This talk will be a part of the “Advanced Introduction” series for graduate students at the the Kurt Gödel Research Center, November 4, 2011.

An old argument, heard perhaps at math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of *pointwise definable models of set theory*, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.