Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

Building on [4], [8] and [9] we study which cardinals are characterizable by a Scott sentence, in the sense that φM characterizes κ, if it has a model of size κ, but not of κ. We show that if אα is characterizable by a Scott sentence and β < ω1, then אα+β is characterizable by a Scott sentence. If 0 < γ < ω1, then the same is true for 2אα+γ . Also, אא0 α is characterizable by a Scott sentence. Characterizable cardinals are closed under countable unions and products. As a corollary we get that for countable α, β, אβ α is characterizable by Scott sentence. Following work of Baumgartner and Malitz it is natural to consider when a cardinal can be homogeneously characterized by a Scott sentence. Many of the issues remain profoundly unclear, however, we do prove that if κ is characterizable, then κ and 2 +) are homogeneously characterizable. Acknowledgment: I would like to thank the Department of Mathematics and Statistics of the University of Melbourne, Australia, for their kind hospitality for the whole academic year 2006-2007. This paper was written while visiting professor Greg Hjorth, my thesis advisor and now a professor at the above university.