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I feel like I understand the general idea behind how Gödel used self-reference to prove that there will always be holes in logical systems, even if you add the self-referential statement to the axioms because you can always create another statement that is true but cannot be proven via Cantor's diagonalization proof. I am confused as to why this implies that other statements, like the twin primes and Goldbach conjectures could also be true but not have proofs. These statements are not self-referential. Do we have proof of any true statement that cannot be proven true in mathematics that isn't self-referential? If not, then why is Gödel's incompleteness Theorem relevant outside of the one case he found?

Edit: I am reading Gödel Escher Bach, and therefore my understanding might be skewed.

user34140
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    The undecidable statements produced by Gödel theorem are also not self-referential. They simply state the nonexistence of numbers with certain arithmetic properties. – MJD Apr 24 '24 at 23:59
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    @MJD wait how are they not self-referential? (what is meant by self-referential?) – confusedcius Apr 25 '24 at 02:23
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    I think the second incompleteness theorem gives a pretty non-self-referential statement, right? This is all a bit beside the point anyway. If someone says "the incompleteness theorem implies that the twin prime conjecture could be true but unprovable", this is an informal statement where the word "could" is doing a lot of heavy lifting. I think it's best interpreted as "it would be nice to exclude that the twin prime conjecture is true but unprovable by proving that every true statement is provable. However, the incompleteness theorem shows we can't do this". – Izaak van Dongen Apr 25 '24 at 09:53
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    This has been discussed many times in the past. See the right-hand sidebar under “related”. In particular see (a) (b). Also props to you for trying to read GEB, but in my opinion it is not a good way to try to actually understand Gödel's incompleteness theorems. There is just way too much Stuff, too many digressions, too many complications, too much mystical woo-woo. Maybe try one of Peter Smith's books about this. They are available for free download. – MJD Apr 26 '24 at 14:57
  • (For example, Hofstadter constructs an elaborate analogy about whether for every phonograph there is a record that will destroy the phonograph when it is played. I don't know what he was hoping to achieve with that, but if he intended that it would assist the reader's understanding of some aspect of Gödel's theorem or to serve as a familiar everyday example, I think it failed.) – MJD Apr 26 '24 at 15:02
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    Gödel's incompleteness theorems tell us that "true but non-provable" statements always exist (under mild conditions). Since the statements Gödel cooks up in order to prove this all look somewhat self-referential (and thus quite unlike most statements we consider in math), it's reasonable to wonder if true "mathematically natural" statements have proofs. It turns out there are plenty of true "mathematically natural" statements which still have no proof (in PA, say). The first example was the Paris-Harrington Theorem – Chris Grossack Apr 26 '24 at 15:37

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The phrase "proven true in mathematics" can have different meanings depending on what axioms one is willing to accept. But let me assume that you mean the most widely accepted axioms, known as ZFC (the Zermelo-Fraenkel axioms of set theory including the axiom of chaoice). Then it is known that the continuum hypothesis, $2^{\aleph_0}=\aleph_1$, is undecidable, i.e., neither provable nor refutable in ZFC. And the continuum hypothesis involves no self-reference.

Andreas Blass
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    +1, nice answer, but is "self-referential" sufficiently well-defined for one to assert that the continuum hypothesis involves no self-reference? Possibily depending on your precise construction of the reals, isn't the continuum hypotheses making assertions about sets that can include the Gödel number of any sentence including all the possible formalisations of the continuum hyptothesis? Isn't it more the point that the proof ot the undecidability of the continuum hypothesis does not appeal to Gödelisation? (I guess I am amplifying @MJD's comment.) – Rob Arthan Apr 25 '24 at 19:34
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    @RobArthan I think "self-referential" is not a property of a sentence but rather a description of how we obtained the sentence. As MJD pointed out in the (currently) first comment under the question, Gödel sentences could be obtained just as arithmetical curiosities. But in fact Gödel obtained them by encoding some self-reference. Conversely, as you said, I can imagine CH being obtained by some coding of some self-referential statement. But that's not how CH was obtained, and in that sense its not self-referential. – Andreas Blass Apr 25 '24 at 19:44
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    I think we are in agreement that "self-referential" is a statement about methodology rather than a statement about mathematical fact. – Rob Arthan Apr 25 '24 at 19:52
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    Thanks for discussing this where I could see it! Noah Schweber has some remarks about possible formalizations of “self-referential” that may be relevant or at least interesting. – MJD Apr 26 '24 at 15:18
  • But the independence of the continuum hypothesis has nothing to do with Godel's incompleteness theorems. – Arvid Samuelsson Apr 29 '24 at 22:34
  • @ArvidSamuelsson I think "nothing to do with" is too strong. Gödel tells us that there are sentences that are neither provable nor refutable in ZFC. And CH turns out to be such a sentence, and to have arisen naturally, i.e., not for the purpose of proving incompleteness. – Andreas Blass Apr 29 '24 at 22:42
  • But the proofs are unrelated. However, to be fair the OP asked for any true but unprovable statements. – Arvid Samuelsson Apr 29 '24 at 22:47