Kurt Gödel Research Center

2013 seminar talk: Vaught's Conjecture, the Generic Morley Tree and Fragment Embeddings

Talk held by Sy-David Friedman (KGRC) at the KGRC seminar on 2013-03-07.

Abstract

If $\varphi$ is a scattered sentence of $L_{\omega_1\omega}$ (i.e., one with at least one model but no perfect set of countable models) then associated to $\varphi$ is its Morley tree. Each node of this tree is a countable theory which is atomic for a countable fragment of $L_{\omega_1\omega}$ containing $\varphi$. The Morley tree has height at most $\omega_1$ and Vaught's conjecture asserts that its height is in fact less than $\omega_1$. Using a generic version of the Morley tree together with a notion of fragment embedding, I'll prove a theorem of Harrington which states that if $\varphi$ is a counterexample to Vaught's conjecture then there are models of $\varphi$ with Scott rank arbitrarily large below $\omega_2$.