This will be a talk for the CUNY Set Theory seminar on September 1, 2017, 10 am. GC 6417.

**Abstract.** The *inner model reflection principle *asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the *ground-model* reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. Both of these principles, expressing a form of width-reflection in constrast to the usual height-reflection, are equiconsistent with ZFC and an outright consequence of the existence of sufficient large cardinals, as well as a consequence (in lightface form) of the maximality principle and also of the inner-model hypothesis. This is joint work with Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, myself and Jonas Reitz.