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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 $$f(0)>0$$[1]. 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 $$\alpha$$.[2]

## 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 $$f(\alpha)=\alpha+1$$.