*Below, all sentences/formulas are first-order and in the language of arithmetic. For simplicity, we conflate numbers and numerals, and conflate sentences/formulas and their Godel numbers.*


Given a formula $\varphi(x)$ and a sentence $\theta$, say that $\theta$ **asserts its own $\varphi$-ness** iff $\mathsf{PA}\vdash\theta\leftrightarrow\varphi(\theta).$ Let $\mathsf{SR}(\theta)$ be the set of $\varphi$ such that $\theta$ asserts its own $\varphi$-ness. Bounded truth predicates show that $\mathsf{SR}(\theta)$ is never empty. I'm curious how much $\mathsf{SR}(\theta)$ could actually depend on $\theta$; in particular, if there's a lot of potential variety here, this might give a meaningful notion of "degree of self-referentiality" of a sentence.


Here's one way to make this precise:



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> Are there sentences $\theta,\theta'$ such that $\mathsf{SR}(\theta)\not\cong\mathsf{SR}(\theta')$ as partial orders?
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The partial ordering I have in mind is provability: $\sigma\le\rho$ iff $\mathsf{PA}\vdash\forall x[\sigma(x)\rightarrow\rho(x)]$. By considering bounded truth predicates, $\mathsf{SR}(\theta)$ always contains a copy of $\mathbb{Z}$: briefly, look at $\tau\_n^+(x)=$ "$x$ is a true $\Sigma\_n$ sentence" and $\tau\_n^-(x)=$ "$x$ is not a false $\Sigma\_n$-sentence" for $n$ sufficiently large. Another "canonical" element is the formula $\varphi(x)\equiv\theta$ (basically, "ignore input, output $\theta$"). Finally, $\mathsf{SR}(\theta)$ is always a distributive lattice. Beyond this, however, I don't see anything useful.



Are there different "levels" of self-referentiality in arithmetic?
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 $\mathsf{SR}(\theta)$.


