The Axiom of Choice for Collections of Finite Sets

Abstract

Some implications among finite versions of the Axiom of Choice are considered. In the first of two chapters some theorems are proven concerning the dependence or independence of these implications on the theory ZFU, the modification of ZF which permits the existence of atoms. The second chapter outlines proofs of corresponding theorems with "ZFU" replaced by "ZF" . The independence proofs involve Mostowski type permutation models in the first chapter and Cohen forcing in the second chapter. The finite axioms considered are C^n , "Every collection of n-element sets has a choice function"; W^n, "Every well-orderable collection of n-element sets has a choice function"; D^n, "Every denumerable collection of n-element sets has a choice function"; and A^n (x), "Every collection Y of n-element sets, with Y ≈ X, has a choice function". The conjunction C^nl &...& C^nk is denoted by CZ where Z = {nl ,...,nk}. Corresponding conjunctions of other finite axioms are denoted similarly by Wz, Dz and Az (X). Theorem: The following are provable in ZFU: W^k1n1+...+krnr ➔ W^n1 v...v W^nr, D^k1n1+...+krnr ➔ D^n1 v...v D^nr, and C^k1n1+...+krnr ➔ C^n1 v W^n2 v...v W^nr