Small Veblen ordinal

Template:Short description In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by Template:Harvtxt is somewhat smaller than the small Veblen ordinal.

There is no standard notation for ordinals beyond the Feferman–Schütte ordinal ${\mathrm{\Gamma }}_0$. Most systems of notation use symbols such as $\psi (\alpha )$, $\theta (\alpha )$, ${\psi }_{\alpha }(\beta )$, some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".

The small Veblen ordinal ${\theta }_{{\mathrm{\Omega }}^{\omega }}(0)$ or $\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$ is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's theorem. It is also the ordinal type of a certain ordering of rooted trees Template:Harv.

The small Veblen ordinal (SVO) is $sup\{\omega =\phi (1),{\epsilon }_0=\phi (1,0),{\mathrm{\Gamma }}_0=\phi (1,0,0),\phi (1,0,0,0),\phi (1,0,0,0,0),\phi (1,0,0,0,0,0),\dots \}$ using the finitary Veblen function. SVO is the least ordinal which is not the sum of a finite number of values of the finitary Veblen function when applied to functions of finite support from ω to SVO. These functions can be encoded as ordinals less than ${\mathrm{SVO}}^{\omega }$ (ordinal exponentiation) and $\mathrm{\Omega }$ can be used as a place holder for SVO. Notice that $\mathrm{SVO}={\omega }^{\mathrm{SVO}}={ϵ}_{\mathrm{SVO}}={\mathrm{\Gamma }}_{\mathrm{SVO}}$.

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