No, it is directly related and is exactly what Godel’s incompleteness theorem says without the self referential part. That is, this covers precisely the idea that if ones system is strong enough to determine the natural numbers then it will have statements that hang. The halting problem and the proof given are proofs that the theorem is true, not that those are the only examples of the theorem.
I really don’t think it is analogous. This result doesn’t exploit any interesting property that the integers have, unlike Godel, it just uses with their size. In addition, this one says ‘certain integers cannot be represented’ whereas Godel says ‘certain statements about integers cannot be proven’.
Godel was working under a philosophical system that assumed that everything could be deconstructed into first order logic. He disagreed and essentially wrote “this statement is false” as a proof that it can not be done. However, he also points out in his proof that there is an infinite number of such statements and that they do not have to be self-referential. Nor was what he was doing exploiting an interesting property of the integers but exploiting an interesting property of the system that creates the integers. This decoder can tell if something is an integer and tell one what integer it is, therefore it is able to evaluate everything needed for Godels proof, therefore it must have not only one but an infinite number of cases for which it hangs.
You’re wrong. You’re arguing from surface similarity rather than detailed internal workings, which is a big no-no in maths. I could have spent about five paragraphs breaking down your argument point by point, but you’ve made my life easy with that final line:
therefore it is able to evaluate everything needed for Godel’s proof, therefore it must have not only one but an infinite number of cases for which it hangs.
This is not the case, there is a simple counter-example. Unary code, in which 0 is 0, 1 is 10, 2 is 110, 3 is 1110, 15 is 1111111111111110, hangs on the string 111111.… and no other. The fact your argument led to this conclusion is a demonstration of how completely wrong it is.
No, it is directly related and is exactly what Godel’s incompleteness theorem says without the self referential part. That is, this covers precisely the idea that if ones system is strong enough to determine the natural numbers then it will have statements that hang. The halting problem and the proof given are proofs that the theorem is true, not that those are the only examples of the theorem.
I really don’t think it is analogous. This result doesn’t exploit any interesting property that the integers have, unlike Godel, it just uses with their size. In addition, this one says ‘certain integers cannot be represented’ whereas Godel says ‘certain statements about integers cannot be proven’.
Godel was working under a philosophical system that assumed that everything could be deconstructed into first order logic. He disagreed and essentially wrote “this statement is false” as a proof that it can not be done. However, he also points out in his proof that there is an infinite number of such statements and that they do not have to be self-referential. Nor was what he was doing exploiting an interesting property of the integers but exploiting an interesting property of the system that creates the integers. This decoder can tell if something is an integer and tell one what integer it is, therefore it is able to evaluate everything needed for Godels proof, therefore it must have not only one but an infinite number of cases for which it hangs.
You’re wrong. You’re arguing from surface similarity rather than detailed internal workings, which is a big no-no in maths. I could have spent about five paragraphs breaking down your argument point by point, but you’ve made my life easy with that final line:
This is not the case, there is a simple counter-example. Unary code, in which 0 is 0, 1 is 10, 2 is 110, 3 is 1110, 15 is 1111111111111110, hangs on the string 111111.… and no other. The fact your argument led to this conclusion is a demonstration of how completely wrong it is.
I got that part, I tried to explain further in another response. If this is all that is being said then I was wrong.