Editing Limit ordinal

{{Short description|Infinite ordinal number class}}[[File:Omega-exp-omega-normal-dark svg.svg|thumb|250px|Representation of the ordinal numbers up to &omega;<sup>&omega;</sup>. Each turn of the spiral represents one power of &omega;. Limit ordinals are those that are non-zero and have no predecessor, such as &omega; or &omega;<sup>2</sup> ]]

In [[set theory]], a '''limit ordinal''' is an [[ordinal number]] that is neither zero nor a [[successor ordinal]]. Alternatively, an ordinal λ is a limit ordinal 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. Every ordinal number is either zero, a successor ordinal, or a limit ordinal.

For example, the smallest limit ordinal is [[ω (ordinal number)|ω]], the smallest ordinal greater than every [[natural number]]. This is a limit ordinal because for any smaller ordinal (i.e., for any natural number) ''n'' we can find another natural number larger than it (e.g. ''n''+1), but still less than ω.  The next-smallest limit ordinal is ω+ω. This will be discussed further in the article.

Using the [[von Neumann definition of ordinals]], every ordinal is the [[well-ordered set]] of all smaller ordinals. The union of a nonempty set of ordinals that has no [[greatest element]] is then always a limit ordinal. Using [[von Neumann cardinal assignment]], every infinite [[cardinal number]] is also a limit ordinal.

==Alternative definitions==
Various other ways to define limit ordinals 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 (omega). The 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, which generalizes to 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>, which generalizes to ω<sup>''n''</sup> for any natural number ''n''. This process can be further iterated as follows to produce:

:<math>\omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega}, \ldots, \varepsilon_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 nonempty 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).

== Indecomposable ordinals ==
{{main article|Indecomposable ordinal}}

'''Additively indecomposable'''

A limit ordinal α is called additively indecomposable if it cannot be expressed as the sum of β < α ordinals less than α. These numbers are any ordinal of the form <math>\omega^\beta</math> for β an ordinal. The smallest is written <math>\gamma_0</math>, the second is written <math>\gamma_1</math>, etc.<ref name=":0">{{Cite web|title=Limit ordinal - Cantor's Attic|url=http://cantorsattic.info/Limit_ordinal#Types_of_Limits|access-date=2021-08-10|website=cantorsattic.info}}</ref>

'''Multiplicatively indecomposable'''

A limit ordinal α is called multiplicatively indecomposable if it cannot be expressed as the product of β < α ordinals less than α. These numbers are any ordinal of the form <math>\omega^{\omega^\beta}</math> for β an ordinal. The smallest is written <math>\delta_0</math>, the second is written <math>\delta_1</math>, etc.<ref name=":0" />

'''Exponentially indecomposable and beyond'''

The term "exponentially indecomposable" does not refer to ordinals not expressible as the exponential product ''(?)'' of β < α ordinals less than α, but rather the [[Epsilon numbers (mathematics)|epsilon numbers]], "tetrationally indecomposable" refers to the zeta numbers, "pentationally indecomposable" refers to the eta numbers, etc.<ref name=":0" />

== See also ==
*[[Ordinal arithmetic]]
*[[Limit cardinal]]
*[[Fundamental sequence (ordinals)]]

==References==
<references/>

==Further reading==
{{refbegin}}
* [[Georg Cantor|Cantor, G.]], (1897), ''Beitrage zur Begrundung der transfiniten Mengenlehre.  II'' (tr.: Contributions to the Founding of the Theory of Transfinite Numbers II), Mathematische Annalen 49, 207-246 [https://archive.org/details/117770262 English translation].
* [[John Horton Conway|Conway, J. H.]] and [[Richard K. Guy|Guy, R. K.]] "Cantor's Ordinal Numbers." In ''The Book of Numbers''. New York: Springer-Verlag, pp.&nbsp;266–267 and 274, 1996.
* Sierpiński, W. (1965). ''[[Cardinal and Ordinal Numbers]]'' (2nd ed.). Warszawa: Państwowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.
{{refend}}

[[Category:Ordinal numbers]]