共查询到20条相似文献,搜索用时 0 毫秒
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Ernest Schimmerling 《Advances in Mathematics》2007,216(1):89-117
The combinatorial principle □(λ) says that there is a coherent sequence of length λ that cannot be threaded. If λ=κ+, then the related principle κ□ implies □(λ). Let κ?ℵ2 and X⊆κ. Assume both □(κ) and κ□ fail. Then there is an inner model N with a proper class of strong cardinals such that X∈N. If, in addition, κ?ℵ02 and n<ω, then there is an inner model Mn(X) with n Woodin cardinals such that X∈Mn(X). In particular, by Martin and Steel, Projective Determinacy holds. As a corollary to this and results of Todorcevic and Velickovic, the Proper Forcing Axiom for posets of cardinality +(ℵ02) implies Projective Determinacy. 相似文献
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We generalize (A), which was introduced in [Sch], to larger cardinals. For a regular cardinal >0 we denote by (A) the statement that and for all regular >,is stationary in It was shown in [Sch] that can hold in a set-generic extension of L. We here prove that can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Rä00] and [Rä01]. is equivalent with the existence of 0#.Mathematics Subject Classification (1991): Primary 03E55, 03E15, Secondary 03E35, 03E60 相似文献
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We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß. 相似文献
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Andreas Liu 《Annals of Pure and Applied Logic》2006,140(1-3):120-127
We use pcf theory to prove results on reflection at singular cardinals. 相似文献
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本文给出划分的限制的一个基本性质,κ 连续性,证明了划分空间上的滤子的κ 完全性,精细性,超滤子性,准超滤子性,等关于限制保持.另外,本文还给出划分空 相似文献
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We deal here with colorings of the pair (μ+, μ), when μ is a strong limit and singular cardinal. We show that there exists a coloring c with no refinement. It follows that the properties of colorings of (μ+, μ) when μ is singular differ in an essential way from the case of regular μ (although the identities may be the same). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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From GCH and Pm(κ)-hypermeasurable (1 <m<gw), we construct a model satisfying 2n = a(n) and 2ω = ω+m for a monotone a:ω→ω satisfying a(n)>n. 相似文献
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Ralf-Dieter Schindler 《Mathematical Logic Quarterly》1999,45(3):361-369
We generalize results of [3] and [1] to hyperprojective sets of reals, viz. to more than finitely many strong cardinals being involved. We show, for example, that if every set of reals in Lω (?) is weakly homogeneously Souslin, then there is an inner model with an inaccessible limit of strong cardinals. 相似文献
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Ralf-Dieter Schindler 《Transactions of the American Mathematical Society》2001,353(2):479-489
We present a technique for coding sets ``into ,' where is the core model below a strong cardinal. Specifically, we show that if there is no inner model with a strong cardinal then any can be made (in the codes) in a reasonable and stationary preserving set generic extension.
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Joel D. Hamkins 《Mathematical Logic Quarterly》2009,55(1):68-86
A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j (κ) > θ and Mκ ? M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal κ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of 2κ as high as desired. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and its dual, the prediction number, as well as two versions of the rearrangement number. When translated appropriately, these yield four new highness notions. We will define these new notions, show some of their basic properties and place them in the computability-theoretic version of Cichoń's diagram. 相似文献
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Heike Mildenberger 《Mathematical Logic Quarterly》2001,47(3):291-298
We consider two models V1, V2 of ZFC such that V1 ⊆ V2, the cofinality functions of V1 and of V2 coincide, V1 and V2 have that same hereditarily countable sets, and there is some uncountable set in V2 that is not covered by any set in V1 of the same cardinality. We show that under these assumptions there is an inner model of V2 with a measurable cardinal κ of Mitchell order κ++. This technical result allows us to show that changing cardinal characteristics without changing cofinalities or ω‐sequences (which was done for some characteristics in [13]) has consistency strength at least Mitchell order κ++. From this we get that the changing of cardinal characteristics without changing cardinals or ω‐sequences has consistency strength Mitchell order ω1, even in the case of characteristics that do not stem from a transitive relation. Hence the known forcing constructions for such a change have lowest possible consistency strength. We consider some stronger violations of covering which have appeared as intermediate steps in forcing constructions. 相似文献
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Ernest Schimmerling 《Proceedings of the American Mathematical Society》2002,130(11):3385-3391
Theorem 4 is a characterization of Woodin cardinals in terms of Skolem hulls and Mostowski collapses. We define weakly hyper-Woodin cardinals and hyper-Woodin cardinals. Theorem 5 is a covering theorem for the Mitchell-Steel core model, which is constructed using total background extenders. Roughly, Theorem 5 states that this core model correctly computes successors of hyper-Woodin cardinals. Within the large cardinal hierarchy, in increasing order we have: measurable Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin, and superstrong cardinals. (The comparison of Shelah versus hyper-Woodin is due to James Cummings.)
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E. Schimmerling J. R. Steel 《Transactions of the American Mathematical Society》1999,351(8):3119-3141
Our main results are: 1) every countably certified extender that coheres with the core model is on the extender sequence of , 2) computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of , 4) (joint with W. J. Mitchell) is universal for mice of height whenever , 5) if there is a such that is either a singular countably closed cardinal or a weakly compact cardinal, and fails, then there are inner models with Woodin cardinals, and 6) an -Erdös cardinal suffices to develop the basic theory of .
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《Annals of Pure and Applied Logic》2020,171(8):102821
Some non-normal κ-complete ultrafilters over a measurable κ with special properties are constructed. Questions by A. Kanamori [4] about infinite Rudin-Frolik sequences, discreteness and products are answered. 相似文献