The Vaught Conjecture: Do Uncountable Models Count?

Abstract

We give a model theoretic proof, replacing admissible set theory by the LopezEscobar theorem, of Makkai’s theorem: Every counterexample to Vaught’s conjecture has an uncountable model which realizes only countably many Lω1,ω-types. The following result is new. Theorem. If a first order theory is a counterexample to the Vaught conjecture then it has 2א1 models of cardinality א1. In this paper we prove several properties of putative counterexamples to the Vaught conjecture. Specifically, these results concern the number of models the counterexample has in power א1. One of these results was proved 30 years ago using admissible model theory; we give a more straightforward argument. The following question guides our discussion. Is the Vaught Conjecture model theory? Here are some possible ways in which this question would have a clear answer. Shelah, Buechler, Newelski have shown using rather difficult techniques from stability theory that the conjecture holds for first order theories that are ‘simple’ from the stability theoretic standpoint: ω-stable or superstable with finite U -rank. If a counterexample were found for a first order theory of slightly greater complexity (e.g. a stable but not superstable first order theory), this would indicate the issue was a model theoretic one. If on the other hand, a uniform proof for sentences of Lω1,ω were given using methods of descriptive set theory, then it would not be a model theoretic problem. The results below give partial answers to the following methodological questions. What specific model theoretic as opposed to descriptive set theoretic techniques can attack the problem? Can one use more direct model theoretic arguments to obtain some result of admissible model theory? I would argue the problem is model theoretic if its solution is different for Lω,ω and Lω1,ω. So we will investigate the differences between properties known about counterexamples to