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 (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, 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 ACω 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 ACω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)
Read more about this topic: Choice Function
Famous quotes containing the words history and/or importance:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“Coming together again after a long day apart can be an experience where joy, relief, anger, and fatigue are all present in different degrees both for the parent and for the child. Because of their importance in marking the resumption of direct contact, reunions deserve as much attention and care as separations to enhance the relationship between parent and child.”
—Alicia F. Lieberman (20th century)