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.