Infinity Without Choice

Abstract

With the Axiom of Choice (AC), if a set A is infinite, then there is a one-to-one function from ω into A, written ω<A, and so there is a function from A onto ω, written ω< A. Without AC, these no longer hold. Therefore, there are many kinds of infinite sets in the absence of AC. We call A Dedekind-infinite if ω< A and weakly Dedekind-infinite if ω< A, otherwise A is called Dedekind-finite and weakly Dedekind-finite, respectively. A is a Dedekind set if A is infinite Dedekind-finite and A is a weakly Dedekind set if A is infinite weakly Dedekindfinite. We investigate the cardinals of such sets and show which properties can be proved from ZF and, provided that ZF is consistent, which properties are consistent with ZF and therefore their negations cannot be proved without AC.