Well, in order to compute isomorphisms between copies of \(\omega\) we need to know the successor relation. For \(\omega\) this relation is computable while for \(\mathcal B\), it computes the Halting set. It turns out that \(\mathbf 0'\), the Turing degree of the Halting set, is the least degree computing isomorphims between any two computable copies of \(\omega\). We say that \(\omega\) has degree of categoricity \(\mathbf 0'\). Degrees of categoricity are a robust measure of the algorithmic complexity of a computable structure: Up to \(\mathbf 0'\) all the copies of \(\omega\) look the same and it can compute isomorphisms between them.

For over ten years degrees of categoricity have been heavily studied by computability theorists. The main questions are:

- Which Turing degrees are degrees of categoricity of computable structures?
- Say d is a degree of categoricity, can we always build two isomorphic computable structures A and B with degree of categoricity d and such that d is the least degree computing isomorphism between A and B? Such degrees are called strong.

In a joint project with Barbara Csima we classified the strong degrees of categoricity above \(\mathbf{0}''\) as the treeable degrees—the degrees of Turing least paths through computable trees in Baire space. This allowed us to give many new examples of degrees of categoricity and provides a new opening to attack the question whether every degree of categoricity is strong. A preprint of this paper can be found on arXiv.

In an old version of this post and an earlier preprint we claimed that every degree of categoricity above \(\mathbf 0 ''\) is strong. Unfortunately, we made a mistake there and do not have a proof of this. A

]]>I recently thought about the Turing degree structure of the degrees that compute every hyperarithmetic sets and could not find much besides the results of Jockusch and Simpson. However, after talking to a few people, I realized that quite a bit is known. Here is an attempt to summarize a few things. I apologize for skipping a lot of the details but hope that anyone can figure those out with the references provided.

**Theorem 1.** Let \(\mathbf d> HYP\). Then \(\mathbf d\) computes a copy of \(\omega_1^{\mathrm{CK}}\) if and only if \(\mathbf d^{(3)}\geq deg(\mathcal O)\).

*Proof sketch.* Assume that \(\mathbf d^{(3)}\geq deg(\mathcal O)\). Then \(\mathbf d^{(3)}\) can compute a copy of \(\omega_1^{\mathrm{CK}}\). One can show that \(\mathbf d\) then computes a copy of \(\omega^2\cdot \omega_1^{\mathrm{CK}}\cong \omega_1^{\mathrm{CK}}\) (see Ash and Knight [Theorem 9.11, 2]). For the other direction assume that \(\mathbf d\) computes a copy of \(\omega_1^{\mathrm{CK}}\). We will show that \(d^{(3)}\) can compute the set of indices of computable well-orderings. One can show that if \(L\) is a well-order and \(o(L)\leq \omega^{\alpha}\), then \(L\) is uniformly \(\Delta^0_{2\alpha}\) categorical, i.e., for \(L\) computable, \(\mathbf 0^{(2\alpha)}\) (\(\mathbf 0^{(2\alpha-1)}\) if \(\alpha\) is finite) can compute an isomorphism between any two computable copies of \(L\), and this is uniform. Let \(C\) be \(\mathbf d\)’s copy of \(\omega_1^{\mathrm{CK}}\). All computable well-orders have order-type less than \(C\), so the set of indices of computable well-orders is \(O=\{e:\exists a \varphi_e \cong C_{<a}\}\) where \(C_{<a}\) is the initial segment of \(C\) up to the element \(a\). Then \(O\) can be defined by

where \(H\) denotes a jump hierarchy on \(C\). Note that it mus not be the case that \(\mathbf d\) computes a jump hierarchy, after all it only computes all hyperarithmetic sets, but it does not necessarily do this uniformly. However, creating a jump hierarchy in this case is \(\Sigma^0_3\) (see Montalbán [3] for more on jump hierarchies). Checking whether \(\varphi^{H^{[a]}}_i\) is an isomorphism is \(\Pi^0_2\). So \(\mathbf d^{(3)}\geq deg(O)=deg(\mathcal O)\).

We have seen, that if we restrict our degrees to degrees that compute \(HYP\) and a copy of \(\omega_1^{\mathrm{CK}}\) then we can not be far below Kleene’s O. What if we can not compute \(\omega_1^{\mathrm{CK}}\)? It turns out we can be quite far away in that case.

**Theorem 2.** There is a degree \(\mathbf d>HYP\) that is hyperarithmetically low, i.e., \(\omega_1^{\mathbf d}=\omega_1^{\mathrm{CK}}\).

