Lightface ${\Sigma }_2^1$-indescribable cardinals

Abstract

${\Sigma }_3^1$-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 "${\Sigma }_3^1$-absoluteness for ccc forcing and ${\omega }_1$ is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface) ${\Sigma }_2^1$-indescribable cardinal; $\kappa$ has this property exactly if it is inaccessible and $H_{\kappa }{\prec }_{{\Sigma }_2}{H}_{{\kappa }^{+}}$.