Bounding and dominating number of families of functions on N

重新审视通过区间结构的$\mathfrak b$和$\mathfrak d$

Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures

本文通过区间结构研究了$\mathfrak b$和$\mathfrak d$的泛化，发现普遍变体在多种设置下保持经典不变量，而存在变体则反转这些不变量。

Higher Dimensional Cardinal Characteristics for Sets of Functions II

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:ω^ω\to ω^ω$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg\mathsf{CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, with possibly one exception, for the bounding numbers there are no $\mathsf{ZFC}$ relations between them beyond those in the higher dime…

More on cardinal invariants of analytic P-ideals

Given an ideal $I$ on $ω$ let $a(I) $ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[ω]^ω$, and denote $b_I$ and$d_I$ the unbounding and dominating numbers of $(ω^ω,\le_I)$. We show that (1) $a(I)>omega$ if $I$ is a summable ideal; (2) $a(Z)=ω$ and $\bar{a}(Z)\le a$ if $Z$ is a tall density ideal, (3) $b\le \bar{a}(I)$, and $b_I=b$ and $d_I=d$, for any analytic P-ideal $I$ on $ω$. Given an analytic $P$-ideal $I$ we investigate the relationship between the Sack, the $I$-bounding, $I$-dominating and $ω^ω$-bounding properties of a…

Covering the Baire space by families which are not finitely dominating

It is consistent (relative to ZFC) that the union of max{b,g} many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U, the cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.

Bounding, splitting, and almost disjointness

We investigate some aspects of bounding, splitting, and almost disjointness. In particular, we investigate the relationship between the bounding number, the closed almost disjointness number, splitting number, and the existence of certain kinds of splitting families.

A Lower Bound for Generalized Dominating Numbers

We show a new proof for the fact that when $κ$ and $λ$ are infinite cardinals satisfying $λ^ κ= λ$, the cofinality of the set of all functions from $λ$ to $κ$ ordered by everywhere domination is $2^λ$. An earlier proof was a consequence of a result about independent families of functions. The new proof follows directly from the main theorem we present: for every $A \subseteq λ$ there is a function $f: {^κλ} \to κ$ such that whenever $M$ is a transitive model of $\textrm{ZF}$ such that ${^κλ} \subseteq M$ and some $g: {^κλ} \to κ$ in $M$ dominates $f$, then $A \in M$. That is, "constructibility…