# PA2 (Peano Arithmetic in 2º order logic and categoricty)

Second order logic implies categoricity in peano arithmetic. But why are the models isomorphic to the standard model of aritmhetic and not to another non standard model, for example?

The point is that there is a single second-order sentence which completely characterizes (up to isomorphism) the structure $(\mathbb{N}, 0, 1, <, +, \times)$. Specifically, the standard model is characterized by the property that every element has only finitely many predecessors - and we can write this in second-order logic as $$\forall n\forall F([\forall m(m<n\implies F(m)<n) \wedge \forall m_1, m_2(m_1\not=m_2\implies F(m_1)\not=F(m_2))]$$ $$\implies \forall m<n\exists k<n(F(k)=m));$$ in English, this is just saying "for every $n$, every injective function from $\{m: m<n\}$ to $\{m: m<n\}$ is surjective." This sentence, when conjoined with the finitely many non-induction axioms of Peano arithmetic, characterizes the standard model up to isomorphism.
• It is said that there are infinite (in metamathematical intuitive sense) numbers, but the sets $n i \in N$ and $n<m$ implies this sets are finite. .Thanks for the answer. Now I have not clear my position about second order logic, maybe it is not only first order set theory in disguise. When I started to interest for theoretical basis for maths (my formation is scientific but not mathematical). But, why the mathematicals are so reticents to second order logic then? Nov 16, 2015 at 6:07