Normal function: Difference between revisions
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(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.") |
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A normal function is |
A normal function is an [[ordinal function]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: |
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* \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) |
* \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) |
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* \(f(\alpha)=\sup f(\beta |
* \(f(\alpha)=\sup \{ f(\beta) | \beta < \alpha \} \) if \(\alpha\) is a [[limit ordinal]]. |
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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. |
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Latest revision as of 00:38, 23 March 2024
A normal function is an ordinal function that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties:
- \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\)
- \(f(\alpha)=\sup \{ f(\beta) | \beta < \alpha \} \) 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.