Properties of the real line and weak forms of the Axiom of Choice

摘要

We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. This paper is a continuation of [6] and [10] and is concerned with Paul Howard's and Jean E. Rubin's project "Consequences of the Axiom of Choice"[7]. Our aim is to clarify the interrelations between weak forms of the Axiom of Choice restricted to the real line R.