Automorphism Properties of Stationary Logic

By means of an Ehrenfeucht-Mostowski construction we obtain an automorphism theorem for a syntactically characterized class of Laa-theories comprising in particular the finitely determinate ones. Examples of Laa-theories with only rigid models show this result to be optimal with respect to a classification in terms of prenex quantifier type: Rigidity is seen to hinge on quantification of type .)V .. stat ... permitting of the parametrization of families of disjoint stationary systems by the elements of the universe. Introduction. The classic theorem of Ehrenfeucht and Mostowski [EhM] shows that any first-order theory which permits infinite models also admits models generated by chains of indiscernibles of an arbitrarily chosen order type. As a consequence, models with large groups of automorphisms are obtained in any infinite cardinality. The study of related techniques of model construction and corresponding automorphism theorems for extensions of first-order logic illuminates a characteristic aspect of these logics' expressive power. L(Q1), the logic with the quantifier "there exist uncountably many", e.g., admits the following analogue of the EM theorem, which is optimal both with respect to the restriction to countable chains of indiscernibles, and with respect to 2W as the maximal number of automorphisms which can be guaranteed. THEOREM 1 (Ebbinghaus[Eb]). Let L = L(Q1), and let SO be a countable vocabulary and T an L[So]-theory with uncountable models. Then there is a countable extension S D SO and a map F from the class C,, of countable linear orderings to the class of (S u C)-structures, C a set of X, many new constant symbols, with the following properties: 1. For I E c I is embedded into the universe of F(I) as a chain of indiscernibles with respect to L[S u C]. 2. F(I) is generated by I as an (S u C)-structure. 3. F(I) I SO = To. We shall here investigate the corresponding behaviour of Laa for models in cardinality N1. For the basic definitions concerning Laa see, for example, [BKM] or [K]. The semantics of aa-quantification is inductively given by the rule: A l= aaXO(X) iff {A' E9,,A I A # /(A')} contains a club; a club being a subset Received November 9, 1990; revised March 15, 1991. ? 1992, Association for Symbolic Logic 0022-4812/92/5701-0016/$01 .70