User blog:Vel!/Ordinal Hyper-E 2.0

Inspired by Aarex's latest extension, Nested Cascading-E Notation, I will revisit E^ and do it right this time. (Someday I'll get a website and this stuff can go in the mainspace.)

Definition
Define \(E_\gamma(\alpha)\):

\[E_\gamma(\alpha) = \max\{n \in \mathbb{N}_0|\exists \beta_2 < \omega^\gamma, \beta_1: \omega^{\gamma+1} \beta_1 + \omega^\gamma \times n + \beta_2 = \alpha\}\]

Also, define \(Q(\alpha) = \{\gamma|E_\gamma(\alpha) \neq 0\}\), which is the set of all positions with non-zero associated entries.

Define \(L(\alpha) = \max(Q(\alpha))\), and \(P(\alpha) = \max\{\gamma < L(\alpha)|\gamma \in Q(\alpha)\}\). In other words, \(L(\alpha)\) is the last non-zero entry and \(P(\alpha)\) is the next-to-last one.