Applications of Fodor's lemma to Vaught's conjecture

Abstract

A tree is said to have unique limits if no two distinct nodes on the same limit level have the same predecessors. If к is a regular cardinal, then a tree of height к is narrow if it is < к branching, has unique limits, and does not embed 2 < w . A set of nodes is stationary if the set of heights is stationary. We prove the following extension of Fodor's theorem: Theorem. If T is narrow and S is a stationary set of nodes, then S has a stationary subset along one branch. If M is a countable model, then say that M is isolated at α if the α-theory of M has only countably many countable models. For φ ϵ L ω1 ω let Mod φ be the collection of isomorphism types of countable models of φ. Theorem. If φ is a counterexample to Vaught's conjecture and for each M ϵ Mod φ , A M is a thin set of countable ordinals at which M is isolated, then ∪ Mϵ Modφ , A M is thin. We use this theorem to prove several new consequences of the existence of a counterexample to Vaught's conjecture, and to prove extensions of some of the known consequences. For example we show: Theorem. If φ is a counterexample, then there is some α < ω 1 such that every nontrivial complete α-theory extending φ is a counterexample. Theorem. If φ is a counterexample, then there is an M ϵ Mod φ such that for every N ϵ Mod φ if o( N) = o( M) , then the o( N )-theory of N is a counterexample.