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1.
Saharon Shelah 《Archive for Mathematical Logic》2003,42(1):1-44
We prove that on many inaccessible cardinals there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal λ is
λ × ω-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra
of cardinality . Lastly, we give further information on guessing of clubs.
Received: 10 March 1992 / First revised version: 11 August 1997 / Second revised version: 12 September 2000 / Published online:
5 November 2002 相似文献
2.
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
3.
Israel Journal of Mathematics - We shall show here that in many successor cardinals λ, there is a Jonsson algebra (in other words Jn(λ), or λ is not a Jonsson cardinal). In... 相似文献
4.
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
5.
Joan Bagaria Joel David Hamkins Konstantinos Tsaprounis Toshimichi Usuba 《Archive for Mathematical Logic》2016,55(1-2):19-35
Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ≥ 3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing \({\mathbb{Q} \in V_\theta}\), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger. 相似文献
6.
It was proved by Dow and Simon that there are 2ω1 (as many as possible) pairwise nonhomeomorphic compact, T2, scattered spaces of height ω1 and width ω. In this paper, we prove that if is an ordinal withω1 < ω2 and θ = κξ: ξ < is a sequence of cardinals such that either κξ = ω or κξ = ω1 for every ξ < , then there are 2ω1 pairwise nonhomeomorphic compact, T2, scattered spaces whose cardinal sequence is θ. 相似文献
7.
In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω2. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary. 相似文献
8.
All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters 下载免费PDF全文
Using the analysis developed in our earlier paper 5 , we show that every uncountable cardinal in Gitik's model of 8 in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of 8 is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory + “All uncountable cardinals are singular” + “Every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter”. 相似文献
9.
Huiling Zhu 《Archive for Mathematical Logic》2013,52(5-6):497-506
In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms. 相似文献
10.
We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly \({\theta}\)-supercompact, for any desired \({\theta}\). In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly \({\theta_\kappa}\)-supercompact cardinals \({\kappa}\), for nearly any desired function \({\kappa\mapsto\theta_\kappa}\). These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals. 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.
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
13.
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.
14.
《Mathematical Logic Quarterly》2018,64(3):207-217
We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ1‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals. 相似文献
15.
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
16.
Jan Tryba 《Israel Journal of Mathematics》1984,49(4):315-324
For a regular cardinal κ a Jónsson model of size κ+ is presented. We notice that every singular Jónsson cardinal κ with uncountable cofinality is the limit of some continuous
sequence of smaller Jónsson cardinals. An analogous statement holds if κ is an inaccessible Jónsson cardinal unless κ is Mahlo.
But we prove that the first Mahlo cardinal cannot be Jónsson. Some additional remarks are included. 相似文献
17.
We prove that the Axiom of Full Reflection at a measurable cardinal is equiconsistent with the existence of a measurable cardinal.
We generalize the result also to larger cardinals such as strong or supercompact cardinals. 相似文献
18.
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
19.
20.
Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf(η) = κ+. Then, in some cardinal‐preserving generic extension there is a superatomic Boolean algebra $\mathcal BUsing Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf(η) = κ+. Then, in some cardinal‐preserving generic extension there is a superatomic Boolean algebra $\mathcal B$ such that $\mathrm{ht}(\mathcal B) = \eta + 1$, $\mathrm{wd}_{\alpha }(\mathcal B) = \kappa$ for every α < η and $\mathrm{wd}_{\eta }(\mathcal B) = \lambda$(i.e., there is a locally compact scattered space with cardinal sequence 〈κ〉η??〈λ〉). Especially, $\langle {\omega }\rangle _{{\omega }_1}{}^{\smallfrown } \langle {\omega }_3\rangle$ and $\langle {\omega }_1\rangle _{{\omega }_2}{}^{\smallfrown } \langle {\omega }_4\rangle$ can be cardinal sequences of superatomic Boolean algebras. 相似文献