Proper forcing and $L(\mathbb{R})$

Abstract

We present two ways in which the model $L(\mathbb{R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a set of ordinals in $V$ cannot be added to $L(\mathbb{R})$ by small forcing. The large cardinal needed corresponds to the consistency strength of $A{D}^{L(\mathbb{R})}$; roughly $\omega$ Woodin cardinals.

14 pages, includes Appendix (pp. 10--13)