Resurrection axioms and uplifting cardinals

  • J. D. Hamkins and T. Johnstone, “Resurrection axioms and uplifting cardinals,” Archive for Mathematical Logic, vol. 53, iss. 3-4, p. p.~463–485, 2014.  
    @ARTICLE{HamkinsJohnstone2014:ResurrectionAxiomsAndUpliftingCardinals,
    AUTHOR = "Joel David Hamkins and Thomas Johnstone",
    TITLE = "Resurrection axioms and uplifting cardinals",
    JOURNAL = "Archive for Mathematical Logic",
    publisher= {Springer Berlin Heidelberg},
    YEAR = "2014",
    volume = "53",
    number = "3-4",
    pages = "p.~463--485",
    month = "",
    note = "",
    url = "http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals",
    eprint = "1307.3602",
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    doi= "10.1007/s00153-014-0374-y",
    issn= {0933-5846},
    abstract = "",
    keywords = "",
    source = "",
    file = F }

Abstract. We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of uplifting cardinal.

Many classical forcing axioms can be viewed, at least informally, as the claim that the universe is existentially closed in its forcing extensions, for the axioms generally assert that certain kinds of filters, which could exist in a forcing extension $V[G]$, exist already in $V$. In several instances this informal perspective is realized more formally: Martin’s axiom is equivalent to the assertion that $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions of the universe, meaning that $H_{\frak{c}}\prec_{\Sigma_1}V[G]$ for all such extensions; the bounded proper forcing axiom is equivalent to the assertion that $H_{\omega_2}$ is existentially closed in all proper forcing extensions, or $H_{\omega_2}\prec_{\Sigma_1}V[G]$; and there are other similar instances.

In model theory, a submodel $M\subset N$ is existentially closed in $N$ if existential assertions true in $N$ about parameters in $M$ are true already in $M$, that is, if $M$ is a $\Sigma_1$-elementary substructure of $N$, which we write as $M\prec_{\Sigma_1} N$. Furthermore, in a general model-theoretic setting, existential closure is tightly connected with resurrection, the theme of this article.

Elementary Fact. If $\mathcal{M}$ is a submodel of $\mathcal{N}$, then the following are equivalent.

  1. The model $\mathcal{M}$ is existentially closed in $\mathcal{N}$.
  2. $\mathcal{M}\subset \mathcal{N}$ has resurrection. That is, there is a further extension $\mathcal{M}\subset\mathcal{N}\subset\mathcal{M}^+$ for which $\mathcal{M}\prec\mathcal{M}^+$.

We call this resurrection because although certain truths in $\mathcal{M}$ may no longer hold in the extension $\mathcal{N}$, these truths are nevertheless revived in light of $\mathcal{M}\prec\mathcal{M}^+$ in the further extension to $\mathcal{M}^+$.

In the context of forcing axioms, we are more interested in the case of forcing extensions than in the kind of arbitrary extension $\mathcal{M}^+$ arising in the fact, and in this context the equivalence of (1) and (2) breaks own, although the converse implication $(2)\to(1)$ always holds, and every instance of resurrection implies the corresponding instance of existential closure. This key observation leads us to the main unifying theme of this article, the idea that

resurrection may allow us to formulate more robust forcing axioms 

than existential closure or than combinatorial assertions about filters and dense sets. We therefore introduce in this paper a spectrum of new forcing axioms utilizing the resurrection concept.

Main Definition. Let $\Gamma$ be a fixed definable class of forcing notions.

  1. The resurrection axiom $\text{RA}(\Gamma)$ is the assertion that for every forcing notion $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, with $\vdash_{\mathbb{Q}}\mathbb{R}\in\Gamma$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.
  2. The weak resurrection axiom $\text{wRA}(\Gamma)$ is the assertion that for every $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.

The main result is to prove that various formulations of the resurrection axioms are equiconsistent with the existence of an uplifting cardinal, where an inaccessible cardinal $\kappa$ is uplifting, if there are arbitrarily large inaccessible cardinals $\gamma$ for which $H_\kappa\prec H_\gamma$.  This is a rather weak large cardinal notion, having consistency strength strictly less than the existence of a Mahlo cardinal, which is traditionally considered to be very low in the large cardinal hierarchy.  One highlight of the article is our development of “the world’s smallest Laver function,” the Laver function concept for uplifting cardinals, and we perform an analogue of the Laver preparation in order to achieve the resurrection axiom for c.c.c. forcing.

