Normal function: Difference between revisions

Normal function (edit)

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Revision as of 20:48, 5 September 2022 (edit) Metachirality (talk \| contribs) (Created page with "A normal function is a function on ordinals that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a limit ordinal.")		Latest revision as of 00:38, 23 March 2024 (edit) (undo) CreeperBomb (talk \| contribs) mNo edit summary
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Line 1: A normal function is ~~a function on~~an [[ordinal~~\|ordinals~~ function]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup \{ f(\beta)\) ~~if and only if~~\| \(\beta < \alpha \} \) ~~and~~if \(\alpha\) is a [[limit ordinal]]. Veblen's fixed point lemma, which is essential for constructing the [[Veblen hierarchy]], guarantees that, not only does every normal function have a [[fixed point]], but the class of fixed points is unbounded and their enumeration function is also normal.