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1.
Arthur W. Apter 《Archive for Mathematical Logic》2007,46(3-4):155-163
It is known that if are such that κ is indestructibly supercompact and λ is 2λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which
points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal,
there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact
cardinal κ in which κ’s strong compactness is indestructible under κ-directed closed forcing.
The author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant. 相似文献
2.
Universal indestructibility for degrees of supercompactness and strongly compact cardinals 总被引:1,自引:1,他引:0
We establish two theorems concerning strongly compact cardinals and universal indestructibility for degrees of supercompactness.
In the first theorem, we show that universal indestructibility for degrees of supercompactness in the presence of a strongly
compact cardinal is consistent with the existence of a proper class of measurable cardinals. In the second theorem, we show
that universal indestructibility for degrees of supercompactness is consistent in the presence of two non-supercompact strongly
compact cardinals, each of which exhibits a significant amount of indestructibility for its strong compactness.
The first author’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. The first
author wishes to thank James Cummings for helpful discussions on the subject matter of this paper. In addition, both authors
wish to thank the referee, for many helpful comments and suggestions which were incorporated into the current version of the
paper. 相似文献
3.
Arthur W. Apter 《Mathematical Logic Quarterly》2009,55(3):271-279
We construct a model for the level by level equivalence between strong compactness and supercompactness in which the least supercompact cardinal κ has its strong compactness indestructible under adding arbitrarily many Cohen subsets. There are no restrictions on the large cardinal structure of our model (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
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
5.
Arthur W. Apter 《Archive for Mathematical Logic》2006,45(7):831-838
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 a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary. 相似文献
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
Arthur W. Apter Saharon Shelah 《Transactions of the American Mathematical Society》1997,349(1):103-128
We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension ZFC + GCH in which, (a) (preservation) for regular, if is supercompact', then is supercompact' and so that, (b) (equivalence) for regular, is strongly compact' iff is supercompact', except possibly if is a measurable limit of cardinals which are supercompact.
13.
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
14.
15.
Arthur W. Apter James Cummings Joel David Hamkins 《Proceedings of the American Mathematical Society》2007,135(7):2291-2300
We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly many normal measures on the least measurable cardinal . This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of strong compactness or supercompactness measures on can be exactly if is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.
16.
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}
17.
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
18.
19.
Arthur W. Apter 《Mathematical Logic Quarterly》2000,46(4):453-459
Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ+ strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, □γ+ω fails but □γ+ω, ω holds. This generalizes a result of Ben‐David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness. 相似文献
20.
Pierre Matet 《Proceedings of the American Mathematical Society》2006,134(7):2147-2152
We show that if holds for every , then is strongly compact.