By John T. Baldwin

Glossy version idea all started with Morley's categoricity theorem: A countable first-order concept that has a distinct (up to isomorphism) version in a single uncountable cardinal (i.e., is specific in cardinality) if and provided that an analogous holds in all uncountable cardinals. during the last 35 years Shelah made nice strides in extending this end result to infinitary good judgment, the place the elemental device of compactness fails. He invented the idea of an summary basic classification to provide a unifying semantic account of theories in first-order, infinitary good judgment and with a few generalized quantifiers. Zilber constructed comparable ideas of infinitary version thought to review complicated exponentiation. This e-book offers the 1st unified and systematic exposition of this paintings. the various examples stretch from natural version idea to module idea and covers of Abelian kinds. Assuming just a first direction in version thought, the publication expounds eventual categoricity effects (for periods with amalgamation) and categoricity in very good periods. Such an important instruments as Ehrenfeucht-Mostowski versions, Galois kinds, tameness, omitting-types theorems, multi-dimensional amalgamation, atomic varieties, sturdy units, susceptible diamonds, and ideal sessions are constructed thoroughly and methodically. The (occasional) reliance on extensions of easy set conception is obviously laid out. The ebook concludes with a collection of open difficulties.

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1) of quasiminimal excellence holds. Proof. We first show the case G = ∅. This proceeds in two stages. Note that for any H |= Σ the elements of dclH (∅) have the form qπH and exp(qπH ). √ That is, dclH (∅) is (Q, Q( 1)). Thus, any partial isomorphism of an H and H ′ must map qπH to qπH ′ and exp(qπH ) to exp(qπH ′ ) for every q. We show this map is actually an L-isomorphism. That is, we show that for any finite sequence of rationals and e = q0 πH , . . , qs πH ∈ H and e′ = q0 πH ′ , . . , qs πH ′ ∈ H ′ , qftpL (exp(e)) = qftpL (exp(e′ )).

2) The only penalty for increasing the size of the language or the L¨ owenheim number is that the size of L′ and the number of types omitted may increase. 18 and its applications. Much of this book is concerned with the spectrum of an AEC: the function which tells us the number of members of K in each cardinality. 17. For any class of models K, I(K, λ) denotes the number of isomorphism types of members of K with cardinality λ. Using the presentation theorem we have a first basic fact about the spectrum.

39 40 5. 3. (K, ≺∗ ) is an AEC with L¨ owenheim Number ℵ1 . 4. (K, ≺∗∗ ) is not an AEC. 5. 2 is ℵ1 . We would like to translate an Lω1 ,ω (Q)-sentence to an AEC with L¨owenhheim number ℵ0 and which has at least approximately the same number of models in each uncountable cardinality. This isn’t quite possible but certain steps can be taken in that direction. This translation will require several steps. We begin here with a fundamental result about Lω1 ,ω (Q); in Chapter 6, we will complete the translation.