Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	申国桢博士，中科院数学院
Inviter:
Title:	A choice-free cardinal equality (II)
Time & Venue:	2019.12.09 09:00-11:00 N602
Abstract:	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 $2^{\mathrm{fin}(\mathfrak{a})}<2^{(\mathrm{fin}(\mathfrak{a}))^2}<2^{(\mathrm{fin}(\mathfrak{a}))^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}$.