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Regarding the first question: The second order theory $PA_2$ is usually defined relative to an ambient universe $V$ of Zermelo-Fraenkel set theory, and as such it only has one model up to isomorphism. In other words, $PA_2$ is a categorical theory from the point of view of any model $V$ of $ZF$ since all of its models are isomorphic to $(\Bbb{N},\mathcal{P}(\omega))$, where $\Bbb{N}$ is the standard model of $PA$ and $\mathcal{P}(\omega)$ is the collection of all subsets of natural numbers (from the point of view of $V$).

On the other hand, $Z_2$ is a first-order approximation of $PA_2$, and by some standard theorems of model theory, it has many $2^\kappa$ nonisomorphic models of cardinality $\kappa$ for each infinite cardinal $\kappa$. In particular, there are continuum-many nonisomorphic countable models of $Z_2$. Each such countable model of $Z_2$ is of the form $(\Bbb{M},\mathcal{F})$, where $\Bbb{M}$ is a standard or nonstandard model of $PA$, and $\mathcal{F}$ is a countable family of subsets of the universe of discourse $M$ of $\Bbb{M}$

Regarding the second question: $ACA$ is much stronger than first order Peano Arithmetic $PA$ since it proves Con($PA$) (the formal consistency of $PA$) and much more (itertions of consistency statements) .

However, $ACA$ is, in turn, much weaker than $Z_2$ since already the fragment known as $\Pi^1_1$-$CA$ of $Z_2$ can prove Con($ACA$).

One way to see this is based on an old result (noticed by a number of people, including Takeuti and Feferman) that $ACA$ is equiconsistent with an extension $PA(T)$ of $PA$ with a distinguished predicate $T$ that codes up the full truth predicate for the standard ambient model of arithmetic. Note that $PA(T)$ includes induction in the extended language of arithmetic augmented by the predicate $T$.

P.S. The subsystem $ATR_0$ of $\Pi^1_1$-$CA$ already proves Con($ACA$).

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Regarding the first question: The second order theory $PA_2$ is usually defined relative to an ambient universe $V$ of Zermelo-Fraenkel set theory, and as such it only has one model up to isomorphism. In other words, $PA_2$ is a categorical theory from the point of view of any model $V$ of $ZF$ since all of its models are isomorphic to $(\Bbb{N},\mathcal{P}(\omega))$, where $\Bbb{N}$ is the standard model of $PA$ and $\mathcal{P}(\omega)$ is the collection of all subsets of natural numbers (from the point of view of $V$).

On the other hand, $Z_2$ is a first-order approximation of $PA_2$, and by some standard theorems of model theory, it has many $2^\kappa$ nonisomorphic models of cardinality $\kappa$ for each infinite cardinal $\kappa$. In particular, there are continuum-many nonisomorphic countable models of $Z_2$. Each such countable model of $Z_2$ is of the form $(\Bbb{M},\mathcal{F})$, where $\Bbb{M}$ is a standard or nonstandard model of $PA$, and $\mathcal{F}$ is a countable family of subsets of the universe of discourse $M$ of $\Bbb{M}$

Regarding the second question: $ACA$ is much stronger than first order Peano Arithmetic $PA$ since it proves Con($PA$) (the formal consistency of $PA$) and much more (itertions of consistency statements) .

However, $ACA$ is, in turn, much weaker than $Z_2$ since already the fragment known as $\Pi^1_1$-$CA$ of $Z_2$ can prove Con($ACA$).

One way to see this is based on an old result (noticed by a number of people, including Takeuti and Feferman) that $ACA$ is equiconsistent with an extension $PA(T)$ of $PA$ with a distinguished predicate $T$ that codes up the full truth predicate for the standard model of arithmetic. Note that $PA(T)$ includes induction in the extended language of arithmetic augmented by the predicate $T$.