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


Itay Neeman and Jindrich Zapletal





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 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 cardinals needed correspond to

the consistency strength of ${AD}^{L({\mathbb R})}$; roughly

$\omega$ Woodin cardinals.