Main Theorem. The following theories are equiconsistent over ZFC:

  1. There is an uplifting cardinal.
  2. $\text{RA}(\text{all})$.
  3. $\text{RA}(\text{ccc})$.
  4. $\text{RA}(\text{semiproper})+\neg\text{CH}$.
  5. $\text{RA}(\text{proper})+\neg\text{CH}$.
  6. For some countable ordinal $\alpha$, the axiom $\text{RA}(\alpha\text{-proper})+\neg\text{CH}$.
  7. $\text{RA}(\text{axiom-A})+\neg\text{CH}$.
  8. $\text{wRA}(\text{semiproper})+\neg\text{CH}$.
  9. $\text{wRA}(\text{proper})+\neg\text{CH}$.
  10. For some countable ordinal $\alpha$, the axiom $\text{wRA}(\alpha\text{-proper})+\neg\text{CH}$.
  11. $\text{wRA}(\text{axiom-A})+\neg\text{CH}$.
  12. $\text{wRA}(\text{countably closed})+\neg\text{CH}$.

The proof outline proceeds in two directions: on the one hand, the resurrection axioms generally imply that the continuum $\frak{c}$ is uplifting in $L$; and conversely, given any uplifting cardinal $\kappa$, we may perform a suitable lottery iteration of $\Gamma$ forcing to obtain the resurrection axiom for $\Gamma$ in a forcing extension with $\kappa=\frak{c}$.

In a follow-up article, currently nearing completion, we treat the boldface resurrection axioms, which allow a predicate $A\subset\frak{c}$ and ask for extensions of the form $\langle H_{\frak{c}},{\in},A\rangle\prec\langle H_{\frak{c}}^{V[g\ast h]},{\in},A^\ast\rangle$, for some $A^\ast\subset\frak{c}^{V[g\ast h]}$ in the extension.  In that article, we prove the equiconsistency of various formulations of boldface resurrection with the existence of a strongly uplifting cardinal, which we prove is the same as a superstrongly unfoldable cardinal.

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact

  • B. Cody, M. Gitik, J. D. Hamkins, and J. A. Schanker, “The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact,” Archive for Mathematical Logic, pp. 1-20, 2015.  
    @article{CodyGitikHamkinsSchanker2015:LeastWeaklyCompact, year= {2015}, issn=
    {0933-5846}, journal= {Archive for Mathematical Logic}, doi=
    {10.1007/s00153-015-0423-1}, title= {The least weakly compact cardinal can be
    unfoldable, weakly measurable and nearly {$\theta$}-supercompact}, publisher=
    {Springer Berlin Heidelberg}, keywords= {Weakly compact; Unfoldable; Weakly
    measurable; Nearly supercompact; Identity crisis; Primary 03E55; 03E35},
    author= {Cody, Brent and Gitik, Moti and Hamkins, Joel David and Schanker,
    Jason A.}, pages= {1--20}, language= {English}, eprint = {1305.5961},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url= {http://jdh.hamkins.org/least-weakly-compact}, }

Abstract.   We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, for nearly any desired function $\kappa\mapsto\theta_\kappa$. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.

In this article, we prove that the least weakly compact cardinal can exhibit any of several much stronger large cardinal properties. Namely, the least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact for any desired $\theta$.

Main Theorem.  Assuming a suitable large cardinal hypothesis, the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$.

Meanwhile, the least weakly compact cardinal can never exhibit these extra large cardinal properties in $L$, and indeed, the existence of a weakly measurable cardinal in the constructible universe is impossible. Furthermore, in each case the extra properties are strictly stronger than weak compactness in consistency strength.

We show in addition a more global result, that the entire class of weakly compact cardinals can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals, and with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, with enormous flexibility in the map $\kappa\mapsto\theta_\kappa$.

Our results therefore extend the `identity-crises’ phenomenon—first identified (and named) by Magidor—which occurs when a given large cardinal property can be made in various models to coincide either with much stronger or with much weaker large cardinal notions. Magidor had proved that the least strongly compact cardinal can be the least supercompact cardinal in one model of set theory and the least measurable cardinal in another. Here, we extend the phenomenon to weak measurability, partial near supercompactness and unfoldability. Specifically, the least weakly measurable cardinal coincides with the least measurable cardinal under the GCH, but it is the least weakly compact cardinal in our main theorem. Similarly, the least cardinal $\kappa$ that is nearly $\kappa^{+}$-supercompact is measurable with nontrivial Mitchell order under the GCH, but it is the least weakly compact cardinal here (and similar remarks apply to near $\kappa^{++}$-supercompactness and so on). The least unfoldable cardinal is strongly unfoldable in $L$, and therefore a $\Sigma_2$-reflecting limit of weakly compact cardinals there, but it is the least weakly compact cardinal in our main theorem. The global results of section 6 show just how malleable these notions are.

