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A '''limit ordinal''' is an [[ordinal number]] which is neither zero nor a [[successor ordinal]].  
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Another way of defining a limit ordinal is to say that λ is a limit ordinal if and only if:
 
:There is an ordinal less than λ and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ.
 
So in the following sequence:
 
:0, 1, 2, ... , ω, ω+1
 
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) we can find another ordinal (natural number) larger than it, but still less than ω.
 
Various other ways to define limit ordinal are:
*It is equal to the [[supremum]] of all the ordinals below it, but is not zero.  (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)
*It is not zero and has no maximum element.
*It can be written in the form ωα for α > 0. That is, in the [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] there is no finite number as last term, and the ordinal is nonzero.
*It is a [[limit point]] of the class of ordinal numbers, with respect to the [[order topology]]. (The other ordinals are [[isolated point]]s.)
 
Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor;
some textbooks include 0 in the class of limit ordinals<ref>for example, Thomas Jech, ''Set Theory''. Third Millennium edition. Springer.</ref> while others exclude it.<ref>for example, Kenneth Kunen, ''Set Theory. An introduction to independence proofs''. North-Holland.</ref>
 
==Examples==
Because the [[Class (set theory)|class]] of ordinal numbers is [[well-order]]ed, there is a smallest infinite limit ordinal; denoted by ω. This ordinal ω is also the smallest infinite ordinal (disregarding ''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 = ω·2, and then we have ω·''n'', for any natural number ''n''. Taking the [[union (set theory)|union]] (the [[supremum]] operation on any [[Set (mathematics)|set]] of ordinals) of all the ω·n, we get ω·ω = ω<sup>2</sup>. This process can be iterated as follows to produce:
 
:<math>\omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega}, \ldots, \epsilon_0 = \omega^{\omega^{\omega^{~\cdot^{~\cdot^{~\cdot}}}}}, \ldots</math>
 
In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still [[countable]] ordinals. However, there is no [[recursively enumerable]] scheme for [[ordinal notation|systematically naming]] all ordinals less than the [[Church–Kleene ordinal]] which is a countable ordinal.
 
Beyond the countable, the [[first uncountable ordinal]] is usually denoted ω<sub>1</sub>. It is also a limit ordinal.
 
Continuing, one can obtain the following (all of which are now increasing in cardinality):
 
:<math>\omega_2, \omega_3, \ldots, \omega_\omega, \omega_{\omega + 1}, \ldots, \omega_{\omega_\omega},\ldots</math>
 
In general, we always get a limit ordinal when taking the union of a set of ordinals that has no [[maximum]] element.
 
The ordinals of the form ω²α, for α > 0, are limits of limits, etc.
 
== Properties ==
The classes of successor ordinals and limit ordinals (of various [[cofinality|cofinalities]]) as well as 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 represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is [[continuous function (topology)|continuous]] in the order topology and this is usually desirable.
 
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 infinite successor ordinal is [[equinumerous]] to a limit ordinal via the [[Hilbert's paradox of the Grand Hotel|Hotel Infinity]] argument.
 
Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level).
 
== See also ==
*[[Ordinal arithmetic]]
*[[Limit cardinal]]
*[[Fundamental sequence (ordinals)]]
 
==References==
<references/>
 
{{DEFAULTSORT:Limit Ordinal}}
[[Category:Ordinal numbers]]

Latest revision as of 02:59, 21 October 2014

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