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If $\varphi$ is allowed to be an arbitrary formula on natural numbers (not required to behave the same way on Gödel codes of equivalent sentences or any such thing), then $\mathsf{SR}(\theta)$ is always equivalent to the preorder on all arbitrary formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\varphi$ gives rise to a formula $\varphi' \in \mathsf{SR}(\theta)$ via $\varphi'(\theta) = \theta$ and $\varphi'(n) = \varphi(f(n))$ for $n \neq \theta$.

We have that $\varphi \leq \psi$ iff $\varphi' \leq \psi'$, and that every formula $\varphi \in \mathsf{SR}(\theta)$ is equivalent to some $\psi'$ (specifically, take $\psi(n) = \varphi(f^{-1}(n))$). Thus, the map $\varphi \mapsto \varphi'$ is an equivalence from the preorder of arbitrary formulas to the preorder $\mathsf{SR}(\theta)$.

If $\varphi$ is not required to behave the same way on Gödel codes of equivalent sentences or any such thing, then $\mathsf{SR}(\theta)$ is always equivalent to the preorder on all arbitrary formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\varphi$ gives rise to a formula $\varphi' \in \mathsf{SR}(\theta)$ via $\varphi'(\theta) = \theta$ and $\varphi'(n) = \varphi(f(n))$ for $n \neq \theta$.

We have that $\varphi \leq \psi$ iff $\varphi' \leq \psi'$, and that every formula $\varphi \in \mathsf{SR}(\theta)$ is equivalent to some $\psi'$ (specifically, take $\psi(n) = \varphi(f^{-1}(n))$). Thus, the map $\varphi \mapsto \varphi'$ is an equivalence from the preorder of arbitrary formulas to the preorder $\mathsf{SR}(\theta)$.

If $\varphi$ is allowed to be an arbitrary formula on natural numbers (not required to behave the same way on Gödel codes of equivalent sentences or any such thing), then $\mathsf{SR}(\theta)$ is always isomorphic to the full order on all arbitrary formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\varphi$ gives rise to a formula $\varphi' \in \mathsf{SR}(\theta)$ via $\varphi'(\theta) = \theta$ and $\varphi'(n) = \varphi(f(n))$ for $n \neq \theta$. We have that $\varphi \leq \psi$ iff $\varphi' \leq \psi'$, and that every formula in $\mathsf{SR}(\theta)$ is equivalent to some $\varphi'$. Thus, the ordering on $\mathsf{SR}(\theta)$ is the same as the ordering on arbitrary formulas.

If $\varphi$ is allowed to be an arbitrary formula on natural numbers (not required to behave the same way on Gödel codes of equivalent sentences or any such thing), then $\mathsf{SR}(\theta)$ is always equivalent to the preorder on all arbitrary formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\varphi$ gives rise to a formula $\varphi' \in \mathsf{SR}(\theta)$ via $\varphi'(\theta) = \theta$ and $\varphi'(n) = \varphi(f(n))$ for $n \neq \theta$. 

We have that $\varphi \leq \psi$ iff $\varphi' \leq \psi'$, and that every formula $\varphi \in \mathsf{SR}(\theta)$ is equivalent to some $\psi'$ (specifically, take $\psi(n) = \varphi(f^{-1}(n))$). Thus, the map $\varphi \mapsto \varphi'$ is an equivalence from the preorder of arbitrary formulas to the preorder $\mathsf{SR}(\theta)$.

If $\varphi$ is allowed to be an arbitrary formula on natural numbers (not required to behave the same way on Gödel codes of equivalent sentences or any such thing), then $\mathsf{SR}(\theta)$ is always equal to the full order on all such formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\varphi$ gives rise to a formula $\varphi' \in \mathsf{SR}(\theta)$ via $\varphi'(\theta) = \theta$ and $\varphi'(n) = \varphi(f(\theta))$ for $n \neq \theta$. We have that $\varphi \leq \psi$ iff $\varphi' \leq \psi'$, and that every formula in $\mathsf{SR}(\theta)$ can be obtained as some $\varphi'$. Thus, the ordering on $\mathsf{SR}(\theta)$ is the same as the ordering on arbitrary formulas.

If $\varphi$ is allowed to be an arbitrary formula on natural numbers (not required to behave the same way on Gödel codes of equivalent sentences or any such thing), then $\mathsf{SR}(\theta)$ is always isomorphic to the full order on all arbitrary formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\varphi$ gives rise to a formula $\varphi' \in \mathsf{SR}(\theta)$ via $\varphi'(\theta) = \theta$ and $\varphi'(n) = \varphi(f(n))$ for $n \neq \theta$. We have that $\varphi \leq \psi$ iff $\varphi' \leq \psi'$, and that every formula in $\mathsf{SR}(\theta)$ is equivalent to some $\varphi'$. Thus, the ordering on $\mathsf{SR}(\theta)$ is the same as the ordering on arbitrary formulas.