Cantor's theorem may fail for finitary partitions

TL;DR

The paper investigates the properties of finitary partitions of infinite sets under ZF set theory, showing that Cantor's theorem may not hold for these partitions and establishing related results.

Contribution

It demonstrates the consistency of surjections from an infinite set onto its finitary partitions without the axiom of choice and proves several theorems about their cardinalities.

Findings

Existence of surjections from A onto finitary partitions without choice

No surjections from A onto finitary partitions if certain partitions exist

Cardinality comparisons between finitary partitions, sequences, and Cartesian powers

Abstract

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite set $A$ and a surjection from $A$ onto $\mathscr{B}(A)$. On the other hand, we prove in $\mathsf{ZF}$ some theorems concerning $\mathscr{B}(A)$ for infinite sets $A$, among which are the following: (1) If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$ and no finite-to-one functions from $\mathscr{B}(A)$ to $A$. (2) For all $n\in\omega$, $|A^n|<|\mathscr{B}(A)|$. (3) $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$, where $\mathrm{seq}(A)$ is the set of all finite sequences of elements of $A$.