The first transfinite ordinal. If you wrote down all the natural numbers on a piece of paper and asked, "what comes next," the answer would be omega-null. It has cardinality aleph-null; however (and here is the part that sucks), so does omega + 1, omega + 2, and so forth. But I'm not a mathematician, and I shouldn't even put that hat on for parlor tricks, so really you should read the writeup below for the gory details.

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.

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