Choice Function - History and Importance

History and Importance

Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem, which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (AC_ω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or AC_ω, some sets can still be shown to have a choice function.

• If is a finite set of nonempty sets, then one can construct a choice function for by picking one element from each member of This requires only finitely many choices, so neither AC or AC_ω is needed.

• If every member of is a nonempty set, and the union is well-ordered, then one may choose the least element of each member of . In this case, it was possible to simultaneously well-order every member of by making just one choice of a well-order of the union, so neither AC nor AC_ω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)