Editing Limit ordinal (section)

{{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.