# Order types of models of theories of ordinals

For $$C$$ a set of ordinals, let $$\mathcal{L}(C)$$ be the language with identity, a relation symbol for less than, function symbols for successor, addition, multiplication, and exponentiation, and a constant for every $$\alpha \in C$$. If $$\beta$$ is an epsilon number with $$\sup (C) \leq \beta$$, let $$\mathsf{T}_\beta(C)$$ be $$\{ \phi \in \mathsf{Sent}(\mathcal{L}(C)) : \mathcal{N}_\beta \vDash \phi \}$$, where $$\mathcal{N}_\beta$$ is the model with domain $$\beta$$ and all symbols in $$\mathcal{L}(C)$$ interpreted standardly. (We require $$\beta$$ to be an epsilon number in order to ensure that the required functions are total.)

Thus $$\mathcal{N}_\omega = \mathcal{N}$$, and $$\mathsf{T}_\omega(\{ 0 \})$$ is true arithmetic. Let us say that a model of $$\mathsf{T}_\beta(C)$$ is standard if its domain is order-isomorphic to $$\beta$$ and nonstandard otherwise. We use $$\vert A \vert$$ to denote the order type of $$A$$.

As is well known, every nonstandard model of true arithmetic has order type $$\omega + \vert A \vert \centerdot (\omega^* + \omega)$$ for $$A$$ a dense linear order without endpoints.

What can we say about the order types of nonstandard models of $$\mathsf{T}_\beta(C)$$ for various choices of $$\beta$$ and $$C$$?

• An easy observation is that by the 'class pigeonhole principle' for any set $C$ there must be some $\beta$ such that $T_\beta(C)$ has arbitrarily large well-ordered models. I have a question related to the issue of well-ordered models of theories here, but that focuses on the well-ordered models. Jun 4 '19 at 0:14

First, I'll discuss the case of empty $$C$$.

Observe that the structures $$(\varepsilon_{\alpha};+,\cdot,\mathsf{exp})$$ are definitionally equivalent with the structures $$HF(\alpha;<)$$ (the structure of hereditarily finite sets with ordinals $$<\alpha$$ as urelements). Namely we consider the bijection $$f\colon HF(\alpha)\to \varepsilon_{\alpha}$$ define by recursion on $$HF(\alpha)$$:

1. $$f\colon \beta\longmapsto \varepsilon_{\beta}$$;
2. $$f\colon \{a_1,\ldots,a_n\}\longmapsto 2^{f(a_1)}+\ldots+2^{f(a_n)}$$, where $$f(a_1)>\ldots>f(a_n)$$.

Using the high expressive power of $$HF(\alpha;<)$$ it is easy to show that we could define the predicates $$f(x)+f(y)=f(z)$$, $$f(x)f(y)=f(z)$$, and $$f(x)^{f(y)}=f(z)$$. And in $$(\varepsilon_{\alpha};+,\cdot,\mathsf{exp})$$ we could easily define the predicates

1. $$\mathsf{Ur}(f^{-1}(x))$$ as "$$x$$ is an $$\varepsilon$$-number";
2. $$f^{-1}(x)\in f^{-1}(y)$$ as "$$y$$ isn't an $$\varepsilon$$-number and $$2^x$$ is in $$2$$-base Cantor normal form for $$y$$" or equivalently as "$$y$$ isn't an $$\varepsilon$$-number and $$2^{x+1}z+2^{x}\le y< 2^{x+1}z+2^{x+1}$$, for some $$z$$";
3. $$f^{-1}(x) as "$$x$$ and $$y$$ are $$\varepsilon$$-numbers and $$x".

