- J. D. Hamkins and T. A. Johnstone, “The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$,” Proc.~Amer.~Math.~Soc., vol. 137, iss. 5, p. 1823–1833, 2009.

[Bibtex]`@ARTICLE{HamkinsJohnstone2009:PFA(aleph_2-preserving), AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.}, TITLE = {The proper and semi-proper forcing axioms for forcing notions that preserve {$\aleph_2$} or {$\aleph_3$}}, JOURNAL = {Proc.~Amer.~Math.~Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {137}, YEAR = {2009}, NUMBER = {5}, PAGES = {1823--1833}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {03E55 (03E40)}, MRNUMBER = {2470843 (2009k:03087)}, MRREVIEWER = {John Krueger}, DOI = {10.1090/S0002-9939-08-09727-X}, URL = {http://wp.me/p5M0LV-3v}, file = F, }`

We prove that the PFA lottery preparation of a strongly unfoldable cardinal $\kappa$ under $\neg 0^\sharp$ forces $\text{PFA}(\aleph_2\text{-preserving})$, $\text{PFA}(\aleph_3\text{-preserving})$ and $\text{PFA}_{\aleph_2}$, with $2^\omega=\kappa=\aleph_2$. The method adapts to semi-proper forcing, giving $\text{SPFA}(\aleph_2\text{-preserving})$, $\text{SPFA}(\aleph_3\text{-preserving})$ and $\text{SPFA}_{\aleph_2}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $\text{SPFA}(\aleph_2\text{-preserving})+\text{SPFA}(\aleph_3\text{-preserving})+\text{SPFA}_{\aleph_2}+2^\omega=\aleph_2$. 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 $\aleph_1$.