The Rigid Relation Principle, a New Weak Choice Principle

TL;DR

The paper introduces the Rigid Relation Principle, a new weak choice principle asserting every set admits a rigid binary relation, which is independent of ZF and not equivalent to the axiom of choice.

Contribution

It establishes the Rigid Relation Principle as a new weak choice principle, independent of ZF and distinct from the axiom of choice.

Findings

The principle is neither equivalent to the axiom of choice nor provable in ZF.

It is provable for sets of reals without the axiom of choice.

The principle introduces a new perspective on weak choice principles.

Abstract

The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals (among other general instances) is provable without the axiom of choice.