Veblen function

In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ₀ is any normal function, then for any non-zero ordinal α, φ_α is the function enumerating the common fixed points of φ_β for β<α. These functions are all normal.

In the special case when φ₀(α)=ω^α this family of functions is known as the Veblen hierarchy. The function φ₁ is the same as the ε function: φ₁(α)= ε_α. If ${\displaystyle \alpha <\beta \,,}$ then ${\displaystyle \varphi _{\alpha }(\varphi _{\beta }(\gamma ))=\varphi _{\beta }(\gamma )\,.}$ From this and the fact that φ_β is strictly increasing we get the ordering: ${\displaystyle \varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )\,}$ if and only if either (${\displaystyle \alpha =\gamma \,}$ and ${\displaystyle \beta <\delta \,}$) or (${\displaystyle \alpha <\gamma \,}$ and ${\displaystyle \beta <\varphi _{\gamma }(\delta )\,}$) or (${\displaystyle \alpha >\gamma \,}$ and ${\displaystyle \varphi _{\alpha }(\beta )<\delta \,}$).

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