Lightface $\Sigma ^1_2$-indescribable cardinals

Abstract:

$\Sigma ^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{\omega _1}}^V \prec _{\Sigma _2}{H_{\omega _1}}^{V^P}$. “$\omega _1$ inaccessible to reals” means that for any real $r$, ${\omega _1}^{L[r]}<\omega _1$. To measure the exact consistency strength of “$\Sigma ^1_3$-absoluteness for ccc forcing and $\omega _1$ is inaccessible to reals”, we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $\Sigma ^1_2$-indescribable cardinal; $\kappa$ has this property exactly if it is inaccessible and $H_\kappa \prec _{\Sigma _2} H_{\kappa ^+}$.