**April 2**

Monroe Eskew,
University of Vienna

**The approximation property and generic embeddings**

The approximation property was introduced by Hamkins for his Gap Forcing Theorem, and it has turned out to be a very useful notion, appearing for example in the partial equiconsistency result of Viale and Weiss on PFA, and in the proof of Woodin's HOD Dichotomy Theorem. In the context of generic embeddings, there can be a useful interplay between elementarity and approximation. We discuss some recent work in this direction: (1) tensions between saturated ideals on $\omega_2$ and the tree property (with Sean Cox), (2) fragility of the strong independence spectra (with Vera Fischer), and (3) mutual inconsistency of Foremanās minimal generic hugeness axioms.