Weak compactness without inaccessibility
The New York City College of Technology (CityTech), CUNY
Weakly compact cardinals are known to have a multitude of equivalent characterizations. The weakly compact cardinals were introduced by Hanf and Tarski in the 1960s as cardinals for which certain infinitary languages satisfied a form of the compactness theorem. Since then, they have been shown to be equivalently characterized as inaccessible cardinals that have, for example, the tree property, or the Keisler extension property, or the $\Pi^1_1$ indescribability property, or the weakly compact embedding property–meaning that they have elementary embeddings with critical point $\kappa$ defined on various transitive sets of size $\kappa$. In this talk we discuss what happens if one drops the requirement that $\kappa$ is inaccessible from weak compactness. We focus especially on the weakly compact embedding property and investigate its relation to the other properties mentioned above. This is joint work with Brent Cody, Sean Cox, and Joel Hamkins, and our results extend prior work of William Boos.
Thomas Johnstone is a professor of mathematics at the New York City College of Technology (City Tech), CUNY. His research in Set Theory includes large cardinals, forcing, and indestructibility.