A Note on Effective Categoricity for Linear Orderings

Abstract

We study effective categoricity for linear orderings. For a computable structure $\mathcal {S}$, the degree of categoricity of $\mathcal {S}$ is the least Turing degree which is capable of computing isomorphisms among arbitrary computable copies of $\mathcal {S}$.

We build new examples of degrees of categoricity for linear orderings. We show that for an infinite computable ordinal $\alpha$, every Turing degree c.e. in and above $\mathbf {0}^{(2\alpha + 2)}$ is the degree of categoricity for some linear ordering. We obtain similar results for linearly ordered abelian groups and decidable linear orderings.

A Appendix: Proof of Proposition 2

For a non-zero countable ordinal $\alpha$, we define the auxiliary formula $Ord_{\alpha }(x,y)$. Suppose that the Cantor normal form of $\alpha$ is equal to

$$\omega ^{\beta _0}\cdot n_0 + \omega ^{\beta _1}\cdot n_1 + \ldots + \omega ^{\beta _t} \cdot n_t,$$

where $\beta _0>\beta _1>\ldots >\beta _t$ and $0<n_i<\omega$ for all i. We set:

It is not difficult to prove the following claim.

Lemma 6

Assume that $\beta$ is a computable ordinal, and $\omega ^{\beta }< \alpha <\omega ^{\beta +1}$. For a well-ordering $\mathcal {A}$ and elements $a,b\in \mathcal {A}$, we have $\mathcal {A}\models Ord_{\alpha }(a,b)$ iff the interval $[a,b[_{\mathcal {A}}$ is isomorphic to $\alpha$. Moreover, the formula $Ord_{\alpha }$ is logically equivalent to a computable $\varSigma _{2\beta +2}$ formula.

Now suppose that $\mathcal {L}_0$ is a computable copy of the ordinal $\omega ^{\beta }$ with the following property: given a pair of elements $a<_{\mathcal {L}_0} b$, one can effectively find the Cantor normal form of the interval $[a,b[_{\mathcal {L}_0}$. We may assume that $\mathcal {L} = \mathcal {L}_0\cdot (1+\eta )$.

We describe the formally $\varSigma ^0_{2\beta +1}$ Scott family $\varPhi$ for the structure $\mathcal {L}$. Let

It is easy to show that $\psi$ is equivalent to a computable $\varPi _{2\beta }$ formula.

First, we define the Scott formula $\phi _a$ for an element $a\in \mathcal {L}$. If a is the least element in $\mathcal {L}$, then set $\phi _a(x) = \forall y (x\le y)$. If a is not the least element and $\mathcal {L} \models \psi (a)$, then define $\phi _a = \psi$. Now assume that $\mathcal {L}\not \models \psi (a)$. We find the element b such that . Let $\gamma$ be the ordinal such that the interval $[b,a[_{\mathcal {L}}$ is isomorphic to $\gamma$. Set

Let $\bar{a} = a_0, a_1, \ldots , a_n$ be a tuple from $\mathcal {L}$ such that $a_0<_{\mathcal {L}} a_1<_{\mathcal {L}} \ldots <_{\mathcal {L}} a_n$. For $i<n$, we set

It is straightforward to prove that $\mathcal {L}\models \phi _{\bar{a}}(\bar{a})$. Moreover, for a tuple $\bar{b}$ from $\mathcal {L}$, $\mathcal {L}\models \phi _{\bar{a}}(\bar{b})$ implies that the tuple $\bar{b}$ is automorphic to $\bar{a}$. Furthermore, the formula $\phi _{\bar{a}}$ is logically equivalent to a computable $\varSigma _{2\beta + 1}$ formula. We use the effective procedure for calculating Cantor normal forms to construct the formulas $\phi _{\bar{a}}$ and to build the desired c.e. Scott family $\varPhi$ consisting of computable $\varSigma _{2\beta +1}$ formulas. Note that the formulas have no parameters. Hence, by Corollary 2, the structure $\mathcal {L}$ is uniformly relatively $\varDelta ^0_{2\beta +1}$ categorical.