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1
Content available remote 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
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Vol. 68, no. 1
1--10
EN
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.
2
Content available remote A new Easton theorem for supercompactness and level by level equivalence
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2017
|
Vol. 65, no. 1
1--10
EN
We establish a new Easton theorem for the least supercompact cardinal κ that is consistent with the level by level equivalence between strong compactness and supercompactness. This theorem is true in any model of ZFC containing at least one supercompact cardinal, regardless if level by level equivalence holds. Unlike previous Easton theorems for supercompactness, there are no limits on the Easton functions F used, other than the usual constraints given by Easton’s theorem and the fact that if δ < κ is regular, then F(δ) < κ In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals.
3
Content available Mixed Levels of Indestructibility
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
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Vol. 63, no. 2
113--122
EN
Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ exhibits mixed levels of indestructibility. Specifically, κ's strong compactness, but not its supercompactness, is indestructible under any κ-directed closed forcing which also adds a Cohen subset of κ. On the other hand, in this model, κ's supercompactness is indestructible under any κ-directed closed forcing which does not add a Cohen subset of κ.
4
Content available remote 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
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Vol. 63, no. 3
185--194
EN
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.
5
Content available remote Singular Failures of GCH and Level by Level Equivalence
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
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Vol. 62, no 1
11--21
EN
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.
6
Content available 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
EN
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.
7
Content available remote Some Remarks on Tall Cardinals and Failures of GCH
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
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Vol. 61, no 2
97--106
EN
We investigate two global GCH patterns which are consistent with the existence of a tall cardinal, and also present some related open questions.
8
Content available remote 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
EN
We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.
9
Content available remote Level by Level Inequivalence, Strong Compactness, and GCH
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2012
|
Vol. 60, no 3
201--209
EN
We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.
10
Content available remote Sandwiching the consistency strength of two global choiceless cardinal patterns
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
Vol. 57, no 3-4
189-197
EN
We provide upper and lower bounds in consistency strength for the theories "ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω" and "ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω 1". In particular, our models for both of these theories satisfy "ZF + ¬ACω + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal".
11
Content available remote 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
EN
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".
12
Content available remote A reduction in consistency strength for universal indestructibility
Apter A. W., Sargsyan G.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
Vol. 55, no 1
1-6
EN
We show how to reduce the assumptions in consistency strength used to prove several theorems on universal indestructibility.
13
Content available remote Universal indestructibility is consistent with two strongly compact cardinals
Apter A. W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
Vol. 53, no 2
131-135
EN
We show that universal indestructibility for both strong compactness and supercompactness is consistent with the existence of two strongly compact cardinals. This is in contrast to the fact that if κ is supercompact and universal indestructibility for either strong compactness or supercompactness holds, then no cardinal λ > κ is measurable.
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