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 $H_\kappa \prec_{\Sigma_2} H_{\kappa^+}$.