The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$

  • 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, pp. 1823-1833, 2009.  
    @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://dx.doi.org/10.1090/S0002-9939-08-09727-X},
    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 $\aleph_1$.

Leave a Reply