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

Arthur W. Apter¹

¹ Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 3, pp. 199-207

Résumé

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”.

Détail

DOI : 10.4064/ba57-3-2

Keywords: force construct model which gch level level equivalence between strong compactness supercompactness along certain additional l like combinatorial principles particular model satisfies following properties diamondsuit delta holds every successor mahlo cardinal delta there stationary subset least supercompact cardinal kappa every delta square delta holds delta carries gap morass weak version square delta holds every infinite cardinal delta there locally defined well ordering universe cal kappa aleph regular cardinal cal restriction kappa definable structure langle kappa rangle parameter formula model constructed amalgamates synthesizes results due author author cummings asper friedman has restrictions structure its class supercompact cardinals may considered part friedmans outer model programme