Set theory without powerset

This is a talk at the Models and Sets Seminar, University of Leeds, November 10, 2021 (virtual).

Many natural set-theoretic structures satisfy the basic axioms of set theory, but not the powerset axiom. These include the collections $H_{\kappa^+}$ of sets whose transitive closure has size at most $\kappa$, forcing extensions of models of ${\rm ZFC}$ by pretame (but not tame) forcing, and first-order models that are morally equivalent to models of the second-order Kelley-Morse set theory (with class choice). It turns out that a reasonable set theory in the absence of the powerset axiom is not simply ${\rm ZFC}$ with the powerset axiom removed. Without the powerset axiom, the Replacement scheme is not equivalent to the Collection scheme, and the various forms of the Axiom of Choice are not equivalent. In this talk, I will give an overview of the properties of a robust set theory without powerset, ${\rm ZFC}^-$, whose axioms are ${\rm ZFC}$ without the powerset axiom, with the Collection scheme instead of the Replacement scheme and the Well-Ordering Principle instead of the Axiom of Choice. While a great deal of standard set theory can be carried out in ${\rm ZFC}^-$, for instance, forcing works mostly as it does in ${\rm ZFC},$ there are several important properties that are known to fail and some which we still don't know whether they hold. For example, the Intermediate Model Theorem fails for ${\rm ZFC}^-$, and so does ground model definability, and it is not known whether ${\rm HOD}$ is definable. I will also discuss a strengthening of ${\rm ZFC}^-$ obtained by adding the Dependent Choice Scheme, and some rather strange ${\rm ZFC}^-$-models.