In this talk we'll discuss some basic ideas behind ZF set theory, and considers some of the philosophical grounds one might have for adopting new axioms to strengthen it. In connection with this, we'll consider Leibniz's place in the history of philosophy of mathematics, and especially the philosophy of set theory, where descendents of his views contributed both to the development of naïve set theory and to the development of certain principles upon which one might want to justify adding new axioms to set theory.
The talk should be readily accessible to everyone.