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For the proof of Gödel's Incompleteness Theorem, most versions of proof use basically self-referential statements.

My question is, what if one argues that Gödel's Incompleteness Theorem only matters when a formula makes self-reference possible?

Is there any proof of Incompleteness Theorem that does not rely on self-referential statements?

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    There are statements that have no hint of self-reference that have been proved to be independent of this or that axiom system, so that very important consequence of the Incompleteness Theorem doesn't rely on self-referential statements.2012-07-17
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    There are [non-constructive versions of the incompleteness theorems](http://math.stackexchange.com/a/1895288/21820) that do not use self-referential statements, simply because they do not even construct any. However, the proofs themselves still use diagonalization.2017-04-11

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