Consider the type (in model theory)$tp(a/M)$ for a first order theory in larger uncountable saturated model $M'$ (of cardinality strictly larger than of the underlying language). By a standard result it can be identified with the orbit of $a$ under the automorphism group $Aut_M(M')$ of $M'$ fixing $M$ pointwise. This is taken as a definition for some infinitary theories in the case we have the monster model. A slightly weaker version can be rephrased in terms of amalgamation.

Rami Grossberg, Classification theory for abstract elementary classes (section 6), Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics 302, AMS, (2002) 165–204, pdf

Rami Grossberg, Monica VanDieren, Galois-stability for tame abstract elementary classes, Journal of Mathematical Logic 6, No. 1 (2006) 25–49, pdf

Last revised on August 9, 2016 at 15:50:07.
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