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1.
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
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Thilo Weinert 《Mathematical Logic Quarterly》2010,56(6):659-665
We introduce the Bounded Axiom A Forcing Axiom (BAAFA). It turns out that it is equiconsistent with the existence of a regular ∑2‐correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom (BPFA) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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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
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In this paper, we consider a class of bounded Reinhardt domains Dα(m, n1,…,nm). The Bergman kernel function K(z,z), the Bergman metric matrix T(z,z), the Cauchy-Szego kernel function S(z,ζ) are obtained. Then we prove that the formal Poisson kernel function is not a Poisson kernel function. At last, we prove that Dαis a quasiconvex domain and Dαis a stronger quasiconvex domain if and only if Dαis a hypersphere. 相似文献
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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
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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
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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
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《中国科学A辑(英文版)》2008,(1)
It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in C~2 is an automorphism. 相似文献
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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
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Pan Yifei Department of Mathematical Sciences Indiana University-Purdue University Fort Wayne Fort Wayne IN - U.S.A. School of Mathematics Informatics Jiangxi Normal University Nanchang China 《中国科学A辑(英文版)》2005,48(Z1)
It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D'Angelo finite type in Cn (n > 1) is an automorphism. 相似文献
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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
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Jason A. Schanker 《Mathematical Logic Quarterly》2011,57(3):266-280
In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection $\mathcal {A}$ containing at most κ+ many subsets of κ, there exists a nonprincipal κ‐complete filter on κ measuring all sets in $\mathcal {A}$. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim 相似文献
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《Annals of Pure and Applied Logic》2014,165(2):620-630
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本文把[1]的结果推广到更广泛的一类Reinhardt域D=D(k1k2…kp) C(1≤p<n),即利用D的解析自同构群Aut(D)下不变函数给出了域D在Aut(D)下不变的Kahler度量. 相似文献