Itay Neeman and Jindrich
Zapletal

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 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.