强独立性及其谱

摘要：

对于$μ, κ$无限，说$\mathcal{A}\subseteq [κ]^κ$是$(μ,κ)$ -极大独立族，如果只要$\mathcal{A}_0$和$\mathcal{A}_1$在$[\mathcal{A}]^{<μ}$中成对不相交非空，则$\bigcap\mathcal{A}_0\backslash\bigcup\mathcal{A}_1 \not= \emptyset$，$\mathcal{A}$在具有此性质的族中的包含下是最大的，并且所有这样的Booelan组合的大小都是$κ$。我们用$\mathfrak{sp}_{\mathfrak i}(μ,κ)$表示这类族的所有基数的集合，如果非空，我们让$\mathfrak{i}_μ(κ)$是它的最小元素。因此，$\mathfrak{i}_μ(κ)$（如果定义了）是高阶Baire空间中$ω$上独立数的自然高阶模拟。本文研究了不可数$μ,κ$的$\mathfrak{sp}_{\mathfrak i}(μ,κ)$。其中，我们证明了：（1）性质$\mathfrak{sp}_{\mathfrak i}(μ,κ)\neq\emptyset$不能由ZFC加大基数决定. (2)相对于可测量的，它是一致的，即：（a）$(\exists κ{>}ω) \, \mathfrak{i}_κ(κ)<2^κ$ ;（b）$(\exists κ{>}ω)\,κ^+<\mathfrak{i}_{ω_1}(κ)<2^κ$。据作者所知，这是第一个例子的$(μ,κ)$ -最大独立家庭的大小严格介于$κ^+$和$2^κ$，不可数$κ$。(3)$\mathfrak{sp}_{\mathfrak i}(μ,κ)$不能是任意的。

英文摘要：

For $μ, κ$ infinite, say $\mathcal{A}\subseteq [κ]^κ$ is a $(μ,κ)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<μ}$ then $\bigcap\mathcal{A}_0\backslash\bigcup\mathcal{A}_1 \not= \emptyset$, $\mathcal{A}$ is maximal under inclusion among families with this property, and moreover all such Booelan combinations have size $κ$. We denote by $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ the set of all cardinalities of such families, and if non-empty, we let $\mathfrak{i}_μ(κ)$ be its minimal element. Thus, $\mathfrak{i}_μ(κ)$ (if defined) is a natural higher analogue of the independence number on $ω$ for the higher Baire spaces. In this paper, we study $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ for $μ,κ$ uncountable. Among others, we show that: (1) The property $\mathfrak{sp}_{\mathfrak i}(μ,κ)\neq\emptyset$ cannot be decided on the basis of ZFC plus large cardinals. (2) Relative to a measurable, it is consistent that: (a) $(\exists κ{>}ω) \, \mathfrak{i}_κ(κ)<2^κ$; (b) $(\exists κ{>}ω)\,κ^+<\mathfrak{i}_{ω_1}(κ)<2^κ$. To the best knowledge of the authors, this is the first example of a $(μ,κ)$-maximal independent family of size strictly between $κ^+$ and $2^κ$, for uncountable $κ$. (3) $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ cannot be quite arbitrary.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。