for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 — for every set S with Hartogs number less than ωα, S is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely. Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization. The former is equivalent in ZF to Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter. Results requiring AC (or weaker forms) but weaker than it One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up. Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC). Equivalently, these statements are true in all models of ZFC but false in some models of ZF. Set theory The ultrafilter lemma (with ZF) can be used to prove the Axiom of choice for finite sets: Given and a collection of non-empty sets, their product is not empty. The union of any countable family of countable sets is countable (this requires countable choice but not the full axiom of choice). If the set A is infinite, then there exists an injection from the natural numbers N to A (see Dedekind infinite). Eight definitions of a finite set are equivalent. Every infinite game in which is a Borel subset of Baire space is determined. Measure theory The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable. The Hausdorff paradox. The Banach–Tarski paradox. Algebra Every field has an algebraic closure. Every field extension has a transcendence basis. Every infinite-dimensional vector field contains an infinite linearly independent subset (this requires dependent choice, but not the full axiom of choice). Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem. The Nielsen–Schreier theorem, that every subgroup of a free group is free. The additive groups of R and C are isomorphic. Functional analysis The Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals The theorem that every Hilbert space has an orthonormal basis. The Banach–Alaoglu theorem about compactness of sets of functionals. The Baire category theorem