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6 Bulletin of the Polish Academy of Sciences. Mathematics

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The consistency of level by level equivalence with V=HOD, the Ground Axiom, and instances of square and diamond

Apter Arthur W.

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 1

1--10

We construct via forcing a model for the level by level equivalence between strong compactness and supercompactness in which both V=HOD and the Ground Axiom (GA) are true. In our model, various versions of the combinatorial principles □ and ♢ hold. In the model constructed, there are no restrictions on the class of supercompact cardinals.

Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

185--194

We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes the author’s result in Arch. Math. Logic 46 (2007), but without the restriction that no cardinal is supercompact up to an inaccessible cardinal.

Singular Failures of GCH and Level by Level Equivalence

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2014

Vol. 62, no 1

11--21

We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is an unbounded set of singular cardinals which witness the only failures of GCH in the universe. In this model, the structure of the class of supercompact cardinals can be arbitrary.

HOD-supercompactness, Indestructibility, and Level by Level Equivalence

Apter A. W., Friedman S.

Bulletin of the Polish Academy of Sciences. Mathematics

2014

Vol. 62, no. 3

197--209

In an attempt to extend the property of being supercompact but not HOD-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not HOD-supercompact holds for the least supercompact cardinal κ0, κ0 is indestructibly supercompact, the strongly compact and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above κ0 but fails below κ0. Additionally, we get the property of being supercompact but not HOD-supercompact at the least supercompact cardinal, in a model where level by level equivalence between strong compactness and supercompactness holds.

The Wholeness Axioms and the Class of Supercompact Cardinals

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 2

101--111

We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.

L-like combinatorial principles and level by level equivalence

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 3-4

199-207

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 κ 0 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 W, i.e., for all κ ≥ ℵ 2 a regular cardinal, W↑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 Aspero 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".