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1.
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
2.
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
3.
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
4.
Arthur W. Apter 《Archive for Mathematical Logic》2005,44(3):387-395
We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, holds for every regular uncountable cardinal , and below the least supercompact cardinal , holds on a stationary subset of . There are no restrictions in our model on the structure of the class of supercompact cardinals.The author wishes to thank the referee for numerous helpful comments and suggestions, which have considerably improved the presentation of the material contained herein. The author also wishes to thank Andreas Blass, the corresponding editor, for a useful suggestion, and Grigor Sargsyan for a very helpful conversation on the subject matter of this paper.Mathematics Subject Classification (2000): 03E35, 03E55 相似文献
5.
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
6.
Carmi Merimovich 《Transactions of the American Mathematical Society》2003,355(5):1729-1772
We define extender sequences, generalizing measure sequences of Radin forcing.
Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing.
We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness.
Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.
7.
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
8.
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
9.
Gunter Fuchs 《Archive for Mathematical Logic》2005,44(8):935-971
A generalization of Příkry's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkry generic sequences reminiscent
of Mathias' criterion for Příkry genericity is provided, together with a maximality theorem which states that a generalized
Příkry sequence almost contains every other one lying in the same extension.
This forcing can be used to falsify the covering lemma for a higher core model if there is an inner model with infinitely
many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of
a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ.
During the research for this paper the author was supported by DFG-Project Je209/1-2. 相似文献
10.
Under the assumption that there exists an elementary embedding(henceforth abbreviated as and in particular under we prove a Coding Lemma for and find certain versions of it which are equivalent to strong regularity of cardinals below . We also prove that a stronger version of the Coding Lemma holds for a stationary set of ordinals below .Mathematics Subject Classification (2000):03E55 相似文献
11.
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
12.
Osamu Takaki 《Archive for Mathematical Logic》2005,44(6):689-709
In this paper, we develop primitive recursive analogues of regular cardinals by using ordinal representation systems for KPi and KPM. We also define primitive recursive analogues of inaccessible and hyperinaccessible cardinals. Moreover, we characterize the primitive recursive analogue of the least (uncountable) regular cardinal. 相似文献
13.
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}
14.
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
15.
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
16.
Arthur W. Apter Grigor Sargsyan 《Proceedings of the American Mathematical Society》2005,133(10):3103-3108
In this note, we provide an affirmative answer to the title question by giving two examples of cardinals satisfying conditions implying they are non-Rowbottom which can be turned into Rowbottom cardinals via forcing. In our second example, our cardinal is also non-Jonsson.
17.
Matthew Foreman 《Advances in Mathematics》2009,222(2):565-2687
Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms.These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction.The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for.The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory. 相似文献
18.
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
19.
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
20.
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