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I feel like godel incompleteness is a cheap trick that mislead to think there are statements in ZFC that can't be proven except that those proven statement are

I feel like godel incompleteness is a cheap trick that mislead to think there are statements in ZFC that can't be proven except that those proven statement are contrived useless meta-statements.

deltasepsilon 4y ago
I'm going to assume you mean that the true unprovable sentences that Gödel constructed are contrived. I don't think I can convince you otherwise, except to say that others thought like you and eventually Paris and Harrington came up with [1]. You can make even more "concrete" statements, see the work of Chaitin, but all this stems from Hilbert's very concrete 10th problem [2]. I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special. Why bother about anything at all... there's the rub. [1] https://en.wikipedia.org/wiki/Paris–Harrington_theorem https://en.wikipedia.org/wiki/Paris–Harrington_theorem [2] https://en.wikipedia.org/wiki/Hilbert's_tenth_problem https://en.wikipedia.org/wiki/Hilbert's_tenth_problem
- SemanticStrengh 4y ago
  > Since Peano arithmetic cannot prove its own consistency by Gödel's second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem. This seems quite circular, where the (claims mine) contrived statements enabling godel 2 incompletude theorem, make it unable to prove a non-contrived statement? Thanks for the reference though, I'll admit that is a topic I'm not familiar enough with to fully challenge it. > I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special. Yeah that's a classic, it's a good thought experiment but that only prove that one must not be careful with the pruning that can enable high level reasoning but more with being careful of fallacious or misleading reasonings. Yes in theory, mathematical chain proofs are only tautologies derived from ZFC/higher order logic, so yes mathematics doesn't say anything new. However in practice, the task of unfolding reasoning chains and being able to refer to past lemnas as abtractions/objects, enable the reader to optimize for cognition, and hence the more proofs advances, the more we can discover, retain, understand and refer, useful tautological yet transitive knowledge.
samth 4y ago
This is not true. For example, the statement "there exists a bijection between the real numbers and the powerset of the integers" is not contrived or meta or useless, but is independent of ZFC.
- SemanticStrengh 4y ago
  independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? Also, I am aware there are non-contrived statements that are can't be proven in ZFC such as https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC#:~:text=A%20statement%20is%20independent%20of,from%20the%20axioms%20of%20ZFC https://en.wikipedia.org/wiki/List_of_statements_independent.... I'm only claiming that the incompleteness theorems only find contrived ones, and that for the non-contrived ones humanity has found, it is NOT that they are mathematically unproveable, it is that ZFC is not enough.
  - JadeNB 4y ago
    > independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? There are plenty of people who find ZFC + ¬CH a reasonable axiomatic system. EDIT: I misread samth's statement (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581), which I think is proveable in ZFC; I thought their statement was ¬CH.
    - SemanticStrengh 4y ago
      It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-zfc-ch https://mathoverflow.net/questions/273861/independence-over-... (Although its CH not the negation of it) what additional axioms be needed there?
      - JadeNB 4y ago
        I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948 https://news.ycombinator.com/item?id=31423948).
mjan22640 4y ago
Halting problem and difficulties in compilation and static analysis are not contrived.

I feel like godel incompleteness is a cheap trick that mislead to think there are statements in ZFC that can't be proven except that those proven statement are contrived useless meta-statements.

deltasepsilon 4y ago

I'm going to assume you mean that the true unprovable sentences that Gödel constructed are contrived. I don't think I can convince you otherwise, except to say that others thought like you and eventually Paris and Harrington came up with [1]. You can make even more "concrete" statements, see the work of Chaitin, but all this stems from Hilbert's very concrete 10th problem [2]. I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special. Why bother about anything at all... there's the rub. [1] https://en.wikipedia.org/wiki/Paris–Harrington_theorem https://en.wikipedia.org/wiki/Paris–Harrington_theorem [2] https://en.wikipedia.org/wiki/Hilbert's_tenth_problem https://en.wikipedia.org/wiki/Hilbert's_tenth_problem

SemanticStrengh 4y ago
> Since Peano arithmetic cannot prove its own consistency by Gödel's second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem. This seems quite circular, where the (claims mine) contrived statements enabling godel 2 incompletude theorem, make it unable to prove a non-contrived statement? Thanks for the reference though, I'll admit that is a topic I'm not familiar enough with to fully challenge it. > I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special. Yeah that's a classic, it's a good thought experiment but that only prove that one must not be careful with the pruning that can enable high level reasoning but more with being careful of fallacious or misleading reasonings. Yes in theory, mathematical chain proofs are only tautologies derived from ZFC/higher order logic, so yes mathematics doesn't say anything new. However in practice, the task of unfolding reasoning chains and being able to refer to past lemnas as abtractions/objects, enable the reader to optimize for cognition, and hence the more proofs advances, the more we can discover, retain, understand and refer, useful tautological yet transitive knowledge.

samth 4y ago

This is not true. For example, the statement "there exists a bijection between the real numbers and the powerset of the integers" is not contrived or meta or useless, but is independent of ZFC.

SemanticStrengh 4y ago
independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? Also, I am aware there are non-contrived statements that are can't be proven in ZFC such as https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC#:~:text=A%20statement%20is%20independent%20of,from%20the%20axioms%20of%20ZFC https://en.wikipedia.org/wiki/List_of_statements_independent.... I'm only claiming that the incompleteness theorems only find contrived ones, and that for the non-contrived ones humanity has found, it is NOT that they are mathematically unproveable, it is that ZFC is not enough.
- JadeNB 4y ago
  > independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? There are plenty of people who find ZFC + ¬CH a reasonable axiomatic system. EDIT: I misread samth's statement (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581), which I think is proveable in ZFC; I thought their statement was ¬CH.
  - SemanticStrengh 4y ago
    It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-zfc-ch https://mathoverflow.net/questions/273861/independence-over-... (Although its CH not the negation of it) what additional axioms be needed there?
    - JadeNB 4y ago
      I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948 https://news.ycombinator.com/item?id=31423948).

mjan22640 4y ago

Halting problem and difficulties in compilation and static analysis are not contrived.