A Generalization of the Trichotomy Principle

Jaruwat Rodbanjong, Pimpen Vejjajiva

Abstract

We write X ≼ Y if there is an injection from a set X into a set Y and
write X ≼ Y if X = ∅ or there is a surjection from Y onto X. For any sets X and
Y , X ≼ Y implies X ≼ Y but the converse cannot be proved without the Axiom
of Choice (AC). The Trichotomy Principle, which states that for any sets X and
Y , X ≼ Y or Y ≼ X, is an equivalent form of AC. Surprisingly, the statement
is still equivalent to AC when ≼ is replaced by ≼. Moreover, it has been shown
that the k-Trichotomy Principle, which states that every family of sets which is of cardinality k contains two distinct sets X and Y such that X ≼ Y , is equivalent to AC when k is any natural number greater than 1. In this paper, we show that the statement is also equivalent to AC when ≼ is replaced by ≼.