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Elementary classes closed under descending intersection

1966
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Proceedings of the American Mathematical Society
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The purpose of this note is to establish the following theorem in the theory of models (announced in [ll]): Theorem 1. If an elementary class 3Z is closed under descending intersection then X. is a A2 class; i.e., 3C is the class of all models of a set of sentences of the form (Vx0 ■ ■ • xm_i) (3yo ■ • • yn-i) M, where M is quantifier-free. At the end, the question of extending Theorem 1 to pseudo-elementary classes will be briefly discussed. Preliminaries. A structure %L = {A, A{){<« is formed

doi:10.1090/s0002-9939-1966-0191822-3
fatcat:iwta72nha5hafm3nbuukfx5iee