Order type omega in mathematics, the order type of the infinite set of natural numbers. The last letter of the Greek alphabet, w, is used to denote this order type; w is thus the first infinite ordinal number. It can be defined as the set of all finite ordinal numbers ordered by magnitude; that is, w % {0,1,2,3 . . . }. A set has order type w provided it is denumerably infinite, has a first element but not a last element, has for each element a unique successor, and has just one element with no immediate predecessor. The set of even numbers ordered by magnitude, {2,4,6,8 . . . }, is of order type w. The set of natural numbers listing first all even numbers and then all odd numbers, {2,4,6,8 . . .; 1,3,5,7 . . . }, is not of order type w, since it has two elements, 1 and 2, with no immediate predecessor. The set of negative integers ordered by magnitude, { . . . –3,–2,–1}, is also not of order type w, since it has no first element. V.K.