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

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- by Joel David Hamkins and Thomas A. Johnstone
- Proc. Amer. Math. Soc.
**137**(2009), 1823-1833 - DOI: https://doi.org/10.1090/S0002-9939-08-09727-X
- Published electronically: December 15, 2008
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## 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 $\aleph _1$.## References

- Mirna D amonja and Joel David Hamkins,
*Diamond (on the regulars) can fail at any strongly unfoldable cardinal*, Ann. Pure Appl. Logic**144**(2006), no. 1-3, 83–95. MR**2279655**, DOI 10.1016/j.apal.2006.05.001 - Ulrich Fuchs. Donder’s version of revised countable support. arXiv:math.LO/9207204.
- Martin Goldstern and Saharon Shelah,
*The bounded proper forcing axiom*, J. Symbolic Logic**60**(1995), no. 1, 58–73. MR**1324501**, DOI 10.2307/2275509 - Joel David Hamkins. A class of strong diamond principles. arXiv:math.LO/0211419.
- Joel David Hamkins,
*The lottery preparation*, Ann. Pure Appl. Logic**101**(2000), no. 2-3, 103–146. MR**1736060**, DOI 10.1016/S0168-0072(99)00010-X - Kai Hauser,
*Indescribable cardinals and elementary embeddings*, J. Symbolic Logic**56**(1991), no. 2, 439–457. MR**1133077**, DOI 10.2307/2274692 - Joel David Hamkins and Thomas A. Johnstone. Indestructible strong unfoldability, submitted.
- Thomas A. Johnstone. Strongly unfoldable cardinals made indestructible.
*Journal of Symbolic Logic*, 73(4):1215–1248, 2008. - Thomas A. Johnstone.
*Strongly unfoldable cardinals made indestructible*. Ph.D. thesis, The Graduate Center of the City University of New York, June 2007. - Richard Laver,
*Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing*, Israel J. Math.**29**(1978), no. 4, 385–388. MR**472529**, DOI 10.1007/BF02761175 - Telis K. Menas,
*On strong compactness and supercompactness*, Ann. Math. Logic**7**(1974/75), 327–359. MR**357121**, DOI 10.1016/0003-4843(75)90009-1 - Tadatoshi Miyamoto,
*A note on weak segments of PFA*, Proceedings of the Sixth Asian Logic Conference (Beijing, 1996) World Sci. Publ., River Edge, NJ, 1998, pp. 175–197. MR**1789737** - Itay Neeman,
*Hierarchies of forcing axioms. II*, J. Symbolic Logic**73**(2008), no. 2, 522–542. MR**2414463**, DOI 10.2178/jsl/1208359058 - Itay Neeman and Ernest Schimmerling,
*Hierarchies of forcing axioms. I*, J. Symbolic Logic**73**(2008), no. 1, 343–362. MR**2387946**, DOI 10.2178/jsl/1208358756 - Andrés Villaveces,
*Chains of end elementary extensions of models of set theory*, J. Symbolic Logic**63**(1998), no. 3, 1116–1136. MR**1649079**, DOI 10.2307/2586730

## Bibliographic Information

**Joel David Hamkins**- Affiliation: Department of Mathematics, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, New York 10016 — and — Department of Mathematics, The College of Staten Island of The City University of New York, Staten Island, New York 10314
- MR Author ID: 347679
- Email: jhamkins@gc.cuny.edu
**Thomas A. Johnstone**- Affiliation: Department of Mathematics, New York City College of Technology of The City University of New York, 300 Jay Street, Brooklyn, New York 11201
- Email: tjohnstone@citytech.cuny.edu
- Received by editor(s): November 20, 2007
- Received by editor(s) in revised form: August 13, 2008
- Published electronically: December 15, 2008
- Additional Notes: The research of the first author has been supported in part by grants from the CUNY Research Foundation and the Netherlands Organization for Scientific Research (NWO), and he is grateful to the Institute of Logic, Language and Computation at Universiteit van Amsterdam for the support of a Visiting Professorship during his 2007 sabbatical there.

Parts of this article are adapted from the second author’s Ph.D. dissertation, The Graduate Center of The City University of New York, June 2007 - Communicated by: Julia Knight
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 1823-1833 - MSC (2000): Primary 03E55, 03E40
- DOI: https://doi.org/10.1090/S0002-9939-08-09727-X
- MathSciNet review: 2470843