Annual Meeting of the Association for Symbolic Logic: Philadelphia 1981

Abstract

S OF PAPERS 899 Indecomposability is a weakening of the concept of measurability; indeed, inaccessible cardinals carrying indecomposable ultrafilters exhibit some of the strong reflection properties of measurable cardinals. Silver asked whether the two notions were actually equivalent for inaccessible cardinals, i.e., whether an inaccessible cardinal carrying an indecomposable ultrafilter must be measurable. We prove that this is not the case: THEOREM 1. Con(ZFC + 3 a measurable cardinal) implies Con(ZFC + 3 an indecomposable ultrafilter over an inaccessible cardinal which is not measurable (in fact, not even weakly compact)). The basic forcing construction can be modified in a number of ways to produce various patterns in the Rudin-Keisler ordering on ultrafilters. Further forcing provides other corollaries; e.g.: THEOREM 2. Con(ZFC + 3 a measurable cardinal) implies Con(ZFC + Martin's Axiom + there is an indecomposable ultrafilter over the continuum but no 81-saturated ideal). KENNETH L. MANDERS, Transfer by definable relations between structures. An m-ary relation R 5 Str(oi1) x ... x Str(oim) between structures of similarity types l.. AIm is finitelyy first-order) definable if R, viewed as a class of m-sorted structures, is definable by a first-order sentence OAR with additional predicates (and possibly an extra sort). A separation base for R is a set i of m-tuples (01 ..,m), q5) a re-sentence, such that: (i) for any 00, . .. ) E= A O|R -A 7' /\ gO (i; (ii) for any or-sentences Xi, i = 1. m, such that OR A , there is a (b1, .., Xb) E A such that Xi F 0k. By the completeness theorem and Craig's trick, any definable R has a decidable separation base, but not necessarily an informative one. In contrast, many classical preservation theorems specify syntactically informative separation bases. THEOREM. There is an effective procedure which generates, for any definable R, a decidable separation base for R by syntactic transformation of AR. The transformation introduces no "extraneous syntactic material" except inequalities between variables. It first constructs a recursive closed game (Svenonius' Theorem); the separation base is constructed from the conjuncts in finite approximations to the game sentence by suitably coordinated positive Boolean combinations and distribution of prefix quantifiers over the Oi of the correct sort. This latter part of the construction generalizes Lindstr6m's theorem on regular rela-

Cite this paper

@article{Kochen1983AnnualMO, title={Annual Meeting of the Association for Symbolic Logic: Philadelphia 1981}, author={Simon Kochen and Hugues Leblanc and Charles D. Parsons}, journal={J. Symb. Log.}, year={1983}, volume={48}, pages={898-910} }