A two-cardinal theorem for homogeneous sets and the elimination of Malitz quantifiers

Abstract

Introduction. The aim of the present paper is to give a first approximation to the problem of finding natural conditions for a theory in the first order language to admit the elimination of certain Malitz quantifiers in the same sense as the substructure completeness theorem does for elimination of elementary quantifiers. In other words, we look for the classes of first order theories which remain essentially the same when we add certain Malitz quantifiers. Thus, the present paper is a contribution to first order model theory, and not to logics with additional quantifiers. Our investigations were inspired by the second author's mostly unpublished work (cf. [TU1] and [TU2]) concerning the eliminability of the quantifiers there are K.a many (= Malitz quantifiers for 1-tuples) which we intended to extend to the general case. Although we did not completely succeed in our design, we obtained some partial results which we present in the following order. ? 1 contains definitions, conventions, central properties related to the quantifiers, and an eliminability condition for Ramsey quantifiers (= Malitz quantifiers in the N0-interpretation). ?2 is devoted to a two cardinal theorem for maximally homogeneous sets (without any reference to quantifiers). ?3 is an application of ?2 to the eliminability problem. ?4 provides a negative answer to a question of Baldwin and Kueker [BK, Question 4]. Open questions are scattered about the paper. Our results were obtained independently of [BK] in September-October 1979. For further historical remarks, see below.