Models Without Indiscernibles

Abstract

For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size 2, with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting over T, H(T), is at least 2, (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly 2b,.) If TX true arithmetic, then H(T) = 2,. If 8 -A (p)'-, then any completion of Peano Arithmetic has a model of size 8 with no set of indiscernibles of size p. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L. ?0. Introduction. The main accomplishment of the research presented here is the lifting of the Specker-MacDowell-Gaifman technology for over arithmetic to a technique for building models from carefully constructed n-types. In Specker and MacDowell [1959] it is shown that every model of Peano Arithmetic, (PA), has a proper elementary extension. Gaifman [1965], [1968] shows how to construct end extension 1-types which give rise to such extensions, investigates the iteration of such extension techniques, and gives a construction of independent extension types which is the direct forebear of the generic types we use here. It is implicit in these papers (and, in fact, is a special case of the M(I) in Gaifman [1976]) that a model of PA generated by 21-many mutually generic will have no two distinct elements with the same 1-type.3 Here we give a direct exposition of such a model, in a way that clearly suggests how to construct a model of PA with no (n + 1)-length sequence of indiscernibles. Carrying through this suggestion involves far more complicated combinatorics than in the case of 2i. What is needed is a certain generalization of finitary Ramsey's Theorem proved independently by Neset'ril and Rodl [1976] and by the present authors. Our original exposition also involved some complicated machinery specially designed for our particular results. We owe a large debt to Haim Gaifman who pointed out some structural properties of our models which are of interest in their own right and which can be used directly to obtain our results. The approach of the present paper is that suggested to us by Gaifman, and we will clearly indicate the specific results which are due to him. Received April 19, 1976; revised September 12, 1977. University of Wisconsin-Milwaukee. This author's research was supported, in part, by an American Mathematical Society Postdoctoral Research Fellowship and by NSF Grant Number