2008 seminar talk: Square Principles and Model Theory
Talk held by Juliette Kennedy (University of Helsinki) at the KGRC seminar on 2008-05-15.
We visit some facts and questions about models of arithmetic inside reduced products, and the more general set-theoretic/model-theoretic results they have led to. A finitary square principle equivalent (in certain important cases) to the universality of an associated reduced product is introduced. This is also equivalent to an isomorphism theorem for ultrapowers of elementarily equivalent models. The principle follows from a cardinal arithmetic assumption, and can be shown to fail under the assumption of strongly compact cardinals. This gives the independence of the model-theoretic principles, under a large cardinal assumption. This is joint work with Saharon Shelah and Jouko Vaananen.