Paris és Enayat is foglalkozott ezzel. A Cohen-Shepherdson-modell is pointwise definable.

http://front.math.ucdavis.edu/1105.4597

Title: Pointwise Definable Models of Set Theory
Authors: Joel David Hamkins, David Linetsky, Jonas Reitz
Categories: math.LO Logic
Comments: 23 pages
MSC: 03E55

Abstract: A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.
Owner: David Linetsky
Version 1: Mon, 23 May 2011 19:53:44 GMT
Version 2: Tue, 19 Jun 2012 03:45:04 GMT