Seminars & Colloquia Calendar

Abstract: Two central themes in logic are how much the universe of sets resembles Gödel's constructible universe \(L\) versus what is possible from forcing and large cardinals. Both  are addressed by using infinite combinatorics to investigate how much  compactness can be obtained in the universe. Compactness is the phenomenon when a given property holding for every smaller substructure of some object implies that property holds for the object itself. This is usually a consequence of large cardinals, and tends to fail in \(L\).

A key instance of compactness is the tree property, which states that every tree of height \(\kappa\) and levels of size less than \(\kappa\) has a cofinal branch. Informally, this principle is a generalization of König's infinity lemma to uncountable cardinals.  It turns out that the tree property and certain strengthenings capture the combinatorial essence of large cardinals. An old project in set theory is to force the tree property (and some strengthenings) at every regular cardinal greater than \(\aleph_1\). I will go over the background and then discuss some recent results giving the state of the art of this project.