Search

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman's completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.

We answer a question of Calderoni and Clay by showing that the conjugation equivalence relation of left orderings of the Baumslag-Solitar groups $\mathrm{BS}(1,n)$ is hyperfinite for any $n$. Our proof relies on a classification of $\mathrm{BS}(1,n)$'s left-orderings via its one-dimensional dynamical realizations. We furthermore use the effectiveness of the dynamical realizations of $\mathrm{BS}(1,n)$ to study algorithmic properties of the left-orderings on $\mathrm{BS}(1,n)$.

We investigate natural variations of behaviourally correct learning and explanatory learning -- two learning paradigms studied in algorithmic learning theory -- that allow us to ``learn'' equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviorally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\pmb\Pi^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\boldsymbol\Pi_\omega^0$-complete set of models. We also give sharp conditions for theories to have a $\boldsymbol\Pi^0_n$-complete set of models. Finally, we determine the Turing degrees needed to witness the completeness.

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $\Sigma_2$ sentences.

We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman-Stanley embedding on Scott sentence complexity and show that it only preserves $\Pi^{\mathrm{in}}_{\alpha}$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott complexities except $\Sigma^{\mathrm{in}}_{3}$ and $\Sigma^{\mathrm{in}}_{\lambda+1}$ for $\lambda$ a limit ordinal. We show that the former can not be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau)$ be the set of countable structures with universe $\omega$ in vocabulary $\tau$ topologized by the Scott topology. We show that an invariant set $X \subseteq Mod(\tau)$ is $\Pi^0_\alpha$ in the effective Borel hierarchy of this topology if and only if it is definable by a $\Pi^p_\alpha$ - formula, a positive $\Pi^0_\alpha$ formula in the infinitary logic $L_{\omega_1,\omega}$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $K$ be positively computably embeddable in $K'$ by $\Phi$, then for every $\Pi^p_\alpha$ formula $\xi$ in the vocabulary of $K'$ there is a $\Pi^p_\alpha$ formula $\xi^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models \xi^\star$ if and only if $\Phi(A) \models \xi$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings., Comment: 17 pages

We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $\mathbf 0''$. They are precisely the \emph{treeable} degrees -- the least degrees of paths through computable trees -- that compute $\mathbf 0''$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $\mathbf d$ with $\mathbf 0^{(\alpha)}\leq \mathbf d\leq \mathbf 0^{(\alpha+1)}$ for $\alpha$ a computable ordinal greater than $2$ is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $\mathbf d$ with $\mathbf 0'\leq \mathbf d\leq \mathbf 0''$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $\mathbf d$ with $\mathbf 0'< \mathbf d< \mathbf 0''$ that is not the degree of categoricity of a rigid structure., Comment: Replaces arXiv:2209.04524 where it was incorrectly claimed that "Every degree of categoricity above 0'' is strong". We do not have a proof of this fact. Changes were made to reflect this

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega$ and that non-standard models of true arithmetic must have Scott rank greater than $\omega$. Other than that there are no restrictions. By giving a reduction via $\Delta^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $\alpha>\omega$ is realized as the Scott rank of a model of $T$.

We study countable structures from the viewpoint of enumeration reducibility. Since enumeration reducibility is based on only positive information, in this setting it is natural to consider structures given by their positive atomic diagram -- the computable join of all relations of the structure. Fixing a structure $\mathcal{A}$, a natural class of relations in this setting are the relations $R$ such that $R^{\hat{\mathcal{A}}}$ is enumeration reducible to the positive atomic diagram of $\hat{\mathcal{A}}$ for every $\hat{ \mathcal{A}}\cong \mathcal{A}$ -- the relatively intrinsically positively enumerable (r.i.p.e.) relations. We show that the r.i.p.e. relations are exactly the relations that are definable by $\Sigma^p_1$ formulas, a subclass of the infinitary $\Sigma^0_1$ formulas. We then introduce a new natural notion of the jump of a structure and study its interaction with other notions of jumps. At last we show that positively enumerable functors, a notion studied by Csima, Rossegger, and Yu, are equivalent to a notion of interpretability using $\Sigma^p_1$ formulas.

We study reductions well suited to compare structures and classes of structures with respect to properties based on enumeration reducibility. We introduce the notion of a positive enumerable functor and study the relationship with established reductions based on functors and alternative definitions.

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by this relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a analytic complete equivalence relation. We then investigate the algorithmic properties of this reduction to show that every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph.

In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures in two such classes, the classes of linear orders and Boolean algebras. We show that if $\mathcal L$ is a computable linear order of Hausdorff rank $n$, then for every bi-embeddable copy of it there is an embedding computable in $2n-1$ jumps from the atomic diagrams. We furthermore show that this is the best one can do: Let $\mathcal L$ be a computable linear order of Hausdorff rank $n\geq 1$, then $\mathbf 0^{(2n-2)}$ does not compute embeddings between it and all its computable bi-embeddable copies. We obtain that for Boolean algebras which are not superatomic, there is no hyperarithmetic degree computing embeddings between all its computable bi-embeddable copies. On the other hand, if a computable Boolean algebra is superatomic, then there is a least computable ordinal $\alpha$ such that $\mathbf 0^{(\alpha)}$ computes embeddings between all its computable bi-embeddable copies. The main technique used in this proof is a new variation of Ash and Knight's pairs of structures theorem.

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $\mathcal A$ as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of $\mathcal A$; the degree of bi-embeddable categoricity of $\mathcal A$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $\mathbf{0}^{(\alpha)}$ for $\alpha$ a computable successor ordinal and $\mathbf{0}^{(\lambda)}$ for $\lambda$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra., Comment: 18 pages

Given a countable scattered linear order $L$ of Hausdorff rank $\alpha < \omega_1$ we show that it has a $d\text{-}\Sigma_{2\alpha+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $\alpha < \omega_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $\alpha$ the class of Hausdorff rank $\alpha$ linear orders is $\pmb \Sigma_{2\alpha+2}$ complete., Comment: 25 pages

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs., Comment: 15 pages

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis., Comment: 17 pages

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $\Delta^0_\alpha$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $\Delta^0_\alpha$ bi-embeddable categoricity and relative $\Delta^0_\alpha$ bi-embeddable categoricity coincide for equivalence structures for $\alpha=1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $\mathbf{0},\mathbf{0}'$, or $\mathbf{0}''$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability., Comment: 18 pages

We study a new notion of reduction between structures called enumerable functors related to the recently investigated notion of computable functors. Our main result shows that enumerable functors and effective interpretability with the equivalence relation computable are equivalent. We also obtain results on the relation between enumerable and computable functors., Comment: 16 pages v2: corrected typos