中科院数学与系统科学研究院
数学研究所
集合论研讨班报告
报告人： 申国桢 博士（中科院数学院）
题 目：A choice-free cardinal equality (II)
时 间：2019.12.05（星期四）, 09:00-11:00
地 点：数学院南楼N602室
摘 要：Fora cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\mathfrak{a}$ and all natural numbers $n$, $2^{(\mathrm{fin}(\mathfrak{a}))^n}=2^{[\mathrm{fin}(\mathfrak{a})]^n}$.On the other hand, it may consistently happen that there exists an infinite cardianl $\mathfrak{a}$ such that $2^{\mathrm{fin}(\mathfrak{a})}<2^{(\mathrm{fin}(\mathfrak{a}))^2}<2^{(\mathrm{fin}(\mathfrak{a}))^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}$.
