On $({\Sigma }_1^2{)}^{uB}$ Absoluteness Between V and HOD

Abstract

We put together Woodin's ${\Sigma }_1^2$ basis theorem of AD${}^{+}$ and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(Σ^2_1)^{\mbox{uB}}$ statement that is true in $V$ is true in $\mbox{HOD}$. Moreover, this is true even if we allow a parameter $C\subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $\mbox{OD}$ and universally Baire. We also investigate whether $(Σ^2_1)^{\mbox{uB}}$ statements are upwards absolute from $\mbox{HOD}$ to $V$ under large cardinal hypotheses, observing that this is true if $\mbox{HOD}$ has a proper class of Woodin cardinals. Finally, we discuss $(\forall^{\mathbb{R}})\, (Σ^2_1)^{\mbox{uB}}$ absoluteness and conclude that this much absoluteness between $\mbox{HOD}$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V=$ Ultimate $L$''.

31 pages