On the elimination of Malitz quantifiers over Archimedian real closed fields

Rapp [5] proves that the theory of the field N of reals in the language with Magidor-Malitz quantifiers Qi, QZ,... is decidable. The quantifier Q] is interpreted in a structure M by: M~Q",xl . . . . ,x,q) iff there exists a set Be=M, card(B)>~% such that for all distinct b 1 . . . . , b ,e B: M ~ q~[b~, ..., b,]; such a set B is called homogeneous for tp. Using Tarski's classical result on real closed fields [6] Rapp's theorem is shown by proving that the quantifiers QI, Q2 . . . . are effectively eliminable over N. Biirger [1] addresses the question whether Rapp's methods can be applied to the class of all uncountable archimedian real closed fields (ARCF). Up to isomorphism this is the class of real closed subfelds of ~ . Bfirger shows that the uniform eliminability of Q], for n > 3, is independent of ZFC. In particular, the continuum hypothesis CH implies the existence of a counterexample to the eliminability of Q3. (Note that by results of Cowles [2], Vinner [7], and Goltz [3], the quantifiers QI and Q2 are always effectively eliminable over uncountable ARCFs.) In view of these results it seems natural to consider models of-q CH, and study the uniform eliminability of Q~, Q~, ... where 2~ card(JR)= o)~. Here we assume that whenever we talk about the quantifier Q~, only fields of cardinality > ~ are considered. We show that again eliminability is independent of ZFC + --7 CH: