---
tag: JQFAE9S2F4
kind: concept
name: Veblen function
msc: ["03E10", "03F15"]
sources:
  - url: https://en.wikipedia.org/wiki/Veblen_function
same_as:
  wikidata: Q921930
  wikipedia: Veblen function
---

## Summary

The Veblen functions $\varphi_\gamma$ are a hierarchy of [[normal functions|ERTAM0SFVB]] on the [[ordinals|0YSCSX5YZC]], each one listing the fixed points of the ones before.
They start from $\varphi_0(\alpha) = \omega^\alpha$, and they give names to large countable ordinals such as the epsilon numbers and the Feferman–Schütte ordinal $\Gamma_0$.
The [[precise definition|6QFEVWKE25]] is by transfinite recursion on the index $\gamma$.
They are a basic tool of ordinal notations and of ordinal analysis in proof theory.

## Explanations

### L1 human

Start with the function $\alpha \mapsto \omega^\alpha$.
Its fixed points, the ordinals with $\omega^\alpha = \alpha$, are the epsilon numbers, and listing them gives a new function $\varphi_1$.
The fixed points of $\varphi_1$ give $\varphi_2$, and so on, also through infinite stages.
Each new function grows so fast that its values outrun everything the earlier functions can name.
For instance, $\varphi_1(0) = \varepsilon_0$, the first ordinal with $\omega^{\varepsilon_0} = \varepsilon_0$.