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1.
Arthur W. Apter 《Mathematical Logic Quarterly》2004,50(1):51-64
We force and construct models in which there are non‐supercompact strongly compact cardinals which aren't measurable limits of strongly compact cardinals and in which level by level equivalence between strong compactness and supercompactness holds non‐trivially except at strongly compact cardinals. In these models, every measurable cardinal κ which isn't either strongly compact or a witness to a certain phenomenon first discovered by Menas is such that for every regular cardinal λ > κ, κ is λ strongly compact iff κ is λ supercompact. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
2.
Arthur W. Apter 《Mathematical Logic Quarterly》2010,56(1):4-12
We construct two models containing exactly one supercompact cardinal in which all non‐supercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness holds. Each universe has only one strongly compact cardinal and contains relatively few large cardinals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
Arthur W. Apter 《Mathematical Logic Quarterly》2003,49(4):375-384
We construct a model in which the strongly compact cardinals can be non‐trivially characterized via the statement “κ is strongly compact iff κ is a measurable limit of strong cardinals”. If our ground model contains large enough cardinals, there will be supercompact cardinals in the universe containing this characterization of the strongly compact cardinals. 相似文献
4.
Arthur W. Apter 《Mathematical Logic Quarterly》2005,51(3):247-253
We establish an Easton theorem for the least supercompact cardinal that is consistent with the level by level equivalence between strong compactness and supercompactness. 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. We also briefly indicate how our methods of proof yield an Easton theorem that is consistent with the level by level equivalence between strong compactness and supercompactness in a universe with a restricted number of large cardinals. We conclude by posing some related open questions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
5.
We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it
is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the nth strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact
cardinals, for the strongly compact cardinals and the cardinals which are both strong cardinals and Woodin cardinals to coincide
precisely. We also show how the techniques employed can be used to prove additional theorems about possible relationships
between Woodin cardinals and strongly compact cardinals.
The first author's research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant. 相似文献
6.
Arthur W. Apter 《Archive for Mathematical Logic》2011,50(7-8):707-712
We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide. 相似文献
7.
Arthur W. Apter 《Mathematical Logic Quarterly》2007,53(1):78-85
If κ < λ are such that κ is indestructibly supercompact and λ is 2λ supercompact, it is known from [4] that
- {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ violates level by level equivalence between strong compactness and supercompactness}
- {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ satisfies level by level equivalence between strong compactness and supercompactness}
8.
Arthur W. Apter 《Mathematical Logic Quarterly》1997,43(3):427-430
We show that it is consistent, relative to a supercompact limit of supercompact cardinals, for the least strongly compact cardinal k to be both the least measurable cardinal and to be > 2k supercompact. 相似文献
9.
Arthur W. Apter 《Mathematical Logic Quarterly》2003,49(6):587-597
We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
Arthur W. Apter 《Mathematical Logic Quarterly》2008,54(3):288-293
Using an idea of Sargsyan, we show how to reduce the consistency strength of the assumptions employed to establish a theorem concerning a uniform level of indestructibility for both strong and supercompact cardinals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
11.
Arthur W. Apter 《Mathematical Logic Quarterly》2006,52(5):457-463
We construct models for the level by level equivalence between strong compactness and supercompactness in which for κ the least supercompact cardinal and δ ≤ κ any cardinal which is either a strong cardinal or a measurable limit of strong cardinals, 2δ > δ + and δ is < 2δ supercompact. In these models, the structure of the class of supercompact cardinals can be arbitrary, and the size of the power set of κ can essentially be made as large as desired. This extends and generalizes [5, Theorem 2] and [4, Theorem 4]. We also sketch how our techniques can be used to establish a weak indestructibility result. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
12.
