A Choice-Free Cardinal Equality

Abstract

For a 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 is proved consistent with $\mathsf{ZF}$ that there exists an infinite cardinal $\mathfrak{a}$ such that

$$2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}{(\mathfrak{a})}^2}<2^{\mathrm{fin}{(\mathfrak{a})}^3}<\cdots <2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}.$$