RANDOM UNARY PREDICATES - ALMOST SURE THEORIES AND COUNTABLE MODELS

Let U-n,U-p be the random unary predicate and T-k be the almost sure f irst-order theory of U-n,U-p under the linear ordering, where k is a p ositive integer and n(-1/k) much less than p(n) much less than (-1/(k1)). For each k, we give an axiomatization for the theory T-k. We find a model M-k of T-k of order type roughly that of Z(k) and show that n o other models of T-k exist of smaller size. (C) 1998 John Wiley & Son s, Inc.