The weakly compact embedding property, Apter-Gitik celebration, CMU 2015

This will be a talk at the Conference in honor of Arthur W. Apter and Moti Gitik at Carnegie Mellon University, May 30-31, 2015.  I am pleased to be a part of this conference in honor of the 60th birthdays of two mathematicians whom I admire very much.

Moti GitikArthur W. Apter

 

 

 

 

 

 

 

 

Abstract. The weakly compact embedding property for a cardinal $\kappa$ is the assertion that for every transitive set $M$ of size $\kappa$ with $\kappa\in M$, there is a transitive set $N$ and an elementary embedding $j:M\to N$ with critical point $\kappa$. When $\kappa$ is inaccessible, this property is one of many equivalent characterizations of $\kappa$ being weakly compact, along with the weakly compact extension property, the tree property, the weakly compact filter property and many others. When $\kappa$ is not inaccessible, however, these various properties are no longer equivalent to each other, and it is interesting to sort out the relations between them. In particular, I shall consider the embedding property and these other properties in the case when $\kappa$ is not necessarily inaccessible, including interesting instances of the embedding property at cardinals below the continuum, with relations to cardinal characteristics of the continuum.

This is joint work with Brent Cody, Sean Cox, myself and Thomas Johnstone.

Slides | Article | Conference web site

Carnegie Mellon University, College of Fine Arts

Singular cardinals and strong extenders

  • A. W. Apter, J. Cummings, and J. D. Hamkins, “Singular cardinals and strong extenders,” Cent. Eur. J. Math., vol. 11, iss. 9, pp. 1628-1634, 2013.  
    @article {ApterCummingsHamkins2013:SingularCardinalsAndStrongExtenders,
    AUTHOR = {Apter, Arthur W. and Cummings, James and Hamkins, Joel David},
    TITLE = {Singular cardinals and strong extenders},
    JOURNAL = {Cent. Eur. J. Math.},
    FJOURNAL = {Central European Journal of Mathematics},
    VOLUME = {11},
    YEAR = {2013},
    NUMBER = {9},
    PAGES = {1628--1634},
    ISSN = {1895-1074},
    MRCLASS = {03E55 (03E35 03E45)},
    MRNUMBER = {3071929},
    MRREVIEWER = {Samuel Gomes da Silva},
    DOI = {10.2478/s11533-013-0265-1},
    URL = {http://jdh.hamkins.org/singular-cardinals-strong-extenders/},
    eprint = {1206.3703},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Brent Cody asked the question whether the situation can arise that one has an elementary embedding $j:V\to M$ witnessing the $\theta$-strongness of a cardinal $\kappa$, but where $\theta$ is regular in $M$ and singular in $V$.

In this article, we investigate the various circumstances in which this does and does not happen, the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing “$\kappa$ is $\mu$-strong”, such that $\mu$ is singular in $Ult(V, E)$.

Inner models with large cardinal features usually obtained by forcing

  • A. W.~Apter, V. Gitman, and J. D. Hamkins, “Inner models with large cardinal features usually obtained by forcing,” Archive for Mathematical Logic, vol. 51, pp. 257-283, 2012.  
    @article {ApterGitmanHamkins2012:InnerModelsWithLargeCardinals,
    author = {Arthur W.~Apter and Victoria Gitman and Joel David Hamkins},
    affiliation = {Mathematics, The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY 10016, USA},
    title = {Inner models with large cardinal features usually obtained by forcing},
    journal = {Archive for Mathematical Logic},
    publisher = {Springer Berlin / Heidelberg},
    issn = {0933-5846},
    keyword = {},
    pages = {257--283},
    volume = {51},
    issue = {3},
    url = {http://jdh.hamkins.org/innermodels},
    eprint = {1111.0856},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    doi = {10.1007/s00153-011-0264-5},
    note = {},
    year = {2012},
    }

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal $\kappa$ for which $2^\kappa=\kappa^+$, another for which $2^\kappa=\kappa^{++}$ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model $W$ with a strongly compact cardinal $\kappa$, such that $H_{\kappa^+}^V\subseteq HOD^W$. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit $\delta$ of ${\lt}\delta$-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Large cardinals with few measures

