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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* \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]. |
* \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]. |
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Revision as of 02:38, 13 September 2022
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:
- \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\)
- \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a limit ordinal.