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I am in process of reading Logic and Godel incomplete proof (not completed) when the following two thought pop up in my head.

I think I can show incompleteness through elementary reasoning

A Proof consists of countable alphabet and finite text length. Hence not all subset (of a power set of countable elements) can have a description. Now I can imagine there existing a theorem which is true but can't be proven in the above setting simply because there is no finite pattern in it. To give an example, it is quite common in maths that we divide the proof into 2 cases even and odd. If instead of 2 there exists unaccountably many cases, then such a theorem couldn't be proved in the above mentioned proof system (I am just suggesting a possibly not demonstrating that such a thing happens as). Hence I don't understand the purpose of Godel theorem and the complicated machinery required to prove it. Off course the theorem is important which implies their is a gap in my understanding.

Did the logicians tried this method and did it actually failed.

The only reason incompleteness works is due to some pathological example of self reference. Taking cue from set theory (Russel paradox) and how Von Nuenmann solved by adjusting the definition of set as hierarchy, I think we think we could also do same in first order logic by making stricter grammar such that no such self-referential formula are allowed (I know easier said than done).

EarthIsNotFlat
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    “The only reason incompleteness works is due to some pathological example of self-reference”. That’s false. There are explicit, not self-referential, first-order statements that can’t be proven first-order in Peano Arithmetic but are true nonetheless. See eg https://en.m.wikipedia.org/wiki/Paris–Harrington_theorem . – Aphelli Oct 21 '20 at 11:48
  • You have a vague intuition about a way that a theorem might be unprovable. I think actually you would only need countably many cases that all needed to be stated, because the length of any single proof is finite. But merely imagining that such a theorem could exist is not a proof that such a theorem does exist. – David K Oct 21 '20 at 11:54
  • See also this post – Mauro ALLEGRANZA Oct 21 '20 at 12:04
  • I would also debate whether there is any self-reference in the system that Gödel proved to be incomplete. The key takeaways for me are: some systems are consistent and complete, but some are not; and a system that is powerful enough to reason about all the integers must be in the "not" case. The system itself does not have to be self-referential, and indeed our usual language for proving things about integers does not allow self-reference. See Gödel's incompleteness theorem - question about self reference. – David K Oct 21 '20 at 12:08
  • Incompleteness is not due to "pathological example of self reference" and there are no "self-referential formulas" in FOL. The technique used by Godel is only a method to manufacture an unprovable formula (unprovable according to some well-specified context). We can simply avoid to "read" the formula as referring to something and the "magic" disappear. What is relevant is that in a consistent formal theory $T$ that can express arithmetical facts there is a formula $G$ such that $T$ does not prove $G$ and $T$ does not prove $\lnot G$. This is the gist of "Incompleteness". – Mauro ALLEGRANZA Oct 21 '20 at 12:19
  • "I can imagine there existing a theorem which is true but can't be proven in the above setting simply because there is no finite pattern in it" OK that's nice - can you prove that one exists? Just imagining something doesn't mean it exists. "I think we think we could also do same in first order logic by making stricter grammar such that no such self-referential formula are allowed" Same thing: try to actually do it. Until you try and fail to fill in the details, the reason we have to go to such efforts to prove incompleteness won't be clear. – Noah Schweber Oct 23 '20 at 20:33

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