A choice-free cardinal equality

TL;DR

This paper proves a cardinal equality involving finite subsets without the axiom of choice and demonstrates the consistency of a strict increasing sequence of power set cardinalities for certain infinite cardinals.

Contribution

It establishes a choice-free proof of a cardinal equality and shows the consistency of a strict hierarchy of power set cardinalities under ZF.

Findings

Proved a cardinal equality without the axiom of choice.

Established the consistency of a strict increasing sequence of power set sizes.

Demonstrated independence results related to cardinal exponentiation.

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 that the following statement is consistent with $\mathsf{ZF}$: there exists an infinite cardinal $\mathfrak{a}$ such that \[ 2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}(\mathfrak{a})^2}<2^{\mathrm{fin}(\mathfrak{a})^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}. \]