Let us denote as $$\mathsf{U}_{\alpha}(C)$$ the elementary theory of $$HF(\alpha;<,\langle c_{\beta}\mid \beta\in C\rangle)$$, where $$c_{\beta}$$ is the constant for the ordinal $$\beta$$. The defined equivalence gives us a one-to-one correspondence $$h$$ between the models of $$\mathsf{T}_{\varepsilon_{\alpha}}(\emptyset)$$ and models of $$\mathsf{U}_{\varepsilon_{\alpha}}(\emptyset)$$. There are two important groups of models of $$\mathsf{U}_{\alpha}(C)$$: those of the form $$HF(A)$$, for some linear order $$A$$ and those that have ill-founded membership relation. The models of the second group are akin of non-standard models of arithmetic, but the models of the first group constitute essentially different phenomenon. Namely it is easy to see that models of $$\mathsf{T}_{\varepsilon_{\alpha}}(\emptyset)$$ of the form $$h^{-1}(HF(A))$$ would always poses a well-founded initial segment of the order type $$\varepsilon_0$$ (latter I'll show that provided that $$\alpha\ge \omega^{\omega}$$ the model $$h^{-1}(HF(A))$$ always start with the initial fragment of the order type $$\varepsilon_{\omega^{\omega}}$$). On the other hand the models of $$\mathsf{T}_{\varepsilon_{\alpha}}(\emptyset)$$ of the form $$h^{-1}(\mathfrak{M})$$, where $$\mathfrak{M}$$ have ill-founded membership predicate would always start with the initial segment of the order type $$\omega+(\omega^{*}+\omega)A$$, where $$A$$ is a dense linear order without end-points (here $$\omega+(\omega^{*}+\omega)A$$ is the order type of the non-standard model of arithmetic produced from $$h^{-1}(\mathfrak{M})$$ by the standard interpretation of arithmetic in hereditarily finite sets). But, I couldn't say much more about the order types of the models of $$\mathsf{T}_{\varepsilon_{\alpha}}(\emptyset)$$ of the latter group.

However the models of $$\mathsf{U}_{\alpha}(\emptyset)$$ of the form $$HF(A)$$ actually are easy to describe. There is a classical result of Ehrenfeucht that structures $$HF(\alpha;<)$$ and $$HF(\beta;<)$$ are elementary equivalent iff either

1. $$\alpha=\beta$$ or
2. $$\alpha,\beta$$ and are of the form $$\alpha=\omega^\omega(1+\alpha')+\gamma$$ and $$\beta=\omega^\omega(1+\beta')+\gamma$$.

By essentially the same argument as Ehrenfeucht's it is easy to show that for any $$\alpha<\omega^{\omega}$$ the structures $$HF(\omega^{\omega}(1+A)+\alpha)$$ and $$HF(\omega^{\omega}(1+A')+\alpha)$$ are elementary equivalent for any linear orders $$A$$ and $$A'$$. That for any $$\alpha<\omega^{\omega}$$ and any linear order $$B$$ if $$HF(B)\equiv HF(\omega^{\omega}(1+A)+\alpha)$$, then $$B$$ is of the form $$\omega^{\omega}(1+A')+\alpha$$. And that for $$\alpha<\omega^{\omega}$$ for any $$A$$ if $$HF(A)\equiv HF(\alpha)$$ then $$A$$ have the order type $$\alpha$$.

When we go back to models of $$T_{\varepsilon_{\alpha}}(\emptyset)$$ this shows that all the models of $$T_{\varepsilon_{\omega^{\omega}(1+\beta)+\alpha}}(\emptyset)$$ from this group are of the order types $$\varepsilon_{\omega^{\omega}(1+A)+\alpha}$$, where $$A$$ is some linear order. Here the order $$\varepsilon_A$$, for a given linear order $$A$$, is defined as the natural order on nested (formal) Cantor normal forms built from $$0$$ and constants $$\varepsilon_a$$, for all $$a\in A$$.

The addition of constants doesn't change the picture too much.

