Normal function

In set theory, a normal function is a (definable) function \(f \colon \text{On} \to \text{On}\) that is strictly increasing and Scott continuous, where \(\text{On}\) denotes the class of ordinals. Some authors also refer to a function \(f \colon \alpha \to \alpha\) for an ordinal \(\alpha\) as a normal function if it satisfies similar conditions, and some also require \(f(0)>0\)H. Levitz, Transfinite Ordinals and Their Notation: For The Uninitiated (p.6). Accessed 2021-05-11.. When we emphasise the domain and the codomain, we call a normal function in the former convention a normal function on \(\text{On}\), and a normal function in the latter convention a normal function on \(\alpha\).O. Veblen, Continuous Increasing Functions of Finite and Transfinite Ordinals, Transactions of the American Mathematical Society, Vol. 9, No. 3 (1908), pp.280--292.
== Explanation ==
We say that \(f\) is strictly increasing if \(\alpha < \beta\) implies \(f(\alpha) < f(\beta)\) holds for any ordinals \(\alpha\) and \(\beta\), and that \(f\) is Scott continuous (or continuous for short) if \(f(\alpha) = \sup\{f(\beta) \mid \beta < \alpha\}\) holds for any limit ordinal \(\alpha \neq 0\).
A trivial example of a normal function is the identity function \(f(\alpha)=\alpha\). Less trivial examples include functions such as \(f(\alpha)=1+\alpha\) or \(f(\alpha)=\omega^\alpha\). The most typical example of a non-normal function is the successor function \(f(\alpha)=\alpha+1\).
== See also ==
* Derivative
== Spurces ==
Category:Set theory
ru:Нормальная функция
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