This will be a talk (in person) for the Logic Seminar of the Mathematics Institute of the Univerisity of Oxford, May 18, 2023 5pm, Wiles Building L3.

**Abstract:** I shall present a new flexible method showing that every countable model of PA admits a pointwise definable end-extension, one in which every point is definable without parameters. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.