Defining The Ordered Pair Using Set Theory
The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property. This was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954, long after Kuratowski discovered his reduction (below). The Kuratowski definition was added in the second edition of Theory of Sets, published in 1970.
If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below.
Read more about this topic: Ordered Pair
Famous quotes containing the words defining the, defining, ordered, pair, set and/or theory:
“Art, if one employs this term in the broad sense that includes poetry within its realm, is an art of creation laden with ideals, located at the very core of the life of a people, defining the spiritual and moral shape of that life.”
—Ivan Sergeevich Turgenev (18181883)
“The industrial world would be a more peaceful place if workers were called in as collaborators in the process of establishing standards and defining shop practices, matters which surely affect their interests and well-being fully as much as they affect those of employers and consumers.”
—Mary Barnett Gilson (1877?)
“The peace conference must not adjourn without the establishment of some ordered system of international government, backed by power enough to give authority to its decrees. ... Unless a league something like this results at our peace conference, we shall merely drop back into armed hostility and international anarchy. The war will have been fought in vain ...”
—Virginia Crocheron Gildersleeve (18771965)
“When strength is yoked with justice, where is a mightier pair than they?”
—Aeschylus (525456 B.C.)
“You have a row of dominoes set up; you knock over the first one, and what will happen to the last one is that it will go over very quickly.”
—Dwight D. Eisenhower (18901969)
“The struggle for existence holds as much in the intellectual as in the physical world. A theory is a species of thinking, and its right to exist is coextensive with its power of resisting extinction by its rivals.”
—Thomas Henry Huxley (182595)