$L$-like Combinatorial Principles and Level by Level Equivalence

Volume 57 / 2009

Arthur W. Apter Bulletin Polish Acad. Sci. Math. 57 (2009), 199-207 MSC: 03E35, 03E55. DOI: 10.4064/ba57-3-2


We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “$L$-like” combinatorial principles. In particular, this model satisfies the following properties:

(1) $\diamondsuit_\delta $ holds for every successor and Mahlo cardinal $\delta $.

(2) There is a stationary subset $S$ of the least supercompact cardinal $\kappa _0$ such that for every $\delta \in S$, $\square_\delta $ holds and $\delta $ carries a gap~$1$ morass.

(3) A weak version of $\square_\delta $ holds for every infinite cardinal $\delta $.

(4) There is a locally defined well-ordering of the universe ${\cal W}$, i.e., for all $\kappa \ge \aleph_2$ a regular cardinal, ${\cal W} \restriction H(\kappa ^+)$ is definable over the structure $\langle H(\kappa ^+), \in \rangle$ by a parameter free formula.

The model constructed amalgamates and synthesizes results due to the author, the author and Cummings, and Asperó and Sy Friedman. It has no restrictions on the structure of its class of supercompact cardinals and may be considered as part of Friedman's “outer model programme”.


  • Arthur W. ApterDepartment of Mathematics
    Baruch College of CUNY
    New York, NY 10010, U.S.A.
    The CUNY Graduate Center, Mathematics
    365 Fifth Avenue
    New York, NY 10016, U.S.A.

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