  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.
