Saharon Shelah
Our main result is that possibly some non-null set of reals cannot be divided to uncountably many non-null sets. We deal also with a non-null set of reals, the graph of any function from it is null and deal with our iterations somewhat more generally.
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In a countable superstable NDOP theory, the existence of a rigid aleph_epsilon-saturated model implies the existence of 2^lambda rigid aleph_epsilon-saturated models of power lambda for every lambda>2^{aleph_0}.
After the main gap theorem was proved (see [Sh:c]), in discussion, Harrington expressed a desire for a finer structure - of finitary character (when we have a structure theorem at all). I point out that the logic L_{infty,aleph_0}(d.q.) (d.q. stands for dimension quantifier) does not suffice: e.g., for T=Th(lambda x 2^ω,E_n)_{n<omega} where (alpha,eta)E_n(beta,nu) =: eta|n=nu|n and for a subset S of 2^omega we define M_S = M | {(alpha,eta): [eta in S -> alpha<omega_1] and [eta in 2^ωbackslash S -> alpha<omega]}. Hence, it seems to me we should try L_{infty,aleph_epsilon}(d.q.) (essentially, in…
We give necessary and sufficient conditions for a polynomially bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be $\aleph_α$-saturated. The conditions are in terms of the value group, residue field, and pseudo- Cauchy sequences of the natural valuation on the real closed field. This is achieved by an analysis of types, leading to the trichotomy. Our characterization provides a construction method for saturated models, using fields of generalized power series.
Assume a complete superstable theory is superstable, and let P be a class of regular types, typically closed under automorphisms of the monster and non-orthogonality. We define the notion of P-NDOP and prove the existence of P-decompositions and derive an analog of Sh401 for superstable theories with P-NDOP. In this context, we also find a sufficient condition on P-decompositions that imply non-isomorphic models. For this, we investigate natural structures on the types in P\intersect S(M) modulo non-orthogonality.
In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.
Assuming three strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\] Under the same assumption, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
We show that Kunen's saturated ideal over $\aleph_1$ is not centered. We also evaluate the extent of saturation of Laver's saturated ideal in terms of $(κ,λ,<ν)$-saturation.
We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) <\mathrm{add}(\mathrm{Meager})=\mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) < \mathrm{non}(\mathrm{Null}) < \mathrm{cof}(\mathrm{Meager})= \mathfrak{d} < \mathrm{cof}(\mathrm{Null}) < 2^{\aleph_0}. \]
$\text{NS}_{ω_1}$ saturated, $Δ_1 ( \{ ω_1 \} )$-definable and a $Δ^1_4$-definable well-order of the reals
本文在假设含一个 Woodin 基数的典范内模型 $M_1$ 存在的前提下，构造了一个使 $NS_{ω_1}$ 为 $\aleph_2$-饱和且 $Δ_1$-可定义、同时存在 $Σ^1_4$-可定义实数良序的模型，从而说明饱和与可定义性假设不能推出射影集具有良好结构性质。
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
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.
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…
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…
Assuming four strongly compact cardinals, it is consistent that all entries in Cichoń's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
Saturation of reduced products
本文研究可数结构约化积的饱和性，证明稳定性决定其饱和程度及与超幂同构的条件，并将结论推广到 $F_\sigma$、分层及解析 P-理想。
Cichoń's maximum with cardinals of the closed null ideal
本文研究了闭零测集理想 $\mathcal{E}$ 的基数特征，证明其一致性与覆盖数可作为独立取值加入 Cichoń 最大值，扩展了经典基数不等式链的一致性结果。
Forcing and cardinal invariants of universally null sets
本文研究普遍空集上的基数不变量，证明了多个相关不变量之间的不等式关系，并探讨了在特定假设下的一致性结果。
This is a continuation of "Some results on nonstationry ideal". The upper bound on precipitousness of NS_lambda^+ for a regular lambda given in this paper is proved to be exact.It is shown that saturatedness of NS_kappa^aleph_0 over inaccessible kappa requires at least o(kappa)=kappa^++.The upper bounds on the strength of NS_kappa precipitous for inaccessible kappa are reduced quite close to the lower bounds.
Countably perfectly meager and countably perfectly null sets
本文通过引入可数完全贫乏集和可数完全零集以强化普遍贫乏集与普遍零集概念，并在连续统假设弱化条件下证明了二者不等价。
Generalizing random real forcing for inaccessible cardinals
本文针对不可达基数$\u03bb$对应的$\u03bb^\u03bb$空间，在附加假设下构造了具备随机实数力迫性质的推广力迫，证明其保持基数与共尾性且不添加无支配实数，填补了随机实数力迫在高阶无穷空间的推广空白。
Saccharinity with ccc
本文利用creatures技术构造了若干Suslin ccc非甜forcing概念，探讨了ZFC与ZF+实数集等于Borel集模null理想闭包的等价性，以及ω₁序列的构造。
Contents of this issue: Workshops on SPM themes; Second workshop on Coverings, Selections and Games in Topology (SPM05); Analysis and Descriptive Set Theory Workshop; Descriptive set theory: Effective methods, equivalence relations; Analysis: typical functions, level sets structure; Organizational and additional details; Research announcements; Cardinal restrictions on some homogeneous compacta; Filters: Topological congruence relations on groups; Inverse Limits and Function Algebras; Ultrafilters and partial products of infinite cyclic groups; More on regular reduced products; Consistency of …
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