The recent history of model theory

- January 2000

Abstract

The influence of Alfred Tarski was decisive in this early stage and in the successive years. This is due not only to his discovery of unquestionable definitions of the notions of truth and definability in a structure, but also to his founding of the basic notions of the theory, such as elementary equivalence and elementary extension. In the fifties and six- ties Jerry Los introduced the ultraproducts, Ronald Fra¨ isse developed the back-and-forth methods and investigated amalgamation properties, and Abraham Robinson started his voluminous contribution to Model Theory, including his celebrated non-standard analysis. Robinson’s non-standard analysis attracted the attention of mathematicians and philoso- phers. But at that time a feeling of exhaustion started pervading the whole theory. Daniel Lascar describes the situation as “un temps d’arret, comme si la machinerie, prete ` a tourner, ne savait quelle direction prendre.” At this point Michael Morley appeared in the scene, causing what can be called a second birth of Model Theory. A theory is said to be categorical at if it has only one model of cardinality up to isomorphism. In 1954 J. Los had asked whether, for every (countable) theory, categoricity at one uncountable cardinal implies categoricity at every other uncountable cardinal. In 1965 M. Morley publishes Categoricity in power (Transactions of the American Math. Society 114, 514-538) where he solves the problem in the armative. He introduces the (topological) spaces of types and defines a rank on types and formulas, now called Mor- ley rank. A theory is called totally trascendental if all types have ordinal Morley rank. M. Morley shows that for countable theories, this is just !-stability: over a countable set there are at most countably many complete types. He also proves that any theory categorical in an uncountable cardinality is !-stable. In the proofs he uses heavily con- structions with indiscernible sequences which had been studied a few years ago by Andrezj Ehrenfeucht and Andrzej Mostowski. The methods were partly combinatorial, based on Ramsey’s theorem, and partly topological. The importance of the results, methods and notions of M. Morley was recognized very soon. M. Morley investigated also the structure of countable models of uncountably

pdf – http://www.ub.edu/modeltheory/documentos/HistoryMT.pdf