Why First Order PA?

Question

All models of Second order Peano axioms are isomorphic. However first order Peano axioms allow for nonstandard models. Why then does it seem most examples of PA uses the first order version of the axioms?

Answer 1

First order logic is complete - that is to say, anything true in every model is provable. Second order logic is not - for any given system of rules of inference, there will always be statements true in every model of second order PA that are not provable. For most logicians, and most mathematicians in general, it's preferable to have a complete but possibly imprecise system rather than a perfectly precise but incomplete one.

Also, it's not the case that all models of the second-order Peano axioms are isomorphic, unless you require that second order quantifiers quantify over true subsets - there are plenty of nonstandard models that pick and choose which subsets exist. For example, the sentence $(\exists X)(\forall Y)\neg(X \subset Y)$ is not satisfiable if every subset exists, but if we consider the universe in which the only set of natural numbers is $\{0\}$ it's true.

So the only way every model of second-order PA is isomorphic is if you insist on using the "true" set of subsets of $\mathbb{N}$ - but that has the same issues as insisting on using "true" $\mathbb{N}$ for first-order PA, besides which it's actually independent of ZFC exactly what the true subsets of $\mathbb{N}$ are, or even how many there are.