Is there a general version of Godel's Incompleteness Theorem which holds for any formal axiomatic system (and not just those capable of modelling basic arithmetic)?

If no, is it absurd to ask why such inherent limitations kick in only when the system becomes strong enough to interpret basic arithmetic? Does this indicate platonic existence of numbers?

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    No, and it is not about strength/weakness alone. There are very weak systems that are incomplete because they allow encoding self-reference, and there are systems that are complete, but incomparable to arithmetic in strength. In some sense, at least, they can be more complex than arithmetic. Moreover, elementary Euclidean geometry, Plato's prime example of an ideal realm, is complete. So incompleteness has nothing to do with platonic existence, whatever that is taken to mean. – Conifold Apr 1 at 20:04
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    @Conifold So the capacity to allow encoding self-reference is "logically equivalent" to the incompleteness phenomenon? – Ajax Apr 1 at 20:42
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    No, it is just sufficient. Incompleteness is a very easy property to get. Any axiomatic system with multiple models is incomplete, group theory, ring theory, theory of vector spaces, homology, etc. Incompleteness of Peano arithmetic is made such a big deal of only because intuitively we feel like we know the model of arithmetic and the Peano axioms seem to exhaust what is needed to set it up, but it turns out not to be enough nonetheless. It is similar with set theory, which is why the undecidability of the continuum hypothesis was a big surprise, and it does not involve self-reference – Conifold Apr 1 at 23:38

No, there is not. See https://en.wikipedia.org/wiki/Complete_theory . For example, Presburger arithmetic (with only addition and not multiplication) is not incomplete, and Gödel's incompleteness theorem does not apply to it.

Gödel's incompleteness theorem arises because arithmetic is capable of modeling itself - within itself - which allows the creation of a Gödel sentence. Presburger arithmetic isn't powerful enough to model itself. Instead of saying Gödel's incompleteness theorem applies to "any formal system powerful enough to model arithmetic," we could say it applies to "any formal system powerful enough to model itself." If a formal system can model arithmetic it can model itself, because arithmetic is powerful enough to model any formal system. (We might very informally say that this is because arithmetic is "Turing-complete"; it is strong enough to work like a computer program, and for any formal system, we can write a computer program to check proofs in that system.)


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