Geodesic
Classification Theory and Stationary Logic
Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 893-907

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Stationary logic L(aa) is obtained for Lωω by adding a quantifier aa which ranges over countable sets and is interpreted to mean “for a closed unbounded set of countable subsets”. The dual quantifier for aa is stat, i.e., stat sφ(s) is equivalent to ¬aa s ¬φ(s). In the study of the L(aa)-model theory of structures a particular well behaved class was isolated, the finitely determinate structures. These are structures in which the quantifier “stat” can be replaced by the quantifier “aa” without changing the validity of sentences. Many structures such as R and all ordinals are finitely determinate. In this paper we will be concerned with finitely determinate first order theories, i.e., those theories all of whose models are finitely determinate. Example 0.1. [5] The theory of dense linear orderings is not finitely determinate. Let S be a stationary costationary subset of ω 1 and where 
Mekler, Alan H. Classification Theory and Stationary Logic. Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 893-907. doi: 10.4153/CJM-1987-044-7
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