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Keywords:

  • Axiom of choice;
  • rigid relation;
  • weak choice principles;
  • MSC (2010) 03E25;
  • 03E35

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.