[bibtex key=HamkinsJohnstone2009:PFA(aleph_2-preserving)]

We prove that the PFA lottery preparation of a strongly unfoldable cardinal $\kappa$ under $\mathrm{\neg }{0}^{\mathrm{♯}}$ forces $\text{PFA}({\mathrm{\aleph }}_2\text{-preserving})$, $\text{PFA}({\mathrm{\aleph }}_3\text{-preserving})$ and ${\text{PFA}}_{{\mathrm{\aleph }}_2}$, with $2^{\omega }=\kappa ={\mathrm{\aleph }}_2$.  The method adapts to semi-proper forcing, giving $\text{SPFA}({\mathrm{\aleph }}_2\text{-preserving})$, $\text{SPFA}({\mathrm{\aleph }}_3\text{-preserving})$ and ${\text{SPFA}}_{{\mathrm{\aleph }}_2}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $\text{SPFA}({\mathrm{\aleph }}_2\text{-preserving})+\text{SPFA}({\mathrm{\aleph }}_3\text{-preserving})+{\text{SPFA}}_{{\mathrm{\aleph }}_2}+2^{\omega }={\mathrm{\aleph }}_2$.  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 ${\mathrm{\aleph }}_3$ to ${\mathrm{\aleph }}_1$.