Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a ⊆-maximal subfamily with the finite intersection property and the principle asserting that if P is a property of finite character then every set has a ⊆-maximal subset of which P holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to Z2 to being weaker than ACA0 and incomparable with WKL0. In particular, we identify a choice principle that, modulo \Sigma^0_2 induction, lies strictly below the atomic model theorem principle AMT and implies the omitting partial types principle OPT.
D. D. Dzhafarov and C. Mummert, Reverse mathematics and equivalents of the axiom of choice, arXiv preprint arXiv:1009.3242. 2010 Sep 16.