Godel's proofs show that any axiomatic system to which arithmetic could be reduced must be either incomplete or self-contradictory.
The proof doesn't depend on any interpretation of the symbolism that he uses, so his paradoxes are, in a sense, syntactical rather than semantic. What he shows is that any system which could encode arithmetical calculations will also be able to encode computations which can be interpreted as tests of the validity of its own theorems, and which are therefore self-referential (since the outcomes of these validity tests are also theorems).
We don't normally think of arithmetic as being intrinsically about truth-telling, or validity. Why is it impossible to do arithmetic without generating valdity tests? If Godel is right, it can't be just an accident. His method depends on being able (in prinicple) to arithmetise all logical validity testing - it seems to make logical validity testing and an essential subset of arithmetical computation isomorphic.
Numbers are truth and truth is numbers.
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