[proofplan] We prove both implications by comparing arithmetical definability with computability from finite iterates of the Turing jump. If $\mathcal M$ satisfies arithmetical comprehension, then the jump $A'$ is obtained by applying comprehension to the arithmetical halting predicate relative to $A$. Conversely, if $\mathcal S$ is closed under jump, then every arithmetically definable set with parameters from $\mathcal S$ is computable from a finite iterated jump of a finite join of those parameters; closure under jump gives that iterated jump belongs to $\mathcal S$, and the $\mathsf{RCA}_0$ ideal closure gives the desired set. [/proofplan]

[step:Use arithmetical comprehension to form every relative jump] Assume $\mathcal M \models \mathsf{ACA}_0$. Let $A \in \mathcal S$. Define the unary predicate $J_A(e)$ on $\mathbb N$ by \begin{align*} J_A(e) \iff \Phi_e^A(e) \text{ halts}. \end{align*} For each $s \in \mathbb N$, let $A\upharpoonright s$ denote the finite binary string of length $s$ whose value at each $t < s$ is $1$ if $t \in A$ and is $0$ if $t \notin A$. The predicate $J_A(e)$ is arithmetical with parameter $A$: there is a primitive recursive computation predicate $H(e,\sigma,s)$, where $\sigma$ is a finite binary string, such that \begin{align*} J_A(e) \iff \exists s \in \mathbb N \, H(e,A\upharpoonright s,s). \end{align*} Since $A \in \mathcal S$ and $\mathcal M$ satisfies arithmetical comprehension, the set \begin{align*} \{e \in \mathbb N : J_A(e)\} \end{align*} belongs to $\mathcal S$. By definition this set is $A'$. Hence $A' \in \mathcal S$ for every $A \in \mathcal S$. [/step]

[step:Reduce arithmetical definitions to finite iterated jumps]Assume conversely that $A' \in \mathcal S$ for every $A \in \mathcal S$. Let $\varphi(n,\vec b,\vec A)$ be an arithmetical formula with free number variable $n$, number parameters $\vec b = (b_1,\dots,b_\ell) \in \mathbb N^\ell$, and set parameters $\vec A = (A_1,\dots,A_k)$ from $\mathcal S$. Number parameters are standard because $\mathcal M$ is an $\omega$-model, so the tuple $\vec b$ can be hardwired into any ordinary computation. Define the oracle $C \subseteq \mathbb N$ as follows. If $k \geq 1$, let \begin{align*} C := A_1 \oplus \cdots \oplus A_k \end{align*} be the finite join computed using a fixed primitive recursive coding of pairs, so that each membership question $t \in A_i$ is computable from $C$ uniformly in $i$ and $t$, and membership in $C$ is computable from the finite tuple $(A_1,\dots,A_k)$. If $k = 0$, let $C := \varnothing$. Since $\mathcal M$ is an $\omega$-model of $\mathsf{RCA}_0$, its second-order part $\mathcal S$ is a [Turing ideal](/page/Turing%20Ideal): it contains all computable sets, is closed under finite joins, and is downward closed under [Turing reducibility](/page/Turing%20Reducibility). Hence $C \in \mathcal S$ in both cases. Define the finite iterated [Turing jump](/page/Turing%20Jump) of $C$ recursively by $C^{(0)} := C$ and, for each $r \in \mathbb N$, by $C^{(r+1)} := (C^{(r)})'$. For sets $D,E \subseteq \mathbb N$, write $D \le_T E$ to mean that $D$ is [Turing reducible](/page/Turing%20Reducibility) to $E$, namely that there is an oracle Turing machine which, with oracle $E$, decides membership in $D$. We use the relativised arithmetical hierarchy theorem, equivalently the relativised form of [Post's theorem](/page/Post%27s%20Theorem): for every arithmetical formula $\varphi(n,\vec b,\vec A)$ whose set parameters are uniformly computable from the oracle $C$, there is some $m \in \mathbb N \cup \{0\}$ and a total computable decision procedure with oracle $C^{(m)}$ deciding $\varphi(n,\vec b,\vec A)$ uniformly in $n$, with the fixed number parameters $\vec b$ built into the program. The hypotheses are satisfied because each set parameter $A_i$ is computable from the finite join $C$, and the number parameters $\vec b$ are standard natural numbers in the $\omega$-model. Equivalently, the set \begin{align*} B := \{n \in \mathbb N : \varphi(n,\vec b,\vec A)\} \end{align*} satisfies $B \le_T C^{(m)}$.[/step]

