On a problem of Foreman and Magidor

A question of Foreman and Magidor asks if it is consistent for every sequence of stationary subsets of the _ns for 1n< to be mutually stationary. We get a positive answer to this question in the context of the negation of the Axiom of Choice. We also indicate how a positive answer to a generalized version of this question in a choiceless context may be obtained.The author wishes to thank James Cummings for helpful correspondence on the subject matter of this paper. The author also wishes to thank the referee and Andreas Blass, the corresponding editor, for helpful comments and suggestions that have been incorporated into this version of the paper. 03E35, 03E55 Supercompact cardinal – Indestructibility – Almost huge cardinal – Mutual stationarity – Symmetric inner modelRevised version: 6 June 2004     

On extendible cardinals and the GCH

We give a characterization of extendibility in terms of embeddings between the structures H _λ . By that means, we show that the GCH can be forced (by a class forcing) while preserving extendible cardinals. As a corollary, we argue that such cardinals cannot in general be made indestructible by (set) forcing, under a wide variety of forcing notions.     

On the singular cardinals problem I

We show how to get a model of set theory in which ℵ_ω is a strong limit cardinal which violates the generalized continuum hypothesis. Generalizations to other cardinals are also given. This research was partially supported by N.S.F. grants at the University of Colorado, Boulder and the University of California, Berkeley.     

All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters

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”.     

On tall cardinals and some related generalizations

We continue the study of tall cardinals and related notions begun by Hamkins in 2009 and answer three of his questions posed in that paper.     

Violating the singular cardinals hypothesis without large cardinals

We extend a transitive model V of ZFC+GCH cardinal preservingly to a model N of ZF + “GCH holds below ℵ _ω ” + “there is a surjection from the power set of ℵ _ω onto λ”, where λ is an arbitrarily high fixed cardinal in V. The construction can be described as follows: add ℵ _n +1 many Cohen subsets of ℵ _n+1 for every n < ω, and adjoin λ many subsets of ℵ _ω which are unions of ω-sequences of those Cohen subsets; then let N be a choiceless submodel generated by equivalence classes of the λ subsets of ℵ _ω modulo an appropriate equivalence relation.     

On the role of supercompact and extendible cardinals in logic

It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Löwenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom schema related to model theory implies the existence of many extendible cardinals.     

On the free subset property at singular cardinals

We give a proof ofTheorem 1. Let be the smallest cardinal such that the free subset property Fr (, ₁)holds. Assume is singular. Then there is an inner model with ₁ measurable cardinals.     

On Shelah’s compactness of cardinals

We deal with the compactness property of cardinals presented by Shelah, who proved a compactness theorem for singular cardinals. We improve that result in eliminating axiom I there and show a new application of that theorem together with a straightforward proof of it for the special case discussed. We discuss compactness for regular cardinals and show some independence results: one of them, a part of which is due to A. Litman, is the independence from ZFC+GCH of the gap-one two cardinal problem for singular cardinals. This paper is based on the author’s M.Sc. thesis written at The Hebrew University under the supervision of Prof. Shelah, to whom he expresses his deep gratitude. An erratum to this article is available at .     

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

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.     

Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model

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.)

     

On the class of measurable cardinals without the axiom of choice

Using techniques of Gitik in conjunction with a large cardinal hypothesis whose consistency strength is strictly in between that of a supercompact and an almost huge cardinal, we obtain the relative consistency of the theory “ZF+⇁AC _w+κ>ω is measurable iffκ is the successor of a singular cardinal”.     

BL代数的fantastic滤子和normal滤子

滤子是研究逻辑代数的有效工具.本文研究了BL代数的fantastic和normal滤子的等价条件,得到了在MV-代数中两种滤子之间的等价性,给出了两个公开问题:"在什么样的合适条件下,一个normal滤子成为一个fantastic滤子?"和"在什么合适的条件下,normal滤子的拓展性成立?"结论成立的一种条件.     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

Measurable cardinals and the continuum hypothesis

Let ZFM be the set theory ZF together with an axiom which asserts the existence of a measurable cardinal. It is shown that if ZFM is consistent then ZFM is consistent with every sentence φ whose consistency is proved by Cohen’s forcing method with a set of conditions of cardinality <k. In particular, if ZFM is consistent then it is consistent with the continuum hypothesis and with its negation. The research of the first named author has been sponsored in part by the Information Systems Branch, Office of Naval Research, Washington, D.C. under Contract F-61052 67 C 0055; the second named author was partially supported by an NAS-NRC post-doctoral fellowship and by National Science Foundation grant GP-5632.     

Indestructibility and destructible measurable cardinals

Say that \({\kappa}\)’s measurability is destructible if there exists a < \({\kappa}\)-closed forcing adding a new subset of \({\kappa}\) which destroys \({\kappa}\)’s measurability. For any δ, let λ_δ =_df The least beth fixed point above δ. Suppose that \({\kappa}\) is indestructibly supercompact and there is a measurable cardinal λ > \({\kappa}\). It then follows that \({A_{1} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ’s measurability is destructible when forcing with partial orderings having rank below λ_δ} is unbounded in \({\kappa}\). On the other hand, under the same hypotheses, \({A_{2} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ′s measurability is indestructible when forcing with either Add(δ, 1) or Add(δ, δ⁺)} is unbounded in \({\kappa}\) as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either \({A_{1} = \emptyset}\) or \({A_{2} = \emptyset}\). In each of these models, both of which have restricted large cardinal structures above \({\kappa}\), every measurable cardinal δ which is not a limit of measurable cardinals is δ⁺ strongly compact, and there is an indestructibly supercompact cardinal \({\kappa}\). In the model in which \({A_{1} = \emptyset}\), every measurable cardinal δ which is not a limit of measurable cardinals is <λ_δ strongly compact and has its <λ_δ strong compactness (and hence also its measurability) indestructible when forcing with δ-directed closed partial orderings having rank below λ_δ. The choice of the least beth fixed point above δ is arbitrary, and other values of λ_δ are also possible.     

Successors of singular cardinals and measurability

It is shown, starting from a model in which κ < λ, κ is 2^λ supercompact, and λ is a measurable cardinal, how to force and obtain a model in which the Axiom of Choice is false and in which the successor of a singular cardinal is measurable.