I was using the first order Peano axioms in my comment. I'm not familiar with the Dedekind axiomatization. Didn't even know such a thing existed but the statement I made holds I believe. The standard model under the first order Peano Axioms exists and I'm assuming it is also a model of the Dedekind axiomatization. The incompleteness theorems thus do apply to the Dedekind axiomatization, right?

There is just one model of Dedekind's axioms up to a unique

isomorphism, which is indeed the standard model. However,

Gödel's incompleteness proofs do not work for Dedekind's theory

because there are uncountable axiom instances. (Dedekind's

theory is nevertheless effective because proof checking is

algorithmically decidable.)

Consequently, the first incompleteness result (inferentially

undecidable) has been proved by other means. See the

following for a proof:

https://papers.ssrn.com/abstract=3603021

The above article explains that Gödel's second

incompleteness result is false because the theory can prove

its own consistency.