##### 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