Published online by Cambridge University Press: 12 March 2014
We study the following conjecture.
Conjecture.Let T be an ω-stable theory with continuum many countable models. Then either i) T has continuum many complete extensions in L1(T), or ii) some complete extension of T in L1has continuum many L1-types without parameters.
By Shelah's proof of Vaught's conjecture for ω-stable theories, we know that there are seven types of ω-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many L2-types.