*Proof sketch.* Consider the set

This set is \(\Sigma^1_1\) so by the Gandy basis theorem it must contain a set \(Y=(e_0,H_0)\) such that \(\omega_1^{Y}=\omega_1^{\mathrm{CK}}\). Thus there is no \(\alpha\) such that \(Y^{(\alpha)}\equiv_T \mathcal O\). See Montalbán [3] for more on Gandy’s basis theorem and jump hierarchies in this context.

Thanks to Antonio Montalbán and Ted Slaman for showing me this.

[1] Jockusch Jr, Carl G., and Stephen G. Simpson. 1976. “A Degree-Theoretic Definition of the Ramified Analytical Hierarchy.” Annals of Mathematical Logic 10 (1): 1–32.

[2] Ash, Christopher J., and Julia F. Knight. 1990. “Pairs of Recursive Structures.” Annals of Pure and Applied Logic 46 (3): 211–34.

[3] Montalbán, Antonio. 2021. Computable Structure Theory: Beyond the Arithmetic. draft. https://math.berkeley.edu/~antonio/CSTpart2_DRAFT.pdf.

]]>We just submitted a new article titled “Positive enumerable functors” and you can find the preprint here or on arxiv. This article is joint work with Barbara Csima and Daniel Yu who proved most of the results presented there in an undergraduate research project he did with Barbara and me over the summer.

We investigate different effectivizations of functors using Turing operators and enumeration operators and compare these notions. The most promising of these notions are *positive enumerable functors*. A positive enumerable functors from \(\mathcal A\) to \(\mathcal B\) consists of two enumeration operators \(\Psi\) and \(\Psi_*\) such that when given an enumeration of the positive diagram \(P(\hat{\mathcal A})\) of a structure isomorphic to \(\hat{\mathcal A}\cong \mathcal A\), \(\Phi\) enumerates the positive diagram of \(F(\hat{\mathcal A})\). The operator \(\Phi_*\) will take care of the isomorphisms by enumerating the graph of \(F(f:\hat{\mathcal A}\to\tilde{\mathcal A})\) given \(P(\hat{\mathcal A})\oplus Graph(f)\oplus P(\tilde{\mathcal A})\).

We show that if \(\mathcal A\) is positive enumerable bi-transformable to \(\mathcal B\) (if there are positve enumerable functors \(F:\mathcal A\to\mathcal B\) and \(G:\mathcal B\to \mathcal A\) such that \(F\) and \(G\) are “enumerable pseudo-inverses”) then \(\mathcal A\) and \(\mathcal B\) have the same enumeration degree spectrum and that positive enumerable functors are independent from the already established enumerable and computable functors.

We submitted the article to a conference and hope to expand it in the future, adding more preservation results and investigating syntactic equivalences.

]]>My newest preprint Degree spectra of analytic complete equivalence relations just appeared on arXiv.

This article investigates the bi-embeddability and elementary bi-embeddability relation on countable structures both from the viewpoint of descriptive set theory and computable structure theory. Two structures are *bi-embeddable* if either is isomorphic to an substructure of the other. They are *elementary bi-embeddable* if either is isomorphic to a elementary substructure of the other.

Our main results are the following.

**Theorem 1.** *The elementary bi-embeddability relation on graphs is a \(\pmb \Sigma^1_1\) complete equivalence relation under Borel reducibility.*

The proof of this result uses Marker extensions with so called minimal models. A structure \(\mathcal A\) is *minimal* if it does not contain an elementary substructure.

Let \(\sim\) be an equivalence relation on a class of structures. Then given a structure \(\mathcal A\) we define the \(\sim\) spectrum of \(\mathcal A\) as the set of all sets Turing equivalent to a structure \(\sim\) equivalent to \(\mathcal A\), i.e., \[ DgSp_\sim(\mathcal A)=\{ X\subseteq \omega: \exists \mathcal B \sim \mathcal A \ X\equiv_T \mathcal A\}.\]

**Theorem 2.** *Every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph, i.e., for every \(\mathcal G\), there is a graph \(\mathcal G'\) such that*
\[ DgSp_\approx(\mathcal G’)=\{ X: X’\in DgSp_\approxeq(\mathcal G)\}.\]

This result is proven by analyzing the reduction given to prove Theorem 1. We show that this reduction induces a computable bi-transformation with a suitable class of structures.

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