$\Sigma_n$-correct Forcing Axioms

Abstract

I introduce a new family of axioms extending ZFC set theory, the $\Sigma_n$-correct forcing axioms. These assert roughly that whenever a forcing name $\dot{a}$ can be forced by a poset in some forcing class $\Gamma$ to have some $\Sigma_n$ property $\phi$ which is provably preserved by all further forcing in $\Gamma$, then $\dot{a}$ reflects to some small name such that there is already in $V$ a filter which interprets that small name so that $\phi$ holds. $\Sigma_1$-correct forcing axioms turn out to be equivalent to classical forcing axioms, while $\Sigma_2$-correct forcing axioms for $\Sigma_2$-definable forcing classes are consistent relative to a supercompact cardinal (and in fact hold in the standard model of a classical forcing axiom constructed as an extension of a model with a supercompact), $\Sigma_3$-correct forcing axioms are consistent relative to an extendible cardinal, and more generally $\Sigma_n$-correct forcing axioms are consistent relative to a hierarchy of large cardinals generalizing supercompactness and extendibility whose supremum is the first-order version of Vopenka's Principle. By analogy to classical forcing axioms, there is also a hierarchy of $\Sigma_n$-correct bounded forcing axioms which are consistent relative to appropriate large cardinals. At the two lowest levels of this hierarchy, outright equiconsistency results are easy to obtain. Beyond these consistency results, I also study when $\Sigma_n$-correct forcing axioms are preserved by forcing, how they relate to previously studied axioms and to each other, and some of their mathematical implications.