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

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