I shall give a tutorial lecture series on Boolean ultrapowers, two or three lectures, at the BLAST conference in Las Cruces, New Mexico, January 5-9, 2015. (The big AMS meeting in San Antonio, reportedly a quick flight, begins on the 10th.)
BLAST is a conference series focusing on
B = Boolean Algebras
L = Lattices, Algebraic and Quantum Logic
A = Universal Algebra
S = Set Theory
T = Set-theoretic and Point-free Topology
In this tutorial, I shall give a general introduction to the Boolean ultrapower construction.
Boolean ultrapowers generalize the classical ultrapower construction on a power-set algebra to the context of an ultrafilter on an arbitrary complete Boolean algebra. Introduced by Vopěnka as a means of undertaking forcing constructions internally to ZFC, the method has many connections with forcing. Nevertheless, the Boolean ultrapower construction stands on its own as a general model-theoretic construction technique, and historically, researchers have come to the Boolean ultrapower concept from both set theory and model theory. An emerging interest in Boolean ultrapowers arises from a focus on well-founded Boolean ultrapowers as large cardinal embeddings.
In this tutorial, we shall see that the Boolean ultrapower construction reveals that two central set-theoretic techniques–forcing and classical ultrapowers–are facets of a single underlying construction, namely, the Boolean ultrapower. I shall provide a thorough introduction to the Boolean ultrapower construction, assuming only an elementary graduate student-level familiarity with set theory and the classical ultrapower and forcing techniques.