# Is second-order logic *with standard semantics* necessary to categorically characterise the natural number structure?

Is second-order logic with standard semantics necessary to categorically characterise the natural number structure?

One can prove that any two models of Dedekind-Peano arithmetic are isomorphic (categoricity theorem), and, by invoking the isomorphism theorem, that one model of Dedekind-Peano arithmetic satisfies A iff any other model of Dedekind-Peano arithmetic satisfies A, for any sentence A (which gives us elementary equivalence).

My question: can one do the same with second-order logic and Henkin semantics or does one require standard semantics for this categoricity result?

I suspect that because Henkin semantics is almost the same as many-sorted first-order semantics, one cannot categorically characterise the natural numbers using Henkin semantics.

• Just to be sure, by second-order logic you have in mind second-order arithmetic? Can you be a bit more precise about what is what (or give a reference), just to avoid possible confusion. – Andrej Bauer Sep 15 '19 at 15:25
• But also, isn't is simply a theorem of second-order logic that any two models of Dedekind-Peano arithmetic are isomorphic? Why do you think we need to refer to models? – Andrej Bauer Sep 15 '19 at 15:26
• More simply, compactness theorem carries over, so you can replicate the usual trick to produce a non standard model. – giuseppe Sep 15 '19 at 18:16

Here's one general fact which in particular kills this question (which is just an instance compactness for Henkin semantics): suppose $$\Sigma$$ is any language and $$T$$ is any second-order $$\Sigma$$-theory which has some infinite Henkin model. Then $$T$$ has Henkin models of arbitrarily large cardinality. This ultimately leans, just as you say, on the fact that Henkin semantics is really just two-sorted first-order logic plus some base axioms.
To clarify: by "Henkin model of $$T$$" I mean a pair $$(S, P)$$ where $$S$$ is a $$\Sigma$$-structure, $$P=(P_i)_{i\in\mathbb{N}}$$ is a family of relations on $$S$$, and $$(S,P)\models T$$ in the sense of Henkin semantics (in particular, every parameter-definable relation on $$S$$ is an element of $$P$$ - here "parameter" refers to both elements of $$S$$ and of $$P$$).