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→Background and formal statement: Added "nonzero" for correctness as several authors consider 0 to be a limit ordinal, but do not require continuity at this point. |
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A [[normal function]] is a [[proper class|class]] function ''f'' from the class Ord of [[ordinal numbers]] to itself such that:
* ''f'' is '''strictly increasing''': ''f''(α) < f(β) whenever α < β.
* ''f'' is '''continuous''': for every nonzero limit ordinal λ, ''f''(λ) = sup { f(α) : α < λ }.
It can be shown that if ''f'' is normal then ''f'' commutes with [[supremum|suprema]]; for any nonempty set ''A'' of ordinals,
:''f''(sup ''A'') = sup {''f''(α) : α ∈ ''A'' }.
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