There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

/pdf/hilbert-s-tenth-problem-1y9h7ahnpm.pdf

Hilbert's tenth problem

Abstract A Turing degree d is the degree of categoricity of a computable structure ${\\cal S}$ if d is the least degree capable of computing isomorphisms among arbitrary computable copies of ${\\cal S}$. A degree d is the strong degree of categoricity of ${\\cal S}$ if d is the degree of categoricity of ${\\cal S}$, and there are computable copies ${\\cal A}$ and ${\\cal B}$ of ${\\cal S}$ such that every isomorphism from ${\\cal A}$ onto ${\\cal B}$ computes d. In this paper, we build a c.e. degree d and a computable rigid structure ${\\cal M}$ such that d is the degree of categoricity of ${\\cal M}$, but d is not the strong degree of categoricity of ${\\cal M}$. This solves the open problem of Fokina, Kalimullin, and Miller [13]. For a computable structure ${\\cal S}$, we introduce the notion of the spectral dimension of ${\\cal S}$, which gives a quantitative characteristic of the degree of categoricity of ${\\cal S}$. We prove that for a nonzero natural number N, there is a computable rigid structure ${\\cal M}$ such that $0\\prime$ is the degree of categoricity of ${\\cal M}$, and the spectral dimension of ${\\cal M}$ is equal to N.

Degrees of categoricity and spectral dimension

Review: Gerald E. Sacks, On the Degrees Less than 0'

Finite computable dimension and degrees of categoricity

Degrees of Categoricity vs. Strong Degrees of Categoricity

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility 
$$\leqslant _c$$

. This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the 
$$\Delta ^0_2$$

 case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by 
$$\leqslant _c$$

 on the 
$$\Sigma ^{-1}_{a}\smallsetminus \Pi ^{-1}_a$$

 equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

https://link.springer.com/content/pdf/10.1007/s00153-020-00710-1.pdf

Classifying equivalence relations in the Ershov hierarchy

A major theme in the study of degree structures of all types has been the question of the decidability or undecidability of their first order theories. This is a natural and fundamental question that is an important goal in the analysis of these structures. In this paper, we study decidability for theories of upper semilattices that arise from the theory of numberings. We use the following approach: given a level of complexity, say $$\varSigma ^0_{\alpha }$$
 , we consider the upper semilattice $$R_{\varSigma ^0_{\alpha }}$$
 of all $$\varSigma ^0_{\alpha }$$
 -computable numberings of all $$\varSigma ^0_{\alpha }$$
 -computable families of subsets of $$\mathbb {N}$$
 . We prove that the theory of the semilattice of all computable numberings is computably isomorphic to first order arithmetic. We show that the theory of the semilattice of all numberings is computably isomorphic to second order arithmetic. We also obtain a lower bound for the 1-degree of the theory of the semilattice of all $$\varSigma ^0_{\alpha }$$
 -computable numberings, where $$\alpha \ge 2$$
 is a computable successor ordinal. Furthermore, it is shown that for any of the theories T mentioned above, the $$\varPi _5$$
 -fragment of T is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on $$\mathbb {N}$$
 , equipped with composition.

Elementary theories and hereditary undecidability for semilattices of numberings

Given a dce degree d, consider the dce sets in d and the corresponding degrees of their Lachlan sets Ishmukhametov provided a systematic investigation of such degrees, and proved that for a given dce degree d > 0, the class of its ce predecessors in which d is ce, denoted as R[d], can consist of either just one element, or an interval of ce degrees After this, Ishmukhametov asked whether there exists a dce degree d for which the class R[d] has no minimal element We give a positive answer to this question

Mars M. Yamaleev

Papers

Degrees of categoricity and spectral dimension

Degrees of Categoricity vs. Strong Degrees of Categoricity

Classifying equivalence relations in the Ershov hierarchy

Elementary theories and hereditary undecidability for semilattices of numberings

On a problem of Ishmukhametov