The inner-model and ground-model reflection principles, CUNY Set Theory seminar, September 2017

This will be a talk for the CUNY Set Theory seminar on September 1, 2017, 10 am. GC 6417.

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Abstract.  The inner model reflection principle asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. Both of these principles, expressing a form of width-reflection in constrast to the usual height-reflection, are equiconsistent with ZFC and an outright consequence of the existence of sufficient large cardinals, as well as a consequence (in lightface form) of the maximality principle and also of the inner-model hypothesis.  This is joint work with Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, myself and Jonas Reitz.

George Leibman

George Joseph Leibman earned his Ph.D. under my supervision in June, 2004 at the CUNY Graduate Center. He was my first Ph.D. student. Being very interested both in forcing and in modal logic, it was natural for him to throw himself into the emerging developments at the common boundary of these topics.  He worked specifically on the natural extensions of the maximality principle where when one considers a fixed definable class $\Gamma$ of forcing notions.  This research engaged with fundamental questions about the connection between the forcing-theoretic properties of the forcing class $\Gamma$ and the modal logic of its forcing validities, and was a precursor of later work, including joint work, on the modal logic of forcing.

George Leibman

George Leibman

 

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George Leibman, “Consistency Strengths of Modified Maximality Principles,” Ph.D. thesis, CUNY Graduate Center, 2004.  ar$\chi$iv

Abstract. The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension $V^{\mathbb{P}}$, and remains true in any further forcing extension of $V^{\mathbb{P}}$, then it is true in all forcing extensions of $V$.  A modified maximality principle $\text{MP}_\Gamma$ arises when considering forcing with a particular class $\Gamma$ of forcing notions. A parametrized form of such a principle, $\text{MP}_\Gamma(X)$, considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set $X$ which depends on the forcing class $\Gamma$ being considered. A stronger necessary form of such a principle, $\square\text{MP}_\Gamma(X)$, occurs when it continues to be true in all $\Gamma$ forcing extensions.

This study uses iterated forcing, modal logic, and other techniques to establish consistency strengths for various modified maximality principles restricted to various forcing classes, including ccc, COHEN, COLL (the forcing notions that collapse ordinals to $\omega$), ${\lt}\kappa$ directed closed forcing notions, etc., both with and without parameter sets. Necessary forms of these principles are also considered.

The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal

  • W. Hamkins Joel D.~and Woodin, “The necessary maximality principle for c.c.c.\ forcing is equiconsistent with a weakly compact cardinal,” MLQ Math.~Log.~Q., vol. 51, iss. 5, pp. 493-498, 2005.  
    @ARTICLE{HamkinsWoodin2005:NMPccc,
    AUTHOR = {Hamkins, Joel D.~and Woodin, W.~Hugh},
    TITLE = {The necessary maximality principle for c.c.c.\ forcing is equiconsistent with a weakly compact cardinal},
    JOURNAL = {MLQ Math.~Log.~Q.},
    FJOURNAL = {MLQ.~Mathematical Logic Quarterly},
    VOLUME = {51},
    YEAR = {2005},
    NUMBER = {5},
    PAGES = {493--498},
    ISSN = {0942-5616},
    MRCLASS = {03E65 (03E55)},
    MRNUMBER = {2163760 (2006f:03082)},
    MRREVIEWER = {Tetsuya Ishiu},
    DOI = {10.1002/malq.200410045},
    URL = {http://dx.doi.org/10.1002/malq.200410045},
    eprint = {math/0403165},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    file = F,
    }

The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.

See related article on the Maximality Principle

A simple maximality principle

  • J. D. Hamkins, “A simple maximality principle,” J.~Symbolic Logic, vol. 68, iss. 2, pp. 527-550, 2003.  
    @article{Hamkins2003:MaximalityPrinciple,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {A simple maximality principle},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {68},
    YEAR = {2003},
    NUMBER = {2},
    PAGES = {527--550},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E35 (03E40)},
    MRNUMBER = {1976589 (2005a:03094)},
    MRREVIEWER = {Ralf-Dieter Schindler},
    DOI = {10.2178/jsl/1052669062},
    URL = {http://projecteuclid.org/getRecord?id=euclid.jsl/1052669062},
    month = {June},
    eprint = {math/0009240},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence$\varphi$ holding in some forcing extension $V^{\mathbb{P}}$ and all subsequent extensions $V^{\mathbb{P}*\mathbb{Q}}$ holds already in $V$. It follows, in fact, that such sentences must also hold in all forcing extensions of $V$. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\Diamond\Box\varphi)\to\Box\varphi$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in $\varphi$, is equiconsistent with the scheme asserting that $V_\delta$ is an elementary substructure of $V$ for an inaccessible cardinal $\delta$, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that $0^\sharp$ exists, that $x^\sharp$ exists for every set $x$, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.