Robust and Stunted Trees

A tree of height and cardinality ω 1 is satiable if it has a new branch in some ω 1-preserving outer model. If V is a standard transitive model of ZFC, say that W is an outer model of V if W ⊇ V is also a standard transitive model of ZFC and V ∩ OR = W ∩ OR. In this section, " tree " will mean normal, everbranching tree of height ω 1 with unique limit nodes. Precise definitions of these terms and others appear in Section 2. There is no uniform characterization of satiable trees. By [S1] we know the following. (1) In the minimum model of ZFC, there is no set containing exactly the satiable trees. (2) If V is a sufficiently non-minimal standard transitive model of ZFC, the set of satiable trees does lie in V. " Sufficiently non-minimal " means that if κ = V ∩ OR and H is the least admissible set with V ∈ H, then κ is definably regular over H. (3) In any case, the set of satiable trees is not uniformly definable. That is, there is no parameter-free formula ϕ that defines the set of satiable trees in any inner or outer model with the same ω 1. Positive results Though these results imply that it is not possible to give a satisfactory characterization of satiable trees of height and cardinality ω 1 in general, (2) leaves open the possibility that there exist " rational " models in which each tree is either satiable or insatiable for a good reason. Indeed, examples of such a rational models are known. Two positive results that follow easily from known results are given in Section 3.