These days, Peano Arithmetic ($PA$) refers to the first-order version of the axioms, where induction is only over formulas referring to natural number variables. Peano's original version of the induction axiom was actually second-order, allowing reference to sets-of-natural-numbers variables. This latter version corresponds to what a category theorist (like me) would call a natural number object. Call this 'second-order $PA$' or $PA\_2$.


However, we also have second-order arithmetic $Z\_2$ ([wikipedia](http://en.wikipedia.org/wiki/Second-order_arithmetic)), which up to some redundant axioms, looks like $PA\_2$ plus a comprehension axiom schema ([details](http://en.wikipedia.org/wiki/Second-order_arithmetic#Induction_and_comprehension_schema) of the schema). We know that $PA\_2$ has a unique model up to isomorphism, and so does $Z\_2$. My first question is then



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> What, if anything, separates a model of $PA\_2$ from a model of $Z\_2$?
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I imagine that there are sets of naturals in $Z\_2$ that cannot be proved to exist in $PA2$, but where is the boundary?


From the Wikipedia page I see that $ACA\_0$, a subsystem of $Z\_2$ with restricted induction, is a conservative extension of $PA$ (and equiconsistent with $PA$), and I infer that $ACA$, the same subsystem plus *unrestricted* induction, would be a conservative extension of $PA\_2$.



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> Is this true? Is $ACA$ equiconsistent with $PA\_2$?
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One reason I am interested in this is that McLarty has proved that the machinery of derived functor cohomology, in the setting appropriate for arithmetic geometry, is do-able in $Z\_2$ (more precisely, derived functor cohomology over a Noetherian scheme, with coefficients in an arbitrary sheaf of modules on the Zariski site, only requires a system equiconsistent with second-order arithmetic). If one can get from working in $Z\_2$ down to $ACA$, then these tools of algebraic geometry are available as soon as one accepts the existence of natural number objects (and, presumably, classical logic).



Z_2 versus second-order PA
$Z\_2$ as it is usually viewed is a first-order theory with two sorts, and as such is not categorical. The difference (apart from terminological issues) is entirely in the semantics that are used. In "full" second-order semantics, the set variables quantify over all subsets of the domain, while in first-order "Henkin" semantics each model has a domain for number variables and a second domain for set quantifiers to range over. 


There are two things that you might mean by $PA\_2$ (I realized this after writing the answer, so I have expanded it). The first option is to have $PA\_2$ include the entire second-order induction *scheme*; let's call that $PA^s\_2$. $PA^s\_2$ and $ACA$ are indeed equiconsistent. Every model of $ACA$ is already a model of $PA^s\_2$, and every model of $PA^2\_2$ extends to a model of $ACA$ by just throwing in the definable sets. In fact, this extends any model of $PA^s\_2$ to a model of $Z\_2$, so these theories are equiconsistent. There is an issue that this could mean "equiconsistent in full second order semantics" or "equiconsistent in first-order semantics", but either way they are pairwise equiconsistent as long as the same semantics is used for both theories. 


The other option is that $PA\_2$ might just have the single second-order induction *axiom*
$$
(\forall x)[0 \in X \land (\forall n)[n \in X \to n+1\in X] \to (\forall n) n \in X].
$$
Let's call that version $PA^i\_2$. Now the semantics matters. In full second-order semantics, any model of $PA^i\_2$ is a model of $PA^s\_2$, so it extends to a model of $Z\_2$. In first-order semantics, $PA^i\_2$ is very weak, because without any comprehension axioms the single second-order induction axiom is not very strong in first-order semantics. $PA^i\_2$ is (syntactically) a subtheory of $\mathsf{RCA}\_0$, one of the weak systems of arithmetic considered in reverse mathematics, and so $PA^i\_2$ has a much lower consistency strength than $Z\_2$ in the first-order setting. 


