On Martin's Axiom and Forms of Choice

摘要

Martin's Axiom MA is the statement that for every well-ordered cardinal kappa < 2(X0), the statement MA(kappa) holds, where MA(kappa) is "if (P, <=) is a c.c.c. quasi order and V is a family of <= kappa dense sets in P, then there is a V-generic filter of P". In ZFC, the fragment MA(X-0) is provable, but not in general in ZF. In this paper, we investigate the interrelation between MA(X-0) and various choice principles.