Equivalents of the axiom of choice

Parts of this note have been discussed elsewhere in the blog, sometimes in a different form (see here and here, for example), but I haven’t examined before the equivalence of 5–7 with choice. In the course of the proof of the equivalence of item 7 with the other statements, a few additional equivalent versions of independent interest will be identified. The equivalence of 1–3 is classic. Zermelo’s original axiomatization of set theory intended to formalize the proof that 2⇒ 1. Statements 2 and 4 are only irrelevantly different. The equivalence of 5 with 1 is due to Levy. On the other hand, the version of 5 where we simply require that each f(β) is finite does not suffice to recover choice. The equivalence of Tychonoff’s theorem and choice is due to Kelley. That the existence of bases implies choice is due to Blass, who proved that 7 implies the axiom of multiple choices. That this statement implies choice is due to Pincus. Pincus’s argument uses the axiom of foundation, and Levy showed that this is essential. The proof I indicate follows a suggestion of Felgner-Jech and uses a result of H. Rubin. Many other results have been studied and shown to be equivalent to choice. For example, the textbook mentions the innocuous looking statement that every nonempty set admits