  • A. W.~Apter, J. Cummings, and J. D. Hamkins, “Large cardinals with few measures,” Proc.~Amer.~Math.~Soc., vol. 135, iss. 7, pp. 2291-2300, 2007.  
    @ARTICLE{ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures,
    AUTHOR = {Arthur W.~Apter and James Cummings and Joel David Hamkins},
    TITLE = {Large cardinals with few measures},
    JOURNAL = {Proc.~Amer.~Math.~Soc.},
    FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {135},
    YEAR = {2007},
    NUMBER = {7},
    PAGES = {2291--2300},
    ISSN = {0002-9939},
    CODEN = {PAMYAR},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {2299507 (2008b:03067)},
    MRREVIEWER = {Tetsuya Ishiu},
    DOI = {10.1090/S0002-9939-07-08786-2},
    URL = {},
    eprint = {math/0603260},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    file = F,
    }

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $\kappa^+$ many normal measures on the least measurable cardinal $\kappa$. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $\lambda$-strong compactness or $\lambda$-supercompactness measures on $P_\kappa(\lambda)$ can be exactly $\lambda^+$, if $\lambda>\kappa$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

Exactly controlling the non-supercompact strongly compact cardinals

  • A. W.~Apter and J. D. Hamkins, “Exactly controlling the non-supercompact strongly compact cardinals,” Journal Symbolic Logic, vol. 68, iss. 2, pp. 669-688, 2003.  
    @ARTICLE{ApterHamkins2003:ExactlyControlling,
    AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
    TITLE = {Exactly controlling the non-supercompact strongly compact cardinals},
    JOURNAL = {Journal Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {68},
    YEAR = {2003},
    NUMBER = {2},
    PAGES = {669--688},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {1976597 (2004b:03075)},
    MRREVIEWER = {A.~Kanamori},
    doi = {10.2178/jsl/1052669070},
    eprint = {math/0301016},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.

Indestructibility and the level-by-level agreement between strong compactness and supercompactness

  • A. W.~Apter and J. D. Hamkins, “Indestructibility and the level-by-level agreement between strong compactness and supercompactness,” J.~Symbolic Logic, vol. 67, iss. 2, pp. 820-840, 2002.  
    @ARTICLE{ApterHamkins2002:LevelByLevel,
    AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
    TITLE = {Indestructibility and the level-by-level agreement between strong compactness and supercompactness},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {67},
    YEAR = {2002},
    NUMBER = {2},
    PAGES = {820--840},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {1905168 (2003e:03095)},
    MRREVIEWER = {Carlos A.~Di Prisco},
    DOI = {10.2178/jsl/1190150111},
    URL = {},
    eprint = {math/0102086},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Can a supercompact cardinal $\kappa$ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above $\kappa$, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.

Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

  • A. W.~Apter and J. D. Hamkins, “Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata,” Math Logic Quarterly, vol. 47, iss. 4, pp. 563-571, 2001.  
    @ARTICLE{ApterHamkins2001:IndestructibleWC,
    AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
    TITLE = {Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata},
    JOURNAL = {Math Logic Quarterly},
    FJOURNAL = {Mathematical Logic Quarterly},
    VOLUME = {47},
    YEAR = {2001},
    NUMBER = {4},
    PAGES = {563--571},
    ISSN = {0942-5616},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {1865776 (2003h:03078)},
    DOI = {10.1002/1521-3870(200111)47:4%3C563::AID-MALQ563%3E3.0.CO;2-%23},
    URL = {},
    eprint = {math/9907046},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

Universal indestructibility

  • A. W.~Apter and J. D. Hamkins, “Universal indestructibility,” Kobe Journal Math, vol. 16, iss. 2, pp. 119-130, 1999.  
    @article {ApterHamkins99:UniversalIndestructibility,
    AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
    TITLE = {Universal indestructibility},
    JOURNAL = {Kobe Journal Math},
    FJOURNAL = {Kobe Journal of Mathematics},
    VOLUME = {16},
    YEAR = {1999},
    NUMBER = {2},
    PAGES = {119--130},
    ISSN = {0289-9051},
    MRCLASS = {03E55 (03E35)},
    MRNUMBER = {1745027 (2001k:03112)},
    MRNUMBER = {1 745 027},
    eprint = {math/9808004},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.