$Σ_n$-correct Forcing Axioms
Ben Goodman · arXiv (Cornell University) · 2024
I introduce a new family of axioms extending ZFC set theory, the $Σ_n$-correct forcing axioms. These assert roughly that whenever a forcing name $\dot{a}$ can be forced by a poset in some forcing class $Γ$ to have some $Σ_n$ property $ϕ$ which is provably preserved by all further forcing in $Γ$, then $\dot{a}$ reflects to some small name such that there is already in $V$ a filter which interprets that small name so that $ϕ$ holds. $Σ_1$-correct forcing axioms turn out to be equivalent to classical forcing axioms, while $Σ_2$-correct forcing axioms for $Σ_2$-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), $Σ_3$-correct forcing axioms are consistent relative to an extendible cardinal, and more generally $Σ_n$-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 $Σ_n$-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 $Σ_n$-correct forcing axioms are preserved by forcing, how they relate to previously studied axioms and to each other, and some of their mathematical implications.