Amorphous sets and dual Dedekind finiteness

TL;DR

This work investigates finiteness notions without the axiom of choice, focusing on amorphous sets and dual Dedekind finiteness. It first exhibits a ZF-consistent example of an amorphous set $A$ whose power set $\mathscr{P}(A)$ is dually Dedekind infinite via a permutation-model construction and Jech–Sochor embedding. It then proves that any strictly (equivalently strongly) amorphous set $A$ satisfies that $\mathscr{P}(A)^n$ is dually Dedekind finite for all $n$, extending prior results and revealing a tight alignment between definability properties and dual finiteness. The paper also connects projective-type pregeometries to dual Dedekind finiteness and settles the strict-strong equivalence, while posing open questions about converses and generalizations to power-sets.

Abstract

A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists an amorphous set $A$ whose power set $\mathscr{P}(A)$ is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss [J. Truss, Fund. Math. 84, 187--208 (1974)]. Nevertheless, we prove in $\mathsf{ZF}$ that, for all strictly amorphous sets $A$ and all natural numbers $n$, $\mathscr{P}(A)^n$ is dually Dedekind finite, which generalizes a result of Goldstern.

Theorems & Definitions (19)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1