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I want to make the following claim. Could you please correct me, thanks ?

Godel's incompleteness theorem relies on the folding of higher order structure with the arithmetic one, leading to self reference which states impossible "liar sentences". This is due to the parsability of the sentences of FOL and its tree data structure. An escape from this is the conception of non-arithmeticizable data structure and the representation of statements of arithmetic within it guaranteeing the higher order structure which secures from such ill self references.

We can represent a lot of data structures in arithmetic, so if our sentences are inside arithmetic, it creates issues. So can we imagine a form of sentences that would be non-parsable/arithmetizable thus avoiding ill-reference ?

Ok Ok
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    G's theorem applies to formal systems. One of the basic features of a formal system is that the syntax is algorithmic. – Mauro ALLEGRANZA May 06 '25 at 06:42
  • Related (probably dupe?): https://math.stackexchange.com/questions/4287007/g%c3%b6dels-incompleteness-theorem-and-theories-which-are-not-recursively-axiomatiza/4287053#4287053 – Noah Schweber May 06 '25 at 20:51
  • @NoahSchweber It seems my question is, is there something that is not a graph with which to do math ? Because so far, everything looks like a graph of a kind to me. Like literally. – Ok Ok May 07 '25 at 05:03

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