Class forcing in its rightful setting

This is a talk at the Kurt Godel Research Seminar, University of Vienna, June 25, 2020 (virtual).

The use of class forcing in set theoretic constructions goes back to the proof Easton's Theorem that ${\rm GCH}$ can fail at all regular cardinals. Class forcing extensions are ubiquitous in modern set theory, particularly in the emerging field of set-theoretic geology. Yet, besides the pioneering work by Friedman and Stanley concerning pretame and tame class forcing, the general theory of class forcing has not really been developed until recently. A revival of interest in second-order set theory has set the stage for understanding the properties of class forcing in its natural setting. Class forcing makes a fundamental use of class objects, which in the first-order setting can only be studied in the meta-theory. Not surprisingly it has turned out that properties of class forcing notions are fundamentally determined by which other classes exist around them. In this talk, I will survey recent results (of myself, Antos, Friedman, Hamkins, Holy, Krapf, Schlicht, Williams and others) regarding the general theory of class forcing, the effects of the second-order set theoretic background on the behavior of class forcing notions and the numerous ways in which familiar properties of set forcing can fail for class forcing even in strong second-order set theories.