On the comparability of cardinals in the absence of the axiom of choice

摘要

It is a well-known result of Hartogs' that the statement "for all sets x and y, x = y or y = x (where 'x = y' means that there is a one-to-one map f : x - y)" is equivalent to the Axiom of Choice (AC) (the latter in the disguise of the well-ordering theorem, i.e., "every set can be well-ordered"). A considerably stronger result by Tarski states that for any natural number n = 2, the statement "if x is a set consisting of n sets, then there exist distinct elements y, z is an element of x such that y = z or z = y" is equivalent to AC.