Pierre Matet, Andrzej Rosłanowski, Saharon Shelah
We show that the reduced cofinality of the nonstationary ideal NS_kappa on a regular uncountable cardinal kappa may be less than its cofinality, where the reduced cofinality of NS_kappa is the least cardinality of any family F of nonstationary subsets of kappa such that every nonstationary subset of kappa can be covered by less than $κ$ many members of F.
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We investigate the cofinality of the strong measure zero ideal for $κ$ inaccessible, and show that it is independent of the size of $2^κ$.
We prove that it is consistent that the covering of the ideal of measure zero sets has countable cofinality.
We answer a question of Shelah by showing that it is consistent that every set of ordinals of cofinality omega_1 in I[omega_2] is nonstationary if and only if it is consistent that that there is a kappa^+ Mahlo cardinal kappa.
In this paper we present a simpler proof of the fact that no inequality between $\mathrm{cof}(\mathcal{SN})$ and $\mathfrak{c}$ can be decided in ZFC by using well-known tecniques and results.
Given an ordinal delta <= lambda and a cardinal theta <= kappa, an ideal J on P_kappa(lambda) is said to be [delta]^{<theta}-normal if given B_e in J for e in P_theta(delta), the set of all a in P_kappa(lambda) such that a in B_e for some e in P_{|a cap theta|}(a cap delta) lies in J. We give necessary and sufficient conditions for the existence of such ideals and we describe the least one and we compute its cofinality.
We study the ideal of meager sets and related ideals.
Yorioka [J. Symbolic Logic 67(4):1373-1384, 2002] introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in ZFC. We construct a matrix iteration of ccc posets to force that, for many ideals in that class, their associated cardinal invariants (i.e. additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with t…
Relative cofinality of ideals
该研究引入理想共尾性不变量的双参数修正版本，纳入理想对相互作用，确定ω上关键理想对的不变量值，二分实数线理想对并研究极大理想的相对共尾性，推进基类结构研究。
We apply Neeman's method of forcing with side conditions to show that PFA does not imply the precipitousness of the nonstationary ideal on $ω_1$.
Let kappa be a regular uncountable cardinal and lambda > kappa a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of P_kappa (lambda) and use this to prove that the nonstationary ideal on P_kappa (lambda) is nowhere precipitous.
PFA and the definability of the nonstationary ideal
该研究在含超紧基数的ZFC模型基础上构造出满足PFA的模型，证明ω₁上的非平稳理想在H_{ℵ₂}中可由参数Π₁-定义，解决了PFA与非平稳理想可定义性的关联问题。
We show that, under the assumption of the existence of $M_1^{\#}$, there exists a model on which the restricted nonstationary ideal $\hbox{NS} \upharpoonright A$ is $\aleph_2$-saturated, for $A$ a stationary co-stationary subset of $ω_1$, while the full nonstationary ideal $\hbox{NS}$ can be made $Δ_1$ definable with $K_{ω_1}$ as a parameter. Further we show, again under the assumption of the existence of $M_1^{\#}$ that there is a model of set theory such that $\hbox{NS}$ is $\aleph_2$-saturated and such that there is lightface $Σ^1_4$-definable well-order on the reals. This result is optimal…
Fragments of SRP, the nonstationary ideal, and diagonal reflection
该研究探讨SRP片段与对角反射原理的关系，发现结合CH的SRP亚完备片段可推出ω₁非平稳理想非饱和且OSR_{ω₂}成立，完整SRP则推出其饱和但不蕴含OSR_{ω₂}，还揭示了两类原理配对的不同模型结果。
We are proving the following: (1) If $\kap$ is a weakly inaccessible then $NS_\kap$ is not $\kap^+$-saturated. (2) If $\kap$ is a weakly inaccessible and $\tet <\kap$ is regular then $NS^\tet_\kap$ is not $\kap^+$-saturated. (3) If $\kap$ is singular then $NS^{cf\kap}_{\kap^+}$ is not $\kap^{++}$-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If $\kap >\aleph_1$ then $NS_\kap$ is not $\kap^+$-saturated. (B) If $\tet^+<\kap$ then $NS^\tet_\kap$ is not $\kap^+$-saturated.
Burke \cite{MR1472122} proved that the generalized nonstationary ideal, denoted NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of $\text{NS}$ to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \emph{$\mathcal{C}$-systems of filters} introduced by Audrito-Steila \cite{AudritoSteila}. First we answer a question of \cite{AudritoSteila}, by proving that $\mathcal…
We define the property of Pi_2-compactness of a statement phi of set theory, meaning roughly that the hard core of the impact of phi on combinatorics of aleph_1 can be isolated in a canonical model for the statement phi. We show that the following statements are Pi_2-compact: ``dominating number = aleph_1,'' ``cofinality of the meager ideal = aleph_1'', ``cofinality of the null ideal = aleph_1'', existence of various types of Souslin trees and variations on uniformity of measure and category = aleph_1. Several important new metamathematical patterns among classical statements of set theory are…
Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$
该数理逻辑研究探讨力迫公理下$\omega_1$非驻定理想的可定义性，证明$\text{BMM}$加武丁基数、$(*)$-公理下其不可被$\Sigma_1(\omega_1)$定义，同时构造出$\text{BPFA}$、$\text{MA}$等公理下其可定义甚至饱和的集合论宇宙。
More about the cofinality and the covering of the ideal of strong measure zero sets
本文通过新组合原理 $\mathrm{DS}(δ)$、关系系统乘积及力迫迭代，改进强测度零集理想 $\mathcal{SN}$ 的共尾度与覆盖数界，并证明多种基数特征分离的一致性。
We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal $κ$, our goal being to witness the nonsaturation of $J$ by the existence of towers (of length possibly greater than $κ^+$).
The cardinal characteristics of the ideal generated by the $F_σ$ measure zero subsets of the reals
本综述研究由实数 $F_σ$ 零测度子集生成的理想 $\mathcal{E}$ 的基数特征，涵盖其加性、覆盖数、一致性与共尾性。
Normal ideals on regular uncountable cardinals are familiar objects. We investigate ideals that are pleasant--while a normal ideal is closed under arbitrary diagonal unions, a pleasant ideal is closed only under diagonal unions indexed by sets that are elements of the ideal. We show any selective ideal extending the nonstationary ideal must be normal.
We investigate generalizations of the topology of the higher Cantor space on $2^κ$, based on arbitrary ideals rather than the bounded ideal on $κ$. Our main focus is on the topology induced by the nonstationary ideal, and we call this topology the nonstationary topology, or also the Edinburgh topology on $2^κ$. It may be of independent interest that as a side result, we show $κ$-Silver forcing to satisfy a strong form of Axiom $A$ not only if $κ$ is inaccessible (which is well-known), but also under the assumption $\diamondsuit_κ$.
$μ$-Clubs of $P_κ(λ)$ : Paradise in heaven
该研究在无大基数内模型等条件下，证明了$P_κ(λ)$与$P_κ(u(κ,λ))$上的$μ$-俱乐部滤子同构，且其对偶理想不满足特定饱和性，推进了无穷基数组合相关研究。
We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show that large pcf(a) implies existence of free sets. An example is that if pp(aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l<omega}). Then we present results complementary to those of section 2 (but not close enough): if IND…
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