[guided]The goal is to show that the set defined by $\varphi$ belongs to $\mathcal S$. Since $\varphi$ may contain alternating number quantifiers, it need not be computable from the original set parameters $A_1,\dots,A_k$. Number parameters cause no additional difficulty: because $\mathcal M$ is an $\omega$-model, each $b_j$ is an ordinary natural number and can be written directly into the index of a Turing machine. First collect all set parameters into one oracle. If $k \geq 1$, define \begin{align*} C := A_1 \oplus \cdots \oplus A_k, \end{align*} where $\oplus$ denotes a fixed primitive recursive finite-join coding. This coding is chosen so that, uniformly in $i$ and $t$, the question $t \in A_i$ is computable from $C$, and membership in $C$ is computable from the finite tuple $(A_1,\dots,A_k)$. If $k = 0$, there are no set parameters to collect, and we define $C := \varnothing$. The second-order part of an $\omega$-model of $\mathsf{RCA}_0$ is a [Turing ideal](/page/Turing%20Ideal): it contains computable sets and is closed under finite joins. Therefore $C \in \mathcal S$ whether $k = 0$ or $k \geq 1$. Now define the jump notation that measures the arithmetical complexity available from $C$. Set $C^{(0)} := C$, and for each $r \in \mathbb N$ define \begin{align*} C^{(r+1)} := (C^{(r)})'. \end{align*} Thus $C^{(m)}$ is the result of applying the [Turing jump](/page/Turing%20Jump) exactly $m$ times to $C$. For sets $D,E \subseteq \mathbb N$, the notation $D \le_T E$ means that $D$ is [Turing reducible](/page/Turing%20Reducibility) to $E$: there is an oracle Turing machine which, with oracle $E$, decides whether an input belongs to $D$. We apply the relativised normal-form theorem for the arithmetical hierarchy, equivalently the relativised form of [Post's theorem](/page/Post%27s%20Theorem). The statement we need is the following: if all set parameters occurring in an arithmetical formula are uniformly computable from an oracle $C$, then the set defined by that formula is computable from a finite iterated jump of $C$. Its relevant hypotheses are satisfied here because every set parameter in $\vec A$ is computable from $C$, and the number parameters $\vec b$ are fixed standard natural numbers. The theorem gives an integer $m \in \mathbb N \cup \{0\}$ such that the truth value of $\varphi(n,\vec b,\vec A)$ is computable uniformly from $C^{(m)}$. In set-theoretic form, if \begin{align*} B := \{n \in \mathbb N : \varphi(n,\vec b,\vec A)\}, \end{align*} then \begin{align*} B \le_T C^{(m)}. \end{align*} This is the exact place where arithmetical formulas are converted into jump computations.[/guided]

[step:Iterate jump closure and use ideal closure to obtain the defining set] Since $C \in \mathcal S$ and $\mathcal S$ is closed under Turing jump, induction on $m$ gives \begin{align*} C^{(m)} \in \mathcal S. \end{align*} The base case is $m = 0$, where $C^{(0)} = C \in \mathcal S$; the induction step uses the assumed jump closure applied to $C^{(r)} \in \mathcal S$ to obtain $C^{(r+1)} = (C^{(r)})' \in \mathcal S$. The set $B$ defined by \begin{align*} B := \{n \in \mathbb N : \varphi(n,\vec b,\vec A)\} \end{align*} is Turing reducible to $C^{(m)}$. Since $\mathcal S$ is the second-order part of an $\omega$-model of $\mathsf{RCA}_0$, it is downward closed under [Turing reducibility](/page/Turing%20Reducibility). Therefore $B \in \mathcal S$. Thus every arithmetically definable subset of $\mathbb N$ with standard number parameters and set parameters from $\mathcal S$ belongs to $\mathcal S$. This is precisely the second-order comprehension scheme of $\mathsf{ACA}_0$ in the $\omega$-model $\mathcal M$. Hence $\mathcal M \models \mathsf{ACA}_0$. [/step]