Talk:Veblen function

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Where is the definition?

I hope someone improves this article so that it defines the Veblen function. Right now it simply seems to state a bunch of properties of this function, without asserting that some set of properties characterizes it. John Baez (talk) 06:08, 4 November 2012 (UTC)[reply]

In the lead it gives the general definition, "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.". In the first section, it defines "Veblen hierarchy" by specifying that "In the special case when φ₀(α)=ω^α, this family of functions is known as the Veblen hierarchy.". JRSpriggs (talk) 10:15, 4 November 2012 (UTC)[reply]

Set theoretic definitions

For the binary Veblen hierarchy, a formal definition using recursion on α and β is:

\phi _{\alpha }(\beta )=\gamma \,\Leftrightarrow \,\operatorname {Ord} (\alpha )\land \operatorname {Ord} (\beta )\land \operatorname {Ord} (\gamma )\land 0<\gamma \land \forall \delta <\gamma (\delta +\delta <\gamma )\land \forall \delta <\alpha (\phi _{\delta }(\gamma )=\gamma )\land \forall \delta <\beta (\phi _{\alpha }(\delta )<\gamma )\land \lnot \exists \rho <\gamma \left(0<\rho \land \forall \delta <\rho (\delta +\delta <\rho )\land \forall \delta <\alpha (\phi _{\delta }(\rho )=\rho )\land \forall \delta <\beta (\phi _{\alpha }(\delta )<\rho )\right)

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OK? JRSpriggs (talk) 23:13, 15 July 2022 (UTC)[reply]