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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

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.

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.