  Assuming that there is no inner model with a strong cardinal, the following
is shown: any subset of \omega_1 can be made \Delta^1_3 (in the codes) by a
reasonable set-forcing; there is a reasonable set-generic extension with a
\Delta^1_3 well-ordering of its reals; 2-step \Sigma^1_4 absoluteness fails
w.r.t. set-sized reasonable forcings.
