Three red herrings around Vaught’s conjecture

Abstract

We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a counterexample such that every model of cardinality א1 is maximal (strengthening a result of Hjorth’s). We also give a new proof of Harrington’s theorem that any counterexample to Vaught’s conjecture has models in א1 of arbitrarily high Scott rank below א2. The three red herrings1 are false leads towards solving Vaught’s conjecture. Here is one strategy for establishing Vaught’s conjecture that there is no sentence of Lω1,ω that has exactly א1 countable models. Hjorth [9, 8], using descriptive set theoretic results of Mackey and others, has established that if there is a counterexample then there is one that has no model in א2. On the other hand, unpublished results of Harrington show that every counterexample has models with arbitrarily large Scott ranks below א2. This supports the notion that one might construct a model of an arbitrary counterexample that has cardinality א2. The resulting contradiction would yield the conjecture. In the introduction we give some background on the conjecture and describe the three red herrings. Recall that Vaught’s conjecture [19] concerns the number of countable models of a countable first-order theory, or more generally, of a sentence in the infinitary logic Lω1ω , where countable conjunctions and disjunctions but only finite strings of quantifiers are allowed for some countable vocabulary τ . The conjecture states: Vaught Conjecture. If φ is a sentence of Lω1ω then φ either has countably many or continuum-many countable models up to isomorphism. The more “absolute” version replaces “continuum-many” by “a perfect set of ” in the conclusion, where a perfect set of countable models is a perfect set of reals, each of which codes a countable model, such that distinct reals in the perfect set code non-isomorphic models. We say a sentence φ of Lω1ω is scattered if it does not have a perfect set of countable models. Morley [20] defined scattered as: for every countable ∗Research partially supported by Simons travel grant G5402 †Research supported by FWF (Austrian Science Fund) Grant P24654-N25. ‡Research supported by the Austrian Science Fund (FWF) Lise Meitner Grant M1410-N25 §Partially supported by NSF grant DMS-1308546 1See http://en.wikipedia.org/wiki/Red_herring for a fairly comprehensive account of the meaning of the red herring idiom.