Covering of the null ideal may have countable cofinality

Template iterations and maximal cofinitary groups

The main result of the present paper is that $\mathfrak a_g$, the minimal size of maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial properties allowing it to be used within a similar template forcing construction. Additionally we obtain that $\mathfrak a_p$, the minimal size of a maximal family of almost disjoint permutations, and $\mathfrak a_e$, the minimal size of a maximal eventually different family, can be of countable cofinality.

On automorphisms groups of structures of countable cofinality

In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in $ω^ω$. (III) There is an $L_{ω_1,ω}$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $κ$ of of cofinality $ω$ (cf. theorem 4.2). The following are equivalent: (I$^*$) There is a model of the Scott sentence of $M$ of size…

在可数余数的奇异基数上的一般描述集合论

Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality

本文在可数余数的奇异基数上发展了描述集合论，研究了λ-波兰空间、λ-博雷尔空间及相关集合的性质，并推广了经典完美集性质。

极限滤子与可数支持对称迭代中的依赖选择

Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations

本文研究了可数支持对称迭代中极限滤子的构造及其在ZF和DC中的应用，证明了特定模型满足ZF和DC，并展示了ω₁-完备滤子对控制AC失败程度的重要性。

Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering

We improve previous work on the consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality building on previous work by Adolf, Cox and Welch. Specifically, we have greatly reduced our reliance on covering properties in the proof. This will allow us to handle sequences in which sets concentrating on points of countable cofinality appear infinitely often. Furthermore we will show that if $κ$ is a Jónsson cardinal with $κ< \aleph_κ$ then the sharp for a model with a strong cardinal exists.

Sigma-Prikry forcing I: The Axioms

We introduce a class of notions of forcing which we call $Σ$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $Σ$-Prikry. We show that given a $Σ$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $Σ$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $Σ$-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $κ$ is the…

On the spectrum of characters of ultrafilters

We show that the character spectrum $Sp_χ(λ)$, for a singular cardinal $λ$ of countable cofinality, may include any prescribed set of regular cardinals between $λ$ and $2^λ$.

$\lambda$-$\Pi^1_1$ 集合上的 $\lambda$-PSP

The $λ$-PSP at $λ$-$Π^1_1$ sets

研究可数共尾强极限基数λ处广义Cantor空间的λ-Π^1_1子集的λ-PSP性质，证明该性质蕴含0†存在或存在具有可测基数的内模型，为广义描述集合论中正则性性质的一致性强度论证提供工具。

Canonical models for aleph_1 combinatorics

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…

Reflection implies the SCH

We prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every stationary family of countable subsets of mu^+ reflect in some subset of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad stationary set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four stationary sets.

Long Borel Hierarchies

We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $ω_2$. This implies that $ω_1$ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than $ω_2$, e.g., $ω$ or $ω_1+ω_1$. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~miller

The structure of random automorphisms of countable structures

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different …

Proper forcing and rectangular Ramsey theorems

I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.

Hechler's theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing, and the statement of the theorem for the meager ideal has been already proved by Bartoszynski and the author.

Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the $c_0$-saturation of $C(K)$, $K$ countable compacta. The second one is the construction of weakly-null sequences generalizing the example of Maurey-Rosenthal.

理想化Miller空集的基数不变量

Cardinal invariants of idealized Miller null sets

本文研究了理想化Miller空集的基数不变量，通过计算典型Borel理想参数化的$σ$-理想，扩展了Cichoń的最大值。

普遍空集上的基数不变量

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。

某类Hausdorff测度的可加性可与Lebesgue测度的可加性不同

The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure

本文通过Tukey归约将Davies-Rogers Hausdorff测度零理想与局部化系统关联，证明其可加性和共尾性可与Lebesgue零理想不同。

Nicely generated and chaotic ideals

We show that if the real line is the disjoint union of kappa meager sets such that every meager set is contained in a countable union of them, then kappa = omega_1. This answers a question addressed by J. Cichon. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for "meager" replaced by "null", as well.

Hurewicz-like tests for Borel subsets of the plane

Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to recognize potentially $\bormxi$ sets.