didn’t have to remind you of basic computability and complexity theory literally every time you reply. As said: Stop the philosophising. The maths are way more watertight than you think. We’re in “God can’t make a triangle with four sides” territory, here, just that computability is a wee bit less intuitive than triangles.
If you want to attack my line of reasoning you could go for solipsism, you could come up with something theological (“god chooses to hide certain aspects of the universe from machines” or whatever). I’m aware of the limits. I didn’t come up with this stuff yesterday and my position isn’t out of the ordinary, either.
its not a “god cant make a triangle of four sides” discussion.
Disregarding the mysterious formal system that “obviously” expresses arithmetic, you always skip my question: then what? how does the laws of the universe being not axiomatizable relate to the brain not using uncomputable functions? This was always the main point of the argument and you keep avoiding giving me an answer.
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.
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.
I know diagonalization proofs, they dont prove what you say they prove.
Not proofs, plural, not the category. This specific one. The details involve a method to enumerate all programs which is the hard part. IIRC the lecturer doesn’t actually get into that, though. Read the original papers if you want nobody found issue with them in nearly 100 years.
Cite any computer science source stating that the existence of hypercomputers are logically impossible.
Church-Turing is a fundamental result of CS, arguably its founding one, and I will not suffer any more denial of it. It’s like asking a physicist to provide a citation for the non-existence of telekinesis: You fucking move something with your mind, then we’ll talk. In the meantime, I’m going to judge you to be nuts.
Feel free to have a look at the criticism section of Wikipedia’s hypercomputation article, though. Feel free to read everything about it but don’t pester me with that nonsense. Would you even have known about it if I didn’t mention off-hand that it was bunk, serves me right I guess.
church-turing is a a thesis, not a logical theorem.
You pointed me to a proof that the halting problem is unsolvable by a Turing Machine, not that hypercomputers are impossible.
The critic Martin Davis mentioned in wikipedia has an article criticizing a kind of attempt at showing the feasibility of hypercomputers. Thats fine. If there was a well-known logical proof of its unfeasibility, his task would be much simpler though. The purely logical argument hasnt been made as far as i know and as far as you were able to show.
You would need to invent a complexity class larger than R, one that contains more than countably infinite programs. Those, too, can be diagonalised, there would still be incomputable functions. Our whole argument would repeat with that complexity class instead of R. Rinse and repeat. By induction, nothing changes, Q.E.D.
And that is contentious, why? If the laws of the universe are formalisable, then the universe is isomorphic to that formalisation and as such also a formal system. We’re not talking being and immanence, here, we’re talking transcendent properties.
How do you express them in ways that do not trigger incompleteness? Hint: You can’t. It’s a sufficient condition, there’s equivalent ones, if I’m not mistaken an infinite number of them, but that doesn’t matter because they’re all equivalent.
These are all things you would understand if I didn’t have to remind you of basic computability and complexity theory literally every time you reply. As said: Stop the philosophising. The maths are way more watertight than you think. We’re in “God can’t make a triangle with four sides” territory, here, just that computability is a wee bit less intuitive than triangles.
If you want to attack my line of reasoning you could go for solipsism, you could come up with something theological (“god chooses to hide certain aspects of the universe from machines” or whatever). I’m aware of the limits. I didn’t come up with this stuff yesterday and my position isn’t out of the ordinary, either.
its not a “god cant make a triangle of four sides” discussion. Disregarding the mysterious formal system that “obviously” expresses arithmetic, you always skip my question: then what? how does the laws of the universe being not axiomatizable relate to the brain not using uncomputable functions? This was always the main point of the argument and you keep avoiding giving me an answer.
…I never said they are not.
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.
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.
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.
It can also be axiomisable but inconsistent. In principle, that is, but as said you’d annoy a lot of physicists.
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.
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.
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.
Are all thinkable phenomena possible? Can there be four-sided triangles?
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.
Not proofs, plural, not the category. This specific one. The details involve a method to enumerate all programs which is the hard part. IIRC the lecturer doesn’t actually get into that, though. Read the original papers if you want nobody found issue with them in nearly 100 years.
Church-Turing is a fundamental result of CS, arguably its founding one, and I will not suffer any more denial of it. It’s like asking a physicist to provide a citation for the non-existence of telekinesis: You fucking move something with your mind, then we’ll talk. In the meantime, I’m going to judge you to be nuts.
Feel free to have a look at the criticism section of Wikipedia’s hypercomputation article, though. Feel free to read everything about it but don’t pester me with that nonsense. Would you even have known about it if I didn’t mention off-hand that it was bunk, serves me right I guess.
church-turing is a a thesis, not a logical theorem. You pointed me to a proof that the halting problem is unsolvable by a Turing Machine, not that hypercomputers are impossible.
The critic Martin Davis mentioned in wikipedia has an article criticizing a kind of attempt at showing the feasibility of hypercomputers. Thats fine. If there was a well-known logical proof of its unfeasibility, his task would be much simpler though. The purely logical argument hasnt been made as far as i know and as far as you were able to show.
You would need to invent a complexity class larger than R, one that contains more than countably infinite programs. Those, too, can be diagonalised, there would still be incomputable functions. Our whole argument would repeat with that complexity class instead of R. Rinse and repeat. By induction, nothing changes, Q.E.D.