Lightface $\Sigma^1_2$-indescribable cardinals

TL;DR

This paper introduces a new large cardinal concept, the lightface _2-indescribable cardinal, to analyze the exact consistency strength of _3-absoluteness for ccc forcing combined with  being inaccessible to reals.

Contribution

It defines the lightface _2-indescribable cardinal and explores its role in measuring the consistency strength of certain set-theoretic principles.

Findings

_2-indescribable cardinals are characterized by -inaccessibility and _2-elementarity.

The introduced cardinal concept provides a precise measure for the consistency strength of _3-absoluteness and -inaccessibility to reals.

The work connects large cardinal properties with forcing absoluteness principles.

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