The axiom of choice asserts that it is possible to choose an element from every nonempty set. More precisely, given any set E, the aim is to define a choice function on E, that is, a function whose domain is P\mathcal{P}\*(E), the set of nonempty subsets of E, and whose codomain is E.
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To choose is to imagine… ------------------------
The defining feature of a choice function is that it assigns to each subset in P\mathcal{P}\(E) one of that subset's elements. For example, if E = {1, 2, 3, 4, 5}, we can define a choice function by assigning its smallest element to every subset of E. This idea extends to any set whose elements can be numbered, that is, any finite or countable set. More generally, this argument shows that we can construct a choice function on well-ordered sets, in which every nonempty subset has a least element. Such an order is then called a well-order*. Sadly, well-orders are relatively rare among the orders we commonly use!
Apart from the argument for well-ordered sets, it is hard to see how one could assert the existence of a choice function on an arbitrary set. That is why, in 1904, Ernst Zermelo introduced an additional axiom alongside the set-theoretic axioms now known as the Zermelo–Fraenkel axioms (ZF): the axiom of choice, which states that every set has a choice function.