A spectrum hierarchy

1. Introduction Let 9 be a finite similarity type, that is, a finite set of (nonlogical) predicate symbols. By an 9'-structure, we mean a relational structure suitable for 9. Let 6 be a first-order sentence (with equality), and let Pl ,. .. , P , be those (nonlogical) predicate symbols in 6 which are not in 9' (these are the extra predicate symbols). Let 6' be the existential second-order sentence 3Pl. .. 3Pm6. The 9'-spectrum (or generalized spectrum) of b' is the class of finite 9'-structures in which CT' is true. This corresponds to TARSKI'S [7] notion of PC, where we restrict our attention to the class of finite structures, When 9' = 8, we can identify the 9'-spectrum of 6' with the set of cardi-nalities of finite structures in which cs is true. This set, called the spectrum of 6, was first considered by H. SCIIOLZ [6]. We show that for each spectrum A , there is a positive integer k such that {nk: n E A) is a spectrum involving only one binary predicate symbol. We use this to show that if there are spectra with certain properties, then there are spectra involving only one binary predicate symbol which have those properties. Define Fk(9') to be the class of those 9-spectra in which all of the extra predicate symbols are k-ary. We show that there is an exact trade-off between the degree of the extra predicate symbols and the cardinality of an " extra universe ". I n the case of spectra, we find that if A is a set of positive integers, and if k 2 2, then A is in 9k+1(0) iff (n[n1lk]: neA} is in Fk(0), where [x] is the greatest integer not exceeding x. We use the trade-off to show that if F P (9) = Fp+l(.Y), then s k (9) = Fp(Y) for each k 2 p. It is an open problem as to whether there is any spectrum not in F2(9'), or, indeed, whether there is any 9-spectrum not obtainable by using only one extra binary predicate symbol. by n. If A is a set, then A is the cardinality of the set. Denote the set of k-tuples <al I. .. , ak) of members of A by A k. If 9' is a finite similarity type and % is an Y-structure (both defined earlier), then we denote the universe of % …