We study the reverse mathematics of the principle stating that,for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey’s lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.
D. D. Dzhafarov and C. Mummert, Reverse mathematics and properties of finite character, Annals of Pure and Applied Logic. 163 (2012), 1243–1251.