Limit ordinal

From Academic Kids

A limit ordinal is an ordinal number which is not a successor ordinal. Intuitively, these are ordinal numbers which cannot be reached via the ordinal successor operation S. In precise terms, we say λ is a limit ordinal if for any α < λ, S(α) < λ. Phrased in yet another way, an ordinal is a limit ordinal if and only if it is equal to the supremum of all the ordinals below it.

(Considerable contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have a predecessor; many mathematicians exclude 0 by requiring limit ordinals to be infinite, but Wikipedia does not).

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; and we denote this by ω. ω is also the smallest infinite ordinal (forgetting the limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω2, and then we have ωn for any n a natural number. Taking the union (the supremum operation on any set of ordinals) of all the ωn, we get ωω = ω2 (more on ordinal arithmetic at the main ordinal number entry). And we can keep going and going, getting

<math>\omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega}, \ldots, \epsilon_0 = \omega^{\omega^{\omega^\ldots}}, \ldots<math>

In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. And even these are only countable ordinals; it is a well-known fact that there is no recursively enumerable scheme of naming just all the countable ordinals! The first uncountable ordinal is usually denoted ω1 and is also a limit ordinal.

And we don't stop there: we have (all of these are increasing in cardinality now!):

<math>\omega_2, \omega_3, \ldots, \omega_\omega, \omega_{\omega_\omega},\dots<math>

In general, we always get a limit ordinal when taking the union of a set of ordinals that has no maximum element.

The term limit derives from using the order topology on the ordinal numbers; limit ordinals correspond precisely to the limit points in this topology.

The classes of successor ordinals and limit ordinals (and if you insist on limit ordinals being infinite, zero) exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals are usually a kind of "turning point" in which we have to use limiting operations such as taking the union over all preceding ordinals (technically we could do anything at limit ordinals, but taking the union is continuous in the order topology and usually this is what we want).

If we use the Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting observation, as cardinal derives from the Latin cardo meaning hinge or turning point!): the proof of this fact is done by simply showing that every successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.

Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level!). More at limit cardinal.


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