# Around Classification Theory of Models by Saharon Shelah

By Saharon Shelah

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Example text

E. it coincides on P 4 with a constructible relation. Indeed for any x, y ∈ P , there exists (according to (1)) a unique σ with σ(x) = y. Thus (x, u) ≡C (y, w) iff one has, for this σ, σ(u) = w. So (x, y, u, w) ∈ / R iff (∃w ∈ P )R(x, y, u, w ) & w = w). The projection of an ω-constructible relation in a universal domain is ω-constructible (appendix A); so ¬R is ω-constructible, as well as R; thus they are both relatively constructible. Let H = AutAut(U/F,C) (P ) = {h ∈ Sym(P ) : (∀σ ∈ Aut(U/F, C)) σh = hσ}.

G. Teubner, Leipzig, 1910; reprinted by Editions Jacques Gabay, 1992.

4 is due to Poizat (cf. 5 is immediate from Tarski’s quantifier-elimination and the fact that constructible subgroups are closed. 2. Definition of internality Definition. D is C-internal if there exists a definable V ⊂ C k and a definable surjective map V → D. The surjective map in question may require parameters. To emphasize this, let us say that D is C-interpreted over F if C, D are F -definable, and there exist F -definable V, f with V ⊂ C k and f : V → D surjective. The reason for the terminology is that f allows to interpret D inside the induced structure on C, with universe V /Ker(f ).