关于与$Q$ -集有关的临界基数
On critical cardinalities related to $Q$-sets
摘要：
在本文中，我们收集了一些已知信息并证明了有关小不可数基数$\mathfrak q_0$的新结果。基数$\mathfrak q_0$被定义为不是$Q$ -集的子集$A\subset \mathbb R$的最小基数$|A|$（如果每个子集$B\subset A$都是$A$中的$F_σ$类型，则子空间$A\subset\mathbb R$被称为$Q$ -集）。给出了$\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$的一个简单证明，并建立了基数$\mathfrak q_0$与其它标准不可数小基数之间的若干严格不等式的一致性.这是通过结合一些已知的强迫结果来完成的。本文的一个新结果是$\mathfrak{p} < \mathfrak{lr} < \mathfrak{q}_0$的相容性，其中$\mathfrak{lr}$表示线性加细数.另一个新的结果是上界$\mathfrak q_0\le\mathrm{non}(\mathcal I)$对$\mathbb R$上的任何$\mathfrak q_0$ -柔性cccc $σ$ -理想$\mathcal I$成立。
英文摘要：
In this note we collect some known information and prove new results about the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$ which is not a $Q$-set (a subspace $A\subset\mathbb R$ is called a $Q$-set if each subset $B\subset A$ is of type $F_σ$ in $A$). We present a simple proof of a folklore fact that $\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$, and also establish the consistency of a number of strict inequalities between the cardinal $\mathfrak q_0$ and other standard small uncountable cardinals. This is done by combining some known forcing results. A new result of the paper is the consistency of $\mathfrak{p} < \mathfrak{lr} < \mathfrak{q}_0$, where $\mathfrak{lr}$ denotes the linear refinement number. Another new result is the upper bound $\mathfrak q_0\le\mathrm{non}(\mathcal I)$ holding for any $\mathfrak q_0$-flexible cccc $σ$-ideal $\mathcal I$ on $\mathbb R$.
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。
