The finitary partitions with $n$ non-singleton blocks of a set

TL;DR

This paper investigates the cardinalities of finitary partitions with a fixed number of non-singleton blocks, proving results in ZF set theory and exploring their consistency with the absence of the axiom of choice.

Contribution

It establishes cardinality equalities for finitary partitions in ZF and demonstrates the consistency of certain inequalities among these cardinalities.

Findings

Proves in ZF that (2^{B_n(a)})^{aleph_0} = 2^{B_n(a)} for infinite a and n.

Shows that 2^{fin(a)^n} = 2^{B_{2^n-1}(a)}.

Establishes the consistency of a strict increasing sequence of cardinalities involving B_n(a).

Abstract

A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{fin}(\mathfrak{a})$ and $\mathscr{B}_{n}(\mathfrak{a})$ be the cardinalities of the set of finite subsets and the set of finitary partitions with exactly $n$ non-singleton blocks of a set which is of cardinality $\mathfrak{a}$, respectively. In this paper, we prove in $\mathsf{ZF}$ (without the axiom of choice) that for all infinite cardinals $\mathfrak{a}$ and all non-zero natural numbers $n$, \[ (2^{\mathscr{B}_{n}(\mathfrak{a})})^{\aleph_0}=2^{\mathscr{B}_{n}(\mathfrak{a})} \] and \[ 2^{\mathrm{fin}(\mathfrak{a})^n}=2^{\mathscr{B}_{2^n-1}(\mathfrak{a})}. \] It is also proved consistent with $\mathsf{ZF}$ that there exists an infinite cardinal $\mathfrak{a}$ such that \[ 2^{\mathscr{B}_{1}(\mathfrak{a})}<2^{\mathscr{B}_{2}(\mathfrak{a})}<2^{\mathscr{B}_{3}(\mathfrak{a})}<\cdots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}. \]