In

ZF set theory, an

ordinal number (by one definition) is a set which contains all its

proper subsets. Keep in mind that, in

ZF, everything is a

set---0 is usally defined as {} (the empty set), 1 as {{}} (the set containing the

empty set), 2 as { {{}}, {} } (the set containing 0 and 1), etc. Thus (one may easily see), all the

natural numbers (as defined above) are

ordinals. We define the

successor S(

*n*) of an ordinal

*n* as S(

*n*) =

*n* union {

*n* }. An ordinal

*a* is said to be `smaller than' an ordinal

*b* if and only if a is an element of b.

omega_0 is then the smallest infinite ordinal. If we make the identifications {} = 0, {{}} = 1, { {{}}, {} } = 2, etc. as above, then omega_0 is the set **N** of all natural numbers. omega_0 is a limit ordinal---that is, omega_0 is not S(*a*) for any ordinal *a*.

omega_0 is often written just omega.