The discussion of problems of contradiction in the development of the socialist economy has not thus far brought scholars to an agreed opinion on this question. The thought has been expressed in this regard that to frame the question as one of defining a basic contradiction of socialism is false and that under socialism only partial and temporary contradictions can occur, which spring up and are resolved in various periods of the development of socialist society.

We present some equivalent conditions for a quasivariety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document} of structures to be generated by a single structure. The first such condition, called the embedding property was found by A.I. Mal′tsev in [6]. It says that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf A}, {\bf B} \in \mathcal {K}}$$\end{document} are nontrivial, then there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf C} \in \mathcal{K}}$$\end{document} (...) such that A and B are embeddable into C. One of our equivalent conditions states that the set of quasi-identities valid in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{K}}$$\end{document} is closed under a certain Gentzen type rule which is due to J. Łoś and R. Suszko [5]. (shrink)

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)

The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is (...) best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups, groups and semigroup conditions which imply orderability, the orderability of free groups, Hölder's Theorem, Mal'tsev's classification of the order types of countable ordered groups. (shrink)

In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable (...) numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice. (shrink)

The keynote idea of the theses is contained in the author’s assumption that modern philosophy doesn’t meet its claiming pretensions: to be universal form of knowledge. First of all philosophy is connected not with knowledge but with ideas and secondly being authentic it “exists only in everyday life”.1 In orderthat philosophy could realize its innate essence corresponding conditions of social being should exist but they are still absent and therefore philosophy is absent as well. Its place is occupied by metaphysics (...) which is free movement of thinking that has no definiteness as the condition of being of any form of thought. That iswhy we watch crisis state of “philosophy” which becomes apparent in vagueness of its subject: “philosophy is not only one, there are many of them”. The author supposes that philosophy as the highest relation of mankind to objective reality is the only one. Philosophy lowered down from the heaven (Platon, Ciceron) in the field of human being must raise people not only to purity of thought, but mainly to purity of their activity both with each other and with nature. Then real interaction of men and nature will take place. This will be the beginning of spiritualization of the Universe. It may exist only under the conditions of social uniformity as historical form of Weltanschauung (world outlook) that removes religious form. Realization of its being supposes fundamental change of social life. It is necessary that private property should be replaced with social property in the sphere of social production. Only under these conditions philosophy may appear and exist as historical form of Weltanschauung. (shrink)

The author elucidates the ontological basis of elementary mathematical theories and thereby assesses their certainty as a foundation for the more complex theories of modern mathematics, such as mathematical analysis and set theory. He adduces arguments in favor of the position of Frege, who held that geometry can provide a sufficiently broad and certain foundation for mathematics.

t. 1. Problemy vospitanii︠a︡ vsestoronne razvitoĭ lichnosti. Dukhovnyĭ mir shkolʹnika. Metodika vospitanii︠a︡ kollektiva.--t. 2. Formirovanie Kommunisticheskikh ubezhdeniĭ molodogo pokolenii︠a︡. Kak vospitat nastoi︠a︡shchego cheloveka. Sto sovetov uchiteli︠u︡.--t. 3. Serdt︠s︡e otdai︠u︡ deti︠a︡m. Rozhdenie grazhdanina. Pisʹma k synu.--t. 4. Pavlyshskai︠a︡ sredni︠a︡i︠a︡ shkola. Razgovor s molodym direktorom shkoly.--t. 5. Statʹi.