Ordinal

An ordinal is the order type of a well-ordered set. Any two finite linear orderings of the same order type are isomorphic.

Finite orderings have order types which are natural numbers: 0,1,2,.... The type of (N,<) is ω. The type of the linear ordering which has all even natural numbers followed by all odd natural numbers is ω×2.

We avoid the Burali-Forti paradox by not letting the ordinals have an ordinal type.

References

Doets, Kees. From Logic to Logic Programming. MIT Press, 1994.