Member Seminar

Speaker: Alejandro Poveda

Title: Compactness and Anticompactness Principles in Set Theory

Abstract: Several fundamental properties in Topology, Algebra or Logic are expressed in terms of Compactness Principles.For instance, a natural algebraic question is the following: Suppose that G is an Abelian group whose all small subgroups are free – Is the group G free? If the answer is affirmative one says that compactness holds; otherwise, we say that compactness fails. Loosely speaking, a compactness principle is anything that fits the following slogan: Suppose that M is a mathematical structure (a group, a topological space, etc) such that all of its small substructures N have certain property $\varphi$; then the ambient structure M has property $\varphi$, as well. Oftentimes when these questions are posed for infinite sets the problem becomes purely set-theoretical and axiom-sensitive. In this talk I will survey the most paradigmatic instances of compactness and present some related results of mine. If time permits, I will hint the proof of a recent result (joint with Rinot and Sinapova) showing that stationary reflection and the failure of the Singular Cardinal Hypothesis can co-exist. These are instances of two antagonist set-theoretic principles: the first is a compactness principle while the second is an anti-compactness one. This result solves a question by M. Magidor from 1982.