Outer Model Satisfiability

This paper solves the anticharacterization problem first identified in [S1]. Perhaps the most striking instance of this problem is that, working relative to just ZFC, there is no uniform first-order definition of the family of subsets of ω 2 that have a closed unbounded (club) subset in some ω 1 and ω 2 preserving outer model. Other examples concern large homogeneous sets for partitions and cofinal branches through trees, as well as the club susbet problem at successors of singular cardinals, cf. [S2] and [S3]. The same phenominon has been considered from a different perspective by S. Fried-man, cf. Theorem 7.28 in [F]. A consequence of the main theorem of this paper is that over a sufficiently non-minimal model of ZFC + " Ramsey cardinals are definably stationary, " all of these characterization problems are solvable.