L-like Combinatorial Principles and Level by Level Equivalence

Arthur W. Apter. "L-like Combinatorial Principles and Level by Level Equivalence." Bulletin of the Polish Academy of Sciences. Mathematics 57.3 (2009): 199-207. <http://eudml.org/doc/281340>.

@article{ArthurW2009,
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) $♢_δ$ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, $◻_δ$ holds and δ carries a gap 1 morass. (3) A weak version of $◻_δ$ holds for every infinite cardinal δ. (4) There is a locally defined well-ordering of the universe , i.e., for all κ ≥ ℵ₂ a regular cardinal, ↾ H(κ⁺) is definable over the structure ⟨H(κ⁺),∈ ⟩ 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”.},
author = {Arthur W. Apter},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {diamond; square; level-by-level equivalence between strong compactness and supercompactness; combinatorial principles; morass; locally defined well-ordering},
language = {eng},
number = {3},
pages = {199-207},
title = {L-like Combinatorial Principles and Level by Level Equivalence},
url = {http://eudml.org/doc/281340},
volume = {57},
year = {2009},
}

TY - JOUR
AU - Arthur W. Apter
TI - L-like Combinatorial Principles and Level by Level Equivalence
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2009
VL - 57
IS - 3
SP - 199
EP - 207
AB - 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) $♢_δ$ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, $◻_δ$ holds and δ carries a gap 1 morass. (3) A weak version of $◻_δ$ holds for every infinite cardinal δ. (4) There is a locally defined well-ordering of the universe , i.e., for all κ ≥ ℵ₂ a regular cardinal, ↾ H(κ⁺) is definable over the structure ⟨H(κ⁺),∈ ⟩ 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”.
LA - eng
KW - diamond; square; level-by-level equivalence between strong compactness and supercompactness; combinatorial principles; morass; locally defined well-ordering
UR - http://eudml.org/doc/281340
ER -