滤子中的塔、基数不变量和Luzin型族
Towers in filters, cardinal invariants, and Luzin type families
摘要：
我们调查$ω$上的过滤器可以包含塔，也就是说，没有任何伪交叉（在$[ω]^ω$）的模有限下降序列。我们证明了以下结果：-许多经典的例子很好的高过滤器不包含塔（ZFC）。- 高解析P-滤子包含任意正则高度的塔是一致的（同时对于许多正则基数也是）。- 所有塔生成非贫乏过滤器是一致的，特别是（一致地）Borel过滤器不包含塔。- “每个超滤器都包含塔。“它独立于ZFC。此外，我们研究了许多可能的逻辑（非）之间的影响存在的塔过滤器，不平等之间的基数不变量的过滤器（$\mbox{add}^*(\mathcal F)$，$\mbox{cof}^*(\mathcal F)$，$\mbox{non}^*(\mathcal F)$，和$\mbox{cov}^*(\mathcal F)$），和存在的Luzin型家庭（大小$\geq ω_2$），即，如果$\mathcal F$是一个过滤器，那么$X\subseteq [ω]^ω$是一个$\mathcal F$-Luzin家庭，如果$\{A\in X:|A\setminus F|=ω\}$是可数的每$F\in \mathcal F$。
英文摘要：
We investigate which filters on $ω$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[ω]^ω$). We prove the following results: - Many classical examples of nice tall filters contain no towers (in ZFC). - It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). - It is consistent that all towers generate non-meager filters, in particular (consistently) Borel filters do not contain towers. - The statement "Every ultrafilter contains towers." is independent of ZFC. Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters ($\mbox{add}^*(\mathcal F)$, $\mbox{cof}^*(\mathcal F)$, $\mbox{non}^*(\mathcal F)$, and $\mbox{cov}^*(\mathcal F)$), and the existence of Luzin type families (of size $\geq ω_2$), that is, if $\mathcal F$ is a filter then $X\subseteq [ω]^ω$ is an $\mathcal F$-Luzin family if $\{A\in X:|A\setminus F|=ω\}$ is countable for every $F\in \mathcal F$.
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