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.