Every totally ordered set (A, <=) is associated with a so-called order type. Two sets A and B 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 ^* ω, although Suppes calls it ω^*. The order type of the rationals is called η. Some sources call the order type of the reals θ, while others call it λ. In general, if α is any order type, then ^* α is the same type ordered backwards.

cardinal number | order isomorphic | ordinal number | totally ordered set