Full Idea
Second-order arithmetic can rule out the non-standard models (with non-standard numbers). Its induction axiom crucially refers to 'any' property, which gives the needed categoricity for the models.
Clarification
'Categoricity' is when all the models are isomorphic to one another
Gist of Idea
Second-order arithmetic covers all properties, ensuring categoricity
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.49
Related Idea
Idea 16321 The compactness theorem can prove nonstandard models of PA [Halbach]