Veblen hierarchy: Difference between revisions
no edit summary
No edit summary |
No edit summary |
||
| (2 intermediate revisions by the same user not shown) | |||
Line 7:
Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]].
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>
== References ==
| |||