[rocqua]

It should be noted that the special sauce that makes arithmetic really difficult is induction.

Induction isn't just reasoning about computation, (i.e simple equations). Instead it is reasoning about reasoning about equations (i.e. reasoning about all equations).

Specifically we have: this formula https://wikimedia.org/api/rest_v1/media/math/render/svg/67e2...

[salthound]

But it should be pointed out that induction is essentially unrelated to Gödel's theorem.

For example, Robinson Arithmetic is a finitely axiomatized theory whose axioms contain only one existential quantifier, but just like full (Peano) arithmetic, it is subject to Gödel's incompleteness theorems.

[rocqua]

Wow, I thought it was required to have second order logic for godel to apply, but this seems to be a first order logic that is still undecidable.

[ratmice]

From the other side of the fence, another thing to look at would be the decidability of presburger, and skolem arithmetic, the following paper then goes over the extensions of these, which render the result undecidable. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.21...