This is a talk at the CUNY Set Theory Seminar on 11 Sept 2015.

The Barwise compactness theorem is a powerful tool, allowing one to prove many interesting results which cannot be gotten just from the ordinary compactness theorem. For example, it can be used to show that every countable transitive model of set theory has an end extension which is a model of $V = L$.

However, the Barwise compactness theorem only applies to transitive sets. What are we to do if we want to have compactness arguments for ill-founded models of set theory?

This is where Barwise’s notion of the admissible cover of a (possibly ill-founded) model of set theory comes in. In this talk, we will see how to construct admissible covers and how they can be used to extend compactness arguments to the ill-founded case.

This talk is a prequel of sorts to my next talk in this seminar. Some of the results discussed in this talk will play a crucial role in the arguments there.