User blog comment:Wythagoras/All my stuff/@comment-10429372-20130715145943/@comment-5529393-20130715215442

Hmm, it looks like one of the slides says that a model is a pointwise definable model of ZFC if and only if it is a prime model of ZFC + V = HOD. So I guess any pointwise definable model of ZFC will satisfy V = HOD.

Different models have different ideas of what omega_1 is, since they contain different functions and hence different bijections. (Similarly for the continuum, for example)  omega_1^L is what L thinks omega_1 is.