Abstract not found

In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the first-order case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galois-stability.

Fellow student, dear friend, colleague

Abstract In this article two game-theoretically flavored approaches to logic are systematically compared: dialogical logic founded by Paul Lorenzen and Kuno Lorenz, and the game-theoretical semantics of Jaakko Hintikka. For classical Propositional logic and for classical First-order logic, an exact connection between ‘intuitionistic dialogues with hypotheses ’ and semantical games is established. Various questions of a philosophical nature are also shown to arise as a result of the comparison, among them the relation between the model-theoretical and proof-theoretical approaches to the philosophy of logic and mathematics.

What do the theorems of Gödel-Deligne, Chevalley-Tarski, Ax-Grothendieck, Tarski-Seidenberg, and Weil-Hrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. In fact, these model theoretic methods often show a pattern that extends across these areas. What are model theoretic methods? Model theory is the activity of a ‘selfconscious’ mathematician. This mathematician distinguishes an object language (syntax) and a class of structures for this language and ‘definable ’ subsets of those structures (semantics). Semantics provides an interpretation of inscriptions in the formal language in the appropriate structures. At its most basic level this allows the recognition that syntactic transformations can clarify the description of the same set of numbers. Thus, x2 − 3x < −6 is rewritten as x < −2 or x> 3; both formulas define the same set of points if they are interpreted in the real numbers. After clarifying these fundamental notions, we give an anachronistic survey of three themes of 20th century model theory: the study of a) properties of first order

What do the theorems of Gödel–Deligne, Chevalley–Tarski, Ax–Grothendieck, Tarski–Seidenberg, and Weil–Hrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. In fact, these model theoretic methods often show a pattern that extends across these areas. What are model theoretic methods? Model theory is the activity of a “selfconscious” mathematician. This mathematician distinguishes an object language (syntax) and a class of structures for this language and “definable ” subsets of those structures (semantics). Semantics provides an interpretation of inscriptions in the formal language in the appropriate structures. At its most basic level this allows the recognition that syntactic transformations can clarify the description of the same set of numbers. Thus, x2 − 3x <−6 is rewritten as x<−2 orx>3; both formulas define the same set of points if they are interpreted in the real numbers. After clarifying these fundamental notions, we give an anachronistic survey of three themes of twentieth century model theory: the study of a) properties of first

1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5