Kurt Gödel Research Center

2013 seminar talk: Around characterizing aleph_1

Talk held by Martin Koerwien (KGRC) at the KGRC seminar on 2013-04-11.

Abstract

An $L_{\omega_1,\omega}$ sentence characterizes a cardinal $\kappa$ if it has a model of size $\kappa$ but no model in $\kappa^+$. We study the known examples of complete sentences that characterize $\aleph_1$ and observe several notable phenomena about them. Our goal is to understand the mechanisms that make a sentence characterize $\aleph_1$. This is related to some recent developments:

(1) Hjorth showed that if there is a counterexample to Vaught's conjecture, there is also one that characterizes $\aleph_1$. So it is tempting to try proving Vaught's conjecture by showing that every counterexample must have a model in $\aleph_2$ (which moreover a result of Harrington's suggests). This however turns out to be a red herring.

(2) While we know the notion of a complete sentence having a model in kappa is absolute for $\kappa=\aleph_1$ and non-absolute for $\kappa=\aleph_3$, even assuming GCH, this is still an open issue for $\kappa=\aleph_2$.