The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph _2$ or $\aleph _3$

Abstract:

We prove that the PFA lottery preparation of a strongly unfoldable cardinal $\kappa$ under $\neg 0^\sharp$ forces $\operatorname {PFA}(\aleph _2\text {-preserving})$, $\operatorname {PFA}(\aleph _3\text {-preserving})$ and $\operatorname {PFA}_{\aleph _2}$, with $2^\omega =\kappa =\aleph _2$. The method adapts to semi-proper forcing, giving $\operatorname {SPFA}(\aleph _2\text {-preserving})$, $\operatorname {SPFA}(\aleph _3\text {-preserving})$ and $\operatorname {SPFA}_{\aleph _2}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $\operatorname {SPFA}(\aleph _2\text {-preserving}) +\operatorname {SPFA}(\aleph _3\text {-preserving})+ \operatorname {SPFA}_{\aleph _2}+2^\omega =\aleph _2$. Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse $\aleph _3$ to $\aleph _1$.