As part of my Torino Lectures in Set Theory, Special Lecture 1 was devoted to ZFC vs HoTT – a possible crisis in foundations of mathematics?

Abstract: The lecture will address the oft-heard issue on “New Foundations” provided by HoTT, primarily for a group of students who are learning for the first time set theory. The students have just learned the intricate details of Gödel’s beautiful argument of consistency of AC + GCH with ZFC and are now entering the big step: learning forcing for the first time in their lives. It will be followed by conversation with special invited guests Mirna Džamonja, Matteo Viale and Fernando Zalamea.