Order Type

Every totally ordered set  is associated with a so-called order type. Two sets  and  are said to have the same order type iff they are order isomorphic (Ciesielski 1997, p. 38; Dauben 1990, pp. 184 and 199; Moore 1982, p. 52; Suppes 1972, pp. 127-129). Thus, an order type categorizes totally ordered sets in the same way that a cardinal number categorizes sets. The term is due to Georg Cantor, and the definition works equally well on partially ordered sets.

The order type of the negative integers is called  (Moore 1982, p. 62), although Suppes (1972, p. 128) calls it . The order type of the rationals is called  (Dauben 1990, p. 152; Moore 1982, p. 115; Suppes 1972, p. 128). Some sources call the order type of the reals  (Dauben 1990, p. 152), while others call it  (Suppes 1972, p. 128).

In general, if  is any order type, then  is the same type ordered backwards (Dauben 1990, p. 153).

REFERENCES:

Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.

Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 242, 1997.

Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.

Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.