Introduction Modern model theory began with Morley’s [Mor65a] categoricity theorem: A first order theory is categorical in one uncountable cardinal 𝜅 (has a unique model of that cardinality) if and only if it is categorical in all uncountable cardinals. This result triggered the change in emphasis from the study of logics to the study of theories. Shelah’s taxonomy of first order theories by the stability classification established the background for most model theoretic researches in the last 35 years. This book lays out of some of the developments in extending this analysis to classes that are defined in non-first order ways. Inspired by [Sac72, Kei71], we proceed via short chapters that can be covered in a lecture or two. There were three streams of model-theoretic research in the 1970’s. For simplic- ity in the discussion below I focus on vocabularies (languages) which contain only countably many relation and function symbols. In one direction workers in alge- braic model theory melded sophisticated algebraic studies with techniques around quantifier elimination and developed connections between model theory and alge- bra. A second school developed fundamental model theoretic properties of a wide range of logics. Many of these logics were obtained by expanding first order logic by allowing longer conjunctions or longer strings of first order quantifiers others added quantifiers for ‘there exist infinitely many’, ‘there exist uncountably many’, ‘equicardinality’, and many other concepts. This work was summarized in the Barwise-Feferman volume [BF85]. The use of powerful combinatorial tools such as the Erd¨ os-Rado theorem on the one hand and the discovery that Chang’s con- jecture on two cardinal models for arbitrary first theories is independent of ZFC and that various two cardinal theorems are connected to the existence of large car- dinals [CK73] caused a sense that pure model theory was deeply entwined both with heavy set-theoretic combinatorics and with (major) extensions of ZFC. In the third direction, Shelah made the fear of independence illusory for the most central questions by developing the stability hierarchy. He split all first order theories into 5 classes. Many interesting algebraic structures fall into the three classes (𝜔-stable, superstable, strictly stable) whose models admit a deep structural analysis. This classification is (set theoretically) absolute as are various fundamental properties of such theories. Thus, for stable theories, Chang’s conjecture is proved in ZFC [Lac72, She78]. Shelah focused his efforts on the test question: compute the func- tion 𝐼(𝑇,𝜅) which gives the number of models of cardinality 𝜅. He achieved the striking main gap theorem. Every theory 𝑇 falls into one of two classes. 𝑇 may be intractable, that is 𝐼(𝑇,𝜅) = 2𝜅, the maximum, for every sufficiently large 𝜅. Or, every model of 𝑇 is decomposed as a tree of countable models and the number of models in 𝜅 is bounded well below 2𝜅. The description of this tree and the proof of the theorem required the development of a far reaching generalization of the Van der Waerden axiomatization of independence in vector spaces and fields. This is vii
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