Additionally, consider for instance the unit circle "S", and the action on "S" by a group "G" consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of "π". Here "G" is countable while "S" is uncountable. Hence "S" breaks up into uncountably many orbits under "G". Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset "X" of "S" with the property that all of its translates by G are disjoint from "X". The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent. Since "X" is not measurable for any rotation-invariant countably additive finite measure on "S", finding an algorithm to select a point in each orbit requires the axiom of choice. See non-measurable set for more details.