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

### Volume 57 / 2009

#### Abstract

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”.