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1
Content available remote

Mixed Levels of Indestructibility

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 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.
2
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On level by level equivalence and inequivalence between strong compactness and supercompactness

100%
Apter A.
Fundamenta Mathematicae
|
2002
|
tom 171
|
nr 1
77-92
EN
We prove two theorems, one concerning level by level inequivalence between strong compactness and supercompactness, and one concerning level by level equivalence between strong compactness and supercompactness. We first show that in a universe containing a supercompact cardinal but of restricted size, it is possible to control precisely the difference between the degree of strong compactness and supercompactness that any measurable cardinal exhibits. We then show that in an unrestricted size universe containing many supercompact cardinals, it is possible to have significant failures of GCH along with level by level equivalence between strong compactness and supercompactness, except possibly at inaccessible levels.
3
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Level by level equivalence and the number of normal measures over $P_{κ}(λ)$

100%
Apter A.
Fundamenta Mathematicae
|
2007
|
tom 194
|
nr 3
253-265
EN
We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures $P_{κ}(λ)$ carries. In the first of these models, $P_{κ}(λ)$ carries $2^{2^{[λ]^{<κ}}}$ many normal measures, the maximal number. In the second of these models, $P_{κ}(λ)$ carries $2^{2^{[λ]^{<κ}}}$ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and hence also $P_{κ}(κ)$) carries only κ⁺ many normal measures. In both of these models, there are no restrictions on the structure of the class of supercompact cardinals.
4
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More Easton theorems for level by level equivalence

100%
Apter A.
Colloquium Mathematicum
|
2012
|
tom 128
|
nr 1
69-86
EN
We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize Theorem 1 in our earlier paper [Math. Logic Quart. 51 (2005)]. In both our ground model and the model witnessing the conclusions of our present theorems, there are no restrictions on the structure of the class of supercompact cardinals.
5
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Singular Failures of GCH and Level by Level Equivalence

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
|
tom 62
|
nr 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 remote

Some Remarks on Tall Cardinals and Failures of GCH

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 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.
7
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Level by Level Inequivalence, Strong Compactness, and GCH

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2012
|
tom 60
|
nr 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.
8
Content available remote

The Wholeness Axioms and the Class of Supercompact Cardinals

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2012
|
tom 60
|
nr 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
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Strong compactness, measurability, and the class of supercompact cardinals

100%
Apter A.
Fundamenta Mathematicae
|
2001
|
tom 167
|
nr 1
65-78
EN
We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. We begin by showing, relative to the appropriate hypotheses, that it is consistent non-trivially for every supercompact cardinal to be the limit of (non-supercompact) strongly compact cardinals. We then show, relative to the existence of a non-trivial (proper or improper) class of supercompact cardinals, that it is possible to have a model with the same class of supercompact cardinals in which every measurable cardinal δ is $2^{δ}$ strongly compact.
10
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How many normal measures can $ℵ_{ω+1}$ carry?

100%
Apter A.
Fundamenta Mathematicae
|
2006
|
tom 191
|
nr 1
57-66
EN
We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for $ℵ_{ω+1}$ to be measurable and to carry exactly τ normal measures, where $τ ≥ ℵ_{ω+2}$ is any regular cardinal. This contrasts with the fact that assuming AD + DC, ${ℵ_{ω+1}}$ is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.
11
Content available remote

L-like Combinatorial Principles and Level by Level Equivalence

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
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tom 57
|
nr 3
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 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 𝓦, i.e., for all κ ≥ ℵ₂ 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 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".
12
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Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
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tom 57
|
nr 3
189-197
EN
We provide upper and lower bounds in consistency strength for the theories "ZF + $¬AC_ω$ + 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 + $¬AC_ω$ + 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 ω₁". In particular, our models for both of these theories satisfy "ZF + $¬AC_ω$ + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal".
13
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Some applications of Sargsyan's equiconsistency method

100%
Apter A.
Fundamenta Mathematicae
|
2012
|
tom 216
|
nr 3
207-222
EN
We apply techniques due to Sargsyan to reduce the consistency strength of the assumptions used to establish an indestructibility theorem for supercompactness. We then show how these and additional techniques due to Sargsyan may be employed to establish an equiconsistency for a related indestructibility theorem for strongness.
14
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A Note on Indestructibility and Strong Compactness

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2008
|
tom 56
|
nr 3
191-197
EN
If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is $2^λ$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], {δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is $2^δ = δ⁺$ supercompact, κ's supercompactness is indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive, and for every measurable cardinal δ, δ is δ⁺ strongly compact iff δ is δ⁺ supercompact.
15
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Universal Indestructibility is Consistent with Two Strongly Compact Cardinals

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom 53
|
nr 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.
16
Content available remote

Supercompactness and partial level by level equivalence between strong compactness and strongness

100%
Apter A.
Fundamenta Mathematicae
|
2004
|
tom 182
|
nr 2
123-136
EN
We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if $δ^{+α}$ is regular, δ is $δ^{+α}$ strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are $δ^{+α}$ strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.
17
Content available remote

Stationary reflection and level by level equivalence

100%
Apter A.
Colloquium Mathematicum
|
2009
|
tom 115
|
nr 1
113-128
EN
We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional "inner model like" properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jónsson cardinal is weakly compact. On the other hand, we force and construct a model for the level by level equivalence between strong compactness and supercompactness in which on a stationary subset of the least supercompact cardinal κ, there are non-weakly compact Mahlo cardinals which reflect stationary sets. We also examine some extensions and limitations on what is possible in our theorems. Finally, we indicate how to ensure in our models that $⋄_δ$ holds for every successor and Mahlo cardinal δ, and below the least supercompact cardinal κ, $◻_δ$ holds on a stationary subset of κ. There are no restrictions in our main models on the structure of the class of supercompact cardinals.
18
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Indestructibility, strongness, and level by level equivalence

100%
Apter A.
Fundamenta Mathematicae
|
2003
|
tom 177
|
nr 1
45-54
EN
We construct a model in which there is a strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing and in which level by level equivalence between strong compactness and supercompactness holds non-trivially.
19
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Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions

100%
Apter A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 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.
20
Content available remote

Indestructibility, strong compactness, and level by level equivalence

100%
Apter A.
Fundamenta Mathematicae
|
2009
|
tom 204
|
nr 2
113-126
EN
We show the relative consistency of the existence of two strongly compact cardinals κ₁ and κ₂ which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ₁. In the model constructed, κ₁'s strong compactness is indestructible under arbitrary κ₁-directed closed forcing, κ₁ is a limit of measurable cardinals, κ₂'s strong compactness is indestructible under κ₂-directed closed forcing which is also (κ₂,∞)-distributive, and κ₂ is fully supercompact.
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