[Maxatar]

This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation.

You can adopt Henkin semantics to give the naturals an interpretation, which is still second order logic, but then you're back to lacking a categorical model of the naturals.

[sam_ezeh]

> So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation.

Can you explain what you mean here? Full semantics for second-order logic has a unique interpretation i.e. the standard natural numbers

[Maxatar]

Interpretation under full second‑order logic is not intrinsic to the logic itself but is always supplied by a richer meta‑theory, usually set theory/ZF. The sentence "All subsets of N" has no standalone meaning in second-order logic, it must be defined inside of the meta-theory, which in turn relies on its own meta‑theory, and so on ad infinitum.

Thus, although full second order Peano axioms are categorical, second order logic by itself never delivers a self‑contained model of the natural numbers. Any actual interpretation of the natural numbers in second order logic requires an infinite regress of background theories.