Indestructible strong unfoldability

  • J. D. Hamkins and T. A. Johnstone, “Indestructible strong unfoldability,” Notre Dame J.~Form.~Log., vol. 51, iss. 3, pp. 291-321, 2010.  
    @ARTICLE{HamkinsJohnstone2010:IndestructibleStrongUnfoldability,
    AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},
    TITLE = {Indestructible strong unfoldability},
    JOURNAL = {Notre Dame J.~Form.~Log.},
    FJOURNAL = {Notre Dame Journal of Formal Logic},
    VOLUME = {51},
    YEAR = {2010},
    NUMBER = {3},
    PAGES = {291--321},
    ISSN = {0029-4527},
    MRCLASS = {03E55 (03E40)},
    MRNUMBER = {2675684 (2011i:03050)},
    MRREVIEWER = {Bernhard A.~K{\"o}nig},
    DOI = {10.1215/00294527-2010-018},
    URL = {http://dx.doi.org/10.1215/00294527-2010-018},
    file = F
    }

Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed + $\kappa^+$-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of ${\lt}\kappa$-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by ${\lt}\kappa$-directed closed forcing. Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability.

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 $\aleph_1$.

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

  • M. D{u{z}}amonja and J. D. Hamkins, “Diamond (on the regulars) can fail at any strongly unfoldable cardinal,” Ann.~Pure Appl.~Logic, vol. 144, iss. 1-3, pp. 83-95, 2006. (Conference in honor of sixtieth birthday of James E.~Baumgartner)  
    @ARTICLE{DzamonjaHamkins2006:DiamondCanFail,
    AUTHOR = {D{\u{z}}amonja, Mirna and Hamkins, Joel David},
    TITLE = {Diamond (on the regulars) can fail at any strongly unfoldable cardinal},
    JOURNAL = {Ann.~Pure Appl.~Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {144},
    YEAR = {2006},
    NUMBER = {1-3},
    PAGES = {83--95},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {03E05 (03E35 03E55)},
    MRNUMBER = {2279655 (2007m:03091)},
    MRREVIEWER = {Andrzej Ros{\l}anowski},
    DOI = {10.1016/j.apal.2006.05.001},
    URL = {http://dx.doi.org/10.1016/j.apal.2006.05.001},
    month = {December},
    note = {Conference in honor of sixtieth birthday of James E.~Baumgartner},
    eprint = {math/0409304},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

If $\kappa$ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which $\Diamond_\kappa(\text{REG})$ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

Unfoldable cardinals and the GCH

  • J. D. Hamkins, “Unfoldable cardinals and the GCH,” J.~Symbolic Logic, vol. 66, iss. 3, pp. 1186-1198, 2001.  
    @article{Hamkins2001:UnfoldableCardinals,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Unfoldable cardinals and the {GCH}},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {66},
    YEAR = {2001},
    NUMBER = {3},
    PAGES = {1186--1198},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E55 (03E35 03E40)},
    MRNUMBER = {1856735 (2002i:03059)},
    MRREVIEWER = {Eva Coplakova},
    DOI = {10.2307/2695100},
    URL = {http://dx.doi.org/10.2307/2695100},
    eprint = {math/9909029},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically, $\kappa$ is $\theta$-unfoldable when for any transitive structure $M$ of size $\kappa$ that contains $\kappa$ as an element, there is an elementary embedding $j:M\to N$ with critical point $\kappa$ for which $j(\kappa)$ is at least $\theta$. Define that $\kappa$ is fully unfoldable, then, when it is $\theta$-unfoldable for every $\theta$. In this paper I show that the embeddings associated with these unfoldable cardinals are amenable to some of the same lifting techniques that apply to weakly compact embeddings, augmented with methods from the strong cardinal context. Using these techniques, I show by set-forcing over any model of ZFC that any given unfoldable cardinal $\kappa$ can be made indestructible by the forcing to add any number of Cohen subsets to $\kappa$. This result contradicts expectations to the contrary that class forcing would be required.