Proper forcing and <i>L</i>(ℝ)

Abstract

Abstract We present two ways in which the model L (ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L (ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (ℝ) : roughly ω Woodin cardinals.