Gödel's Theorem

Re:Gödel's Theorem by undershirt on Wednesday January 30, 2019 @09:12PM (#19041675)

This seems to throw more shade on the problem for me. There is a profound sense that there is no mode of reasoning (nevermind just the mechanical deductivism) that can completely capture all expressible truths without reaching contradiction.

The nature of knowing what is true is to capture a slice of reality in a model, and a model by its essence must leave out details, considering only elements which are important to its function. It seems that all of language, including anything we create with it, is an incorrect but useful model of Truth. Different models describing different things are expected to produce contradictions, but this math thing was talking about itself and it couldn't even do THAT consistently, right?

If Godel's theorem seems to be blown out of proportion as a synecdoche for the hundreds of years of collective malaise about unattainable truth, it is at least appreciable (and relevant to this wider domain) that Godel devastated the world by showing how logic fails at proving the consistency of a simple mathematical system (correct me if I'm wrong—it's not just axiomatic systems, "there ain't no such animal" of a consistency proof).

Anyway, there is a popular distinction between lowercase-t truth and uppercase-T Truth. The uppercase form represents knowledge we can never understand (outside our Circumference), and the lowercase form represents what we understand now. Since our understanding constantly changes over time, I imagine truth being a function of time, t. As t approaches infinity, truth(t) approaches this limit of Truth.

I highly recommend this poem (analyzed by Nerdwriter) on the expressibility of truth as well: https://youtu.be/55kqNg88JqI

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Re:Gödel's Theorem by brobdingnagians on Wednesday January 30, 2019 @10:02PM (#19041923)

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.

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Re:Gödel's Theorem by smallnamespace on Thursday January 31, 2019 @01:09AM (#19042777)

> 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'...

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Re:Gödel's Theorem by heavenlyblue on Thursday January 31, 2019 @12:49PM (#19046850)

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.

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Re:Gödel's Theorem by undershirt on Wednesday January 30, 2019 @11:15PM (#19042274)

> 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 [2]: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

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Re:Gödel's Theorem by clairity on Wednesday January 30, 2019 @09:58PM (#19041903)

> “There is a profound sense that there is no mode of reasoning (nevermind just the mechanical deductivism) that can completely capture all expressible truths without reaching contradiction.”

but you’re extrapolating beyond godël’s incompleteness theorem here, which only makes that sort of assertion for axiomatic systems, as the article exhorts not to do, do you not?

what does seem self-evident is that logical systems cannot encompass the whole of human experience (yet?) so we continue to search for a more complete system of thinking. godël’s work probably is only incidental to this, not a completely explanatory theory.

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Re:Gödel's Theorem by rocqua on Thursday January 31, 2019 @12:35AM (#19042624)

What logical system isnt axiomatic? I can't envision such a thing.

The only way out I see is uncountably many axioms.

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