It is well-known that the order type of any countable non-standard model of arithmetic $\mathfrak{A}$ is $\omega+(\omega^*+\omega)\eta$. My question is what could be said about the order types of ordinal notation systems within non-standard models of arithmetic?

The question is motivated by my answer to the question Order types of models of theories of ordinals about the order types of non-standard models of ordinal arithmetic: some group of this non-standard models corresponded to the ordinal notation systems within non-standard models of arithmetic, and I could say very little about their order types.

Now I'll provide a bit more details. Let us consider some standard computable ordinal notation system, say Cantor ordinal notation for the ordinals $<\varepsilon_0$ (although the same question could be asked for different ordinal notation systems). The ordinal notation system essentially is an arithmetical formula $x\prec y$ such in the standard model it defines a well-ordering with the order type $\varepsilon_0$. For each $\alpha<\varepsilon_0$ let me denote as $\hat\alpha$ the Gödel number for the ordinal $\alpha$ within the ordinal notation system. And for $\alpha<\varepsilon_0$ and a model $\mathfrak{A}\models \mathsf{PA}$ let me denote as $(\alpha)^{\mathfrak{A}}$ the order type in $\mathfrak{A}$ of the arithmetical relation $\prec\upharpoonright_{\hat\alpha}$ that is the restriction of $\prec$ to the elements $\prec \hat\alpha$.

It is easy to see that for each model of arithmetic $\mathfrak{A}\models \mathsf{PA}$ with the order type $A$ and an ordinal $\alpha<\omega^{\omega}$ with the Cantor normal form $\omega^{k_1}+\ldots +\omega^{k_n}$ the order type $(\alpha)^{\mathfrak{A}}$ is precisely $A^{k_1}+\ldots+ A^{k_n}$.

However I have no clue of what happening even with $(\omega^{\omega})^{\mathfrak{A}}$. Does $(\omega^{\omega})^{\mathfrak{A}}$ depends only on the order type of $\mathfrak{A}$? Even if the latter isn't the case, is $(\omega^{\omega})^{\mathfrak{A}}$ the same for all countable non-standard models $\mathfrak{A}$? In the other side of the spectrum of possibilities, one could imagine that $(\omega^{\omega})^{\mathfrak{A}}$ reflects a lot of information about the model $\mathfrak{A}$. Could we recover model $\mathfrak{A}$ from the order type $(\omega^{\omega})^{\mathfrak{A}}$? Could we recover $\mathsf{SSy}(\mathfrak{A})$ from the order type $(\omega^{\omega})^{\mathfrak{A}}$?

Of course I would be interested in the answers to the same kind of questions for other ordinals $\alpha$ and for more restricted classes of models $\mathfrak{A}$ (e.g. models of true arithmetic, or recursively saturated models, etc.).

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