Ordinal Numbers

The set N of natural numbers is naturally ordered: 1, 2, 3, 4, ... But this is not the only order the numbers may be arranged in. For example, the famous Sarkovskii's theorem lists the natural numbers starting with all the odd numbers, followed by the odd numbers times the increased powers of two (a power of 2 at a time) and closing by the powers of 2 in the decreasing order:

3, 5, 7, ...,
2·3, 2·5, 2·7, ...,
2²·3, 2²·5, 2²·7, ...,
... 2³, 2², 2, 1.

(The theorem states that if a real function f: R → R has an orbit of period n and m comes after n in the above ordering, then f has an orbit of period m.)

Another curious order comes from the enumeration of rational numbers. If rn is the nth rational number according to a particular enumeration we may define a total dense order on the set N of natural numbers.

However, here we are concerned only with well orderings of N. The term for an ordering of a well ordered set is ordinal number or just ordinal.

The natural order of N is denoted ω and is the first transfinite ordinal. Every positive integer is a finite ordinal.

If a well-ordered set A with the ordinal α is similar to a subset of a well-ordered set B with the ordinal β then, by definition, α ≤ β. In particular, for every finite n, n ≤ ω. However, since there is no injection, let alone an order-preserving one, from N into a finite set, we may claim that n ≠ ωand n < ω.