By John T. Baldwin

This new quantity of the Ω-series is written as an creation to first order balance conception. it really is prepared round the the spectrum challenge: calculate the variety of types a primary order idea T has in each one uncountable cardinal. to unravel this challenge a generalization of the suggestion of algebraic independence "nonforking" used to be built. during this textual content the summary houses of this relation (in distinction to different books which start with the technical description). the $64000 notions of orthogonality and regularity are rigorously constructed: this equipment is then utilized to the spectrum challenge. whole proofs of the Vaught conjecture for omega-stable theories are awarded the following for the 1st time in booklet shape. enormous attempt has been made by way of the writer to supply a lot wanted examples. particularly, the ebook comprises the 1st book of Shelah's notorious instance exhibiting the need of his ways to resolve Vaught's conjecture for omega-stable theories. The connections of summary balance idea with algebra really with the speculation of modules are emphasised.

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Groundwork The next result follows immediately from the previous exercise. Both results are easily proved by induction on the complexity of formulas. See [Chang & Keisler 1973]. 24 Lemma. 25 Definition. Let E C M be linearly ordered by <, which need not be a definable relation in M. The ordered sequence (E, <) is a sequence of order indiscernibles if for any φ(x) and any pair e, e' of finite sequences from E, which are both in increasing order, (= φ(e) <-* φ(e'). The following result is proved using the compactness theorem and Ramsey's theorem.

7 it is rather easy to see that if φ(x) is strongly minimal in T then the dependence relation defined by algebraic closure allows us to assign a dimension to the strongly minimal set. If T is not No-categorical, it is easy to conclude that infinitely many distinct finite dimensions are possible. Thus when φ contains no parameters one concludes that T has NO countable models. 34, some finite inessential extension of every NI but not N0-categorical theory has infinitely many countable models. To prove that T itself has infinitely many models is much more difficult.

Formula of the form 0(0; 6 — y) or is inconsistent. 10 Exercise. p. formula without parameters then φ(M) is a submodule of Mn. p. p. definable subgroups where the meet of two such subgroups is their intersection and the join is defined by (φ + ψ)(v) = (3w)φ(w) Λ ψ(v — w). The following exercise is crucial to our development. 11 Exercise. p. formula φ and any α, 6, either 0(z;α) and 0(x;6) are equivalent or they are contradictory. Here is the main theorem of this section. 12 Theorem. Fix an R- module M.

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