For each set of ordinals $$C\subseteq \varepsilon_{\alpha}$$ the structure $$(\varepsilon_{\alpha};+,\cdot,\mathsf{exp},\langle c_{\beta}\mid \beta\in C\rangle)$$ is definitionally equivalent to the structure $$HF(\alpha;<,\langle c_{\beta}\mid \beta \in \mathbb{H}(C)\rangle)$$, where $$\mathbb{H}(C)$$ is the least set of ordinals such that each $$\beta\in C$$ is the value $$t(\varepsilon_{\beta_1},\ldots,\varepsilon_{\beta_n})$$, for some $$\beta_1,\ldots,\beta_n\in \mathbb{H}(C)$$ and $$t(x_1,\ldots,x_n)$$ is a term build of operations $$+,\cdot,\mathsf{exp}$$. Further observe that in $$HF(\alpha;<)$$ (and actually even $$(\alpha,<)$$) given a constant $$c_{\beta}$$ for the ordinal $$\beta$$ with the Cantor normal form $$\omega^{\omega+\beta_1}+\ldots+\omega^{\omega +\beta_n}+\omega^{k_1}+\ldots+\omega^{k_n}$$ any other ordinal in the interval $$[\omega^{\omega+\beta_1}+\ldots+\omega^{\omega +\beta_n},\omega^{\omega+\beta_1}+\ldots+\omega^{\omega +\beta_n}+\omega^{\omega})$$ is definable in $$HF(\alpha;<,c_{\beta})$$. Henceforth, over $$HF(\alpha;<)$$ the constants for ordinals from a set $$C\subseteq \alpha$$ are definitionally equivalent to the set of constants for ordinals from $$\{\omega^{\omega}(1+\beta)\mid \beta\in \mathbb{B}(C)\}$$, where $$\mathbb{B}(C)$$ is the least set of ordinals such that each $$\beta\in C$$ either $$<\omega^{\omega}$$ or there is $$\gamma\in \mathbb{B}(C)$$ such that $$\omega^{\omega}(1+\gamma)\le \beta<\omega^{\omega}(1+\gamma+1)$$. Therefore $$(\varepsilon_{\alpha};+,\cdot,\mathsf{exp},\langle c_{\beta}\mid \beta\in C\rangle)$$ is definitionally equivalent to the structure $$HF(\alpha;<,\langle c_{\omega^{\omega}(1+\beta)}\mid \beta\in \mathbb{B}(\mathbb{H}(C))\rangle)$$.

It is fairly simple to adopt Ehrenfeucht's result to the case with constants for ordinals of the form $$\omega^{\omega}(1+\beta)$$. For a given ordinal $$\beta$$, set $$D\subseteq \beta$$, and ordinal $$\alpha<\omega^{\omega}$$ let $$\gamma=\omega^{\omega}(1+\beta)+\alpha$$ and let us look at orders $$A$$ such that there is a model of $$\mathsf{U}_{\gamma}(\omega^{\omega}(1+D))$$ which constant-free part is of the form $$HF(A)$$. It is easy to show that here the possible order types of $$A$$ are of the form $$\omega^{\omega}+\sum_{\delta\in D}\omega^{\omega}A_{\delta}+\alpha,$$ for some non-empty linear orders $$A_{\delta}$$. Thus for each $$C\subseteq \varepsilon_{\gamma}$$ the theory $$\mathsf{T}_{\varepsilon_{\gamma}}(C)$$ have models with the order types $$\varepsilon_A$$, where $$A$$ is of the form $$\omega^{\omega}+\sum_{\delta\in \mathbb{B}(\mathbb{H}(C))}\omega^{\omega}A_{\delta}+\alpha,$$ for some non-empty linear orders $$A_{\delta}$$. And as in the case of empty $$C$$, any model $$\mathfrak{M}$$ of $$\mathsf{T}_{\varepsilon_{\gamma}}(C)$$ which order type isn't of this form should correspond to a structure with ill-founded sets and hence in $$\mathfrak{M}$$ there would be an initial interval with the order type $$\omega+(\omega^*+\omega)A$$ for some dense linear order $$A$$ without end-points.

[1] A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae, 49:129–141, 1961.