Arthur W. Apter 《Archive for Mathematical Logic》2008,47(2):101-110
If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded in κ. The two aforementioned phenomena, however, need not occur in a universe with an indestructibly supercompact cardinal and
sufficiently few large cardinals. In particular, we show how to construct a model with an indestructibly supercompact cardinal
κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry fewer than the maximal number of normal measures. We also, however, show how to construct a model with an indestructibly
supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry the maximal number of normal measures. If we weaken the requirements on indestructibility, then this last result
can be improved to obtain a model with an indestructibly supercompact cardinal κ in which every measurable cardinal δ < κ carries the maximal number of normal measures.
A. W. Apter’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. In addition, the
author wishes to thank the referee, for helpful comments, corrections, and suggestions which have been incorporated into the
current version of the paper. 相似文献
13.
We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds,
along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due
to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals.
A. W. Apter’s research was partially supported by PSC-CUNY Grants and CUNY Collaborative Incentive Grants. J. Cummings’s research
was partially supported by NSF Grant DMS-0400982. 相似文献
14.
We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim 相似文献
15.
Arthur W. Apter Victoria Gitman Joel David Hamkins 《Archive for Mathematical Logic》2012,51(3-4):257-283
We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ ?=?κ +, another for which 2 κ ?=?κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that ${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$ . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH?+?V?=?HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results. 相似文献
16.
Arthur W. Apter 《Mathematical Logic Quarterly》2013,59(4-5):371-377
If are such that δ is indestructibly supercompact and γ is measurable, then it must be the case that level by level inequivalence between strong compactness and supercompactness fails. We prove a theorem which points to this result being best possible. Specifically, we show that relative to the existence of cardinals such that κ1 is λ‐supercompact and λ is inaccessible, there is a model for level by level inequivalence between strong compactness and supercompactness containing a supercompact cardinal in which κ’s strong compactness, but not supercompactness, is indestructible under κ‐directed closed forcing. In this model, κ is the least strongly compact cardinal, and no cardinal is supercompact up to an inaccessible cardinal. 相似文献
17.
From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact
and strong cardinals precisely coincide. In this model, it is the case that no strongly compact cardinal is supercompact.
Received: 16 December 1998 / Revised version: 29 March 1999 相似文献
18.
Arthur W. Apter 《Mathematical Logic Quarterly》2011,57(3):261-265
Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ < κ∣δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and Vδ?Ti} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing four models in which Ai = ?? for i = 1, …, 4. In each of these models, there is an indestructibly supercompact cardinal κ, and no cardinal δ > κ is inaccessible. We show it is also the case that if κ is indestructibly supercompact, then Vκ?T1, so by reflection, B1 = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and Vδ?T1} is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B1 = ??. On the other hand, assuming κ is supercompact and no cardinal δ > κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, Vδ?T1. It is thus not possible to prove in ZFC that Bi = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and Vδ?Ti} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim 相似文献
19.
Arthur W. Apter 《Mathematical Logic Quarterly》2009,55(3):228-236
If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ ‐strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non‐weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a Mahlo cardinal in which the least supercompact cardinal κ is also the least strongly compact cardinal, κ 's strongness is indestructible under κ ‐strategically closed forcing, κ 's supercompactness is indestructible under κ ‐directed closed forcing not adding any new subsets of κ, and δ is Mahlo and reflects stationary sets iff δ is weakly compact. In this model, no strong cardinal δ < κ is indestructible under δ ‐strategically closed forcing. It therefore follows that it is relatively consistent for the least strong cardinal κ whose strongness is indestructible under κ ‐strategically closed forcing to be the same as the least supercompact cardinal, which also has its supercompactness indestructible under κ ‐directed closed forcing not adding any new subsets of κ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
20.
Arthur W. Apter 《Mathematical Logic Quarterly》1996,42(1):211-218
Assuming Con(AD), a model in which there are unboundedly many regular cardinals below Θ and in which the only regular cardinals below Θ are limit cardinals was previously constructed. Using a large cardinal hypothesis far beyond Con(AD), we construct in this paper a model in which there is a proper class of regular cardinals and in which the only regular cardinals in the universe are limit cardinals. Mathematics Subject Classification: 03E55, 03E60. 相似文献