Veblen hierarchy: Difference between revisions

 

Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>GS dimensional Veblen extensions</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref>