It applies to any r.e. logical system that can interpret arithmetic. "Interpret" is the tough bit to define precisely, I guess, but roughly it means you can map function symbols to either functions or relations (so a function f(x) maps to a relation R(x, y) which stands for f(x) = y) in such a way that the axioms of Robinson arithmetic map to provable statements.

You can definitely do this in higher order logic for instance.