论文标题
过滤器和理想独立性
Filters and Ideal Independence
论文作者
论文摘要
一个家庭$ \ MATHSCR {i} \ subseteq [ω]^ω$，使得对于所有有限的$ \ {x_i \} _ {i \ in n} \ subseteq \ subseteq \ mathcal i $ and $ a \ in \ sebscr i $ $ a \ setminus \ bigcup_ {i <n} x_i $是无限的，据说是理想的独立性。一个理想的独立家族，最大程度地说是一个最大的理想独立家庭，这种家族的最低基数表示为$ \ mathfrak {s} _ {mm} $。 我们表明$ \ mathfrak {u} \ leq \ mathfrak {s} _ {mm} $，特别是建立了$ \ mathfrak {s} _ {mm} $和$ \ mathfrak {i} $的独立性。考虑到任意套件的无数枢机主教的$ c $，我们展示了如何通过强迫C $中每个$λ\的最大理想独立家庭$λ$同时搭配，从而确立了$ c \ subseteq \ subseteq \ hox \ hbox \ hbox {spec}}（spec}（spec}）（\ mathfrak {s} {s} {s} _ {mmmmmmmmmm}））。假设$ \ mathsf {ch} $，我们构建了一个最大的理想独立家族，在用任何适当的，$^ωΩ$结合强迫后，它仍然保持最大，$ p $ - 点保存强迫概念并评估$ \ mathfrak {s} _ {mmm} _ {mmm} $在几个精心研究的锻造伪造分机中。
A family $\mathscr{I} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted $\mathfrak{s}_{mm}$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which in particular establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $λ$ for each $λ\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^ωω$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions.
