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Like the sister comment by clairity, I agree that you are extrapolating beyond the meaning of the theorem. You can prove the truth value of the statement, but n

Like the sister comment by clairity, I agree that you are extrapolating beyond the meaning of the theorem. You can prove the truth value of the statement, but not in the axiomatic system, it requires having an extended system. So it would be more of a moral of the story that sometimes you just don't know enough, but with more information or context you could prove it is true (but even that moral is probably off). It states more about provability in the context of a single axiomatic system for constructed self-referential statements rather than truth in general.

undershirt 8y ago
> It states more about provability in the context of a single axiomatic system for constructed self-referential statements rather than truth in general. Thanks. And oops, I didn't know that the result was proven in a larger system. Did some searching and found this short answer[1] and a longer description[2] that I liked. [1]:https://math.stackexchange.com/a/120018/19178 https://math.stackexchange.com/a/120018/19178 [2]:https://mathoverflow.net/a/24919 https://mathoverflow.net/a/24919 This statement in particular helped: > The Incompleteness Theorem prevents ZFC from proving its own consistency and for that we need to have an additional axiom, giving us a stronger theory which can then prove ZFC is consistent
smallnamespace 8y ago
> sometimes you just don't know enough, but with more information or context you could prove it is true Yes, but that larger system itself contains statements that are true but unprovable in itself, and so on in an infinite regression. And before one hand-waves it away as a mere 'hole' that we can safely ignore, Chaitin extended Godel's work to show that there are in fact an infinitude of 'relevant' mathematical statements that are unprovable within any formal system S: https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin's_incompleteness_theorem https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin'...
- heavenlyblue 8y ago
  Then if you're discussing the topic in the context of "whether our current mathematics is useless in the context of Godel's theorem" - then we can easily define progress in mathematics as "how many contradictions we had found today" and thus continue living with an objective till the heat death of the universe.

undershirt 8y ago

> It states more about provability in the context of a single axiomatic system for constructed self-referential statements rather than truth in general. Thanks. And oops, I didn't know that the result was proven in a larger system. Did some searching and found this short answer[1] and a longer description[2] that I liked. [1]:https://math.stackexchange.com/a/120018/19178 https://math.stackexchange.com/a/120018/19178 [2]:https://mathoverflow.net/a/24919 https://mathoverflow.net/a/24919 This statement in particular helped: > The Incompleteness Theorem prevents ZFC from proving its own consistency and for that we need to have an additional axiom, giving us a stronger theory which can then prove ZFC is consistent

smallnamespace 8y ago

> sometimes you just don't know enough, but with more information or context you could prove it is true Yes, but that larger system itself contains statements that are true but unprovable in itself, and so on in an infinite regression. And before one hand-waves it away as a mere 'hole' that we can safely ignore, Chaitin extended Godel's work to show that there are in fact an infinitude of 'relevant' mathematical statements that are unprovable within any formal system S: https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin's_incompleteness_theorem https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin'...

heavenlyblue 8y ago
Then if you're discussing the topic in the context of "whether our current mathematics is useless in the context of Godel's theorem" - then we can easily define progress in mathematics as "how many contradictions we had found today" and thus continue living with an objective till the heat death of the universe.