The Hanf number for a set S of sentences in \ is the least infinite cardinal \ such that for all \, if \ has models in all infinite cardinalities less than \, then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \. The same argument proves that \ is the Hanf number for Scott sentences of hyperarithmetical structures.

The author of the article studies the creative legacy of Yuriy Mamleev, trying to establish the correlation between of his philosophical and fiction writings. The conventional editorial distinction between philosophical and literary texts as different and autonomous categories is rejected. Instead of the idea, that philosophical and fictional aspects of texts by Mamleev are inextricably linked, it is claimed that they should be considered together. In the author’s opinion, these texts present different perspectives and implementations of his oeuvre that contains (...) the rational dimension and on the other hand – irrational dimension of artistic images and obscure allusions. Mamleev’s writings considered by means of 5 crucial “metathemes”: Аbyss and the “Ultimate relationships” concerned with it, the Mystery which has two levels – one cognizable and the another – uncoognizable in principal, the “root of self-being” and metaphysic of “I”, death and immortality, Russia and the Russian soul which are existed between West and East with special historical and cosmological mission. It is concluded that these 5 metathemes pervade all Mamleev’s texts and comprise his spiritual search. The characters of Mamleev’s fiction are mostly people of post-Soviet world, who ask questions with no set answers to become acquainted with the secrets of being. At the same time they are concerned with their own existence, their self destiny. Their actions are extraordinary and illogical. Supposedly Mamleev’s artistic images not only correspond with his philosophical ideas, for instance – utrism or the “Ultimate relationships”, but also reveal them from unexpected perspective. The author determines that distinction between philosophy and literature of Mamleev is incorrect, as these areas are conflated. Any distinction of this kind will ruin the logic of Mamleev’s thinking and outlook. Thus, metaphysical realism can be viewed as the crossing, distinction blurring area, productive and creative transgression. Or as the situation when philosophy is more than it is itself. (shrink)

This paper calculates, in a precise way, the complexity of the index sets for three classes of computable structures: the class $K_{\omega _{1}^{\mathit{CK}}}$ of structures of Scott rank $\omega _{1}^{\mathit{CK}}$ , the class $K_{\omega _{1}^{\mathit{CK}}+1}$ of structures of Scott rank $\omega _{1}^{\mathit{CK}}+1$ , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete $\Sigma _{1}^{1},\,I(K_{\omega _{1}^{\mathit{CK}}})$ is m-complete $\Pi _{2}^{0}$ relative to Kleen's O, and $I(K_{\omega _{1}^{\mathit{CK}}+1})$ is m-complete $\Sigma _{2}^{0}$ relative to O.

In the paper A. I. Malcev's problem on the characterization of axioms for classes with strong homomorphisms is being solved.

The theory of constructive (recursive) models follows from works of Froehlich, Shepherdson, Mal'tsev, Kuznetsov, Rabin, and Vaught in the 50s. Within the framework of this theory, algorithmic properties of abstract models are investigated by constructing representations on the set of natural numbers and studying relations between algorithmic and structural properties of these models. This book is a very readable exposition of the modern theory of constructive models and describes methods and approaches developed by representatives of the Siberian school of algebra (...) and logic and some other researchers (in particular, Nerode and his colleagues). The main themes are the existence of recursive models and applications to fields, algebras, and ordered sets (Ershov), the existence of decidable prime models (Goncharov, Harrington), the existence of decidable saturated models (Morley), the existence of decidable homogeneous models (Goncharov and Peretyat'kin), properties of the Ehrenfeucht theories (Millar, Ash, and Reed), the theory of algorithmic dimension and conditions of autostability (Goncharov, Ash, Shore, Khusainov, Ventsov, and others), and the theory of computable classes of models with various properties. Future perspectives of the theory of constructive models are also discussed. Most of the results in the book are presented in monograph form for the first time. The theory of constructive models serves as a basis for recursive mathematics. It is also useful in computer science, in particular, in the study of programming languages, higher level languages of specification, abstract data types, and problems of synthesis and verification of programs. Therefore, the book will be useful for not only specialists in mathematical logic and the theory of algorithms but also for scientists interested in the mathematical fundamentals of computer science. The authors are eminent specialists in mathematical logic. They have established fundamental results on elementary theories, model theory, the theory of algorithms, field theory, group theory, applied logic, computable numberings, the theory of constructive models, and the theoretical computer science. (shrink)

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," (...) which generalize the older results and which include positive results on when a certain homogeneous model of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees. (shrink)

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t. 1. Domarksistskai︠a︡ teorii︠a︡ poznanii︠a︡ --t. 2. Sot︠s︡ialʹno-kulturnai︠a︡ priroda poznanii︠a︡.