DECIDABLE CLASSES OF NUMBER-THEORETIC SENTENCES by I. J. HEATH in Leicester (England) A. Method of Construction 1. Limited Constituents Terminology. Let h ' denote the class of natural numbers 0, 1 , 2 , . . . We say that P ( x l , . . . , x,) is a predicate if P : N" -+ ( 0 , l}, and that f(xl, . . . , x,) is a function if f : N n -+ N . We find it expedient to treat predicates and functions together under the single term Constituents. We usually denote a set of constituents in the form: P, Q , . . . ; f , g , . . ., where P, Q , . . . are the predicates in the set, and f , g , . . . are the functions in the set. The reader may consult [3] for omitted proofs and details. L i m i t e r s a n d t h e i r r e p r e s e n t a t i v e functions. Let B,,, . . . be an increasing sequence of finite Booman algebras of subsets of N , such that, for every positive integer 1, we have { I -1) E 8 1 . The B l are called limiters. Let B1x denote {BE B1: x E B } , and let p1x be any function which selects a unique element of B1x, i.e. Vx(,!lzx E Bzx) and V x , y ( B p = Bly + = Bly), then 812 is called a representative function for 231. I n particular, if minA denotes the least member of A , then minBzx is a representative function for BZ.