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.

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 unraveling the long-hidden issues of the most famous free speech case of all time, noted author I.F. Stone ranges far and wide over Roman as well as Greek history to present an engaging and rewarding introduction to classical antiquity and its relevance to society today. The New York Times called this national best-seller an "intellectual thriller.".

Islamic ethics; Muslims; conduct of life.

Книга предназначена для студентов, преподавателей и научных сотрудников высших учебных заведений, занимающихся вопросами педагогики и психологии.

The revolution in science and technology affects all aspects of the life of society. First of all, it influences the development of the productive forces and changes the character of the technological process and the place of the immediate producer in that process. The transformation of science into a direct productive force raises the effectiveness of socialist social production, the productivity of labor, and people's well-being.

We study the matrices, reduced matrices and algebras associated to the systems SAT of structural annotated logics. In previous papers, these systems were proven algebraizable in the finitary case and the class of matrices analyzed here was proven to be a matrix semantics for them.We prove that the equivalent algebraic semantics associated with the systems SAT are proper quasivarieties, we describe the reduced matrices, the subdirectly irreducible algebras and we give a general decomposition theorem. As a consequence we obtain a (...) decision procedure for these logics. (shrink)