Ordinal

An ordinal number is a type of number that describes the position of an element in a well-ordered sequence — first, second, third, and so on into the transfinite. In set theory, ordinals are identified with the order types of well-ordered sets, and the von Neumann construction defines each ordinal as the set of all smaller ordinals: 0 = ∅, 1 = {0}, 2 = {0, 1}, and so on. The ordinals extend beyond the finite into the transfinite: ω is the first infinite ordinal, followed by ω+1, ω+2, and ultimately uncountable ordinals that index the stages of the von Neumann universe.

The ordinal hierarchy is the backbone of transfinite recursion, the structural principle by which mathematical objects are constructed in stages. The Axiom of Replacement guarantees that the image of a set under a function is a set, enabling the construction of ordinals beyond ω and ensuring that the transfinite hierarchy does not collapse at finite limits.

The ordinal is not a number in the sense of quantity. It is a number in the sense of position — and the discovery that position can extend beyond the finite is one of the strangest achievements of modern mathematics.