Sergey S. Goncharov

The computable dimension of ordered abelian groups

Abstract Let G be a computable ordered abelian group. We show that the computable dimension of G is either 1 or ω , that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable presentation which admits a computable basis.

Trivial, strongly minimal theories are model complete after naming constants

We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(M M ) is a model complete £ M -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0-decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is Σ 0 5.

Friedberg Numberings of Families of n-Computably Enumerable Sets

We establish a number of results on numberings, in particular, on Friedberg numberings, of families of d.c.e. sets. First, it is proved that there exists a Friedberg numbering of the family of all d.c.e. sets. We also show that this result, patterned on Friedbergs famous theorem for the family of all c.e. sets, holds for the family of all n-c.e. sets for any n>2. Second, it is stated that there exists an infinite family of d.c.e. sets without a Friedberg numbering. Third, it is shown that there exists an infinite family of c.e. sets (treated as a family of d.c.e. sets) with a numbering which is unique up to equivalence. Fourth, it is proved that there exists a family of d.c.e. sets with a least numbering (under reducibility) which is Friedberg but is not the only numbering (modulo reducibility).

Computability and Numberings

The theory of computable numberings is one of the main parts of the theory of numberings. The papers of H. Rogers [36] and R. Friedberg [21] are the starting points in the systematical investigation of computable numberings. The general notion of a computable numbering was proposed in 1954 by A.N. Kolmogorov and V.A. Uspensky (see [40, p. 398]), and the monograph of Uspensky [41] was the first textbook that contained several basic results of the theory of computable numberings. The theory was developed further by many authors, and the most important contribution to it and its applications was made by A.I. Malt’sev, Yu.L. Ershov, A. Lachlan, S.S. Goncharov, S.A. Badaev, A.B. Khutoretskii, V.L. Selivanov, M. Kummer, M.B. Pouer-El, I.A. Lavrov, S.D. Denisov, and many other authors.

On a question of Frank Stephan

For many TxtEX-learnable computable families of recursively enumerable sets, all their computable numberings are equivalent with respect to the reduction via the functions recursive in the halting problem. We show that this holds for every TxtEX-learnable computable family of recursively enumerable sets, but, in general, the converse is not true.

Hanf number for Scott sentences of computable structures

The Hanf number for a set S of sentences in

Intrinsic bounds on complexity and definability at limit levels

\mathcal {L}_{\omega _1,\omega }

On automorphic tuples of elements in computable models

Lω1,ω (or some other logic) is the least infinite cardinal