Elementary classes closed under descending intersection

R. L. Vaught
1966 Proceedings of the American Mathematical Society  
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
more » ... A, A{){<« is formed by a nonempty set A = | 31 j and finitary relations R$ among the elements of A, ior £Kt. We also may write iA, Aj, S/)t-SB, we write 21 >-*58 or 58*-<2l. If ft is a nonempty set of similar structures, then the notions U{2I/2l£ft} and fl{21/216ft} are defined in the obvious way, provided that ft is upward or downward directed, respectively, by the notion C, and provided in the second case that D { | 211 /3t£ft} f^O. Under these conditions, we speak of an ascending union or a descending intersection. Theorem 1 was conjectured by M. Rabin. It improves a result of A. Robinson [7]. Robinson showed that an elementary class 3C closed under arbitrary intersection is closed under ascending union; later, Los-Suszko [4] and Chang [2] established that an elementary class
doi:10.1090/s0002-9939-1966-0191822-3 fatcat:iwta72nha5hafm3nbuukfx5iee