User blog:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal

Okay, I promised to blog about ordinal notations. I thought I would start by going over the basic notations up to the Bachmann-Howard ordinal.

I will start with the basic \(\psi\) function, invented by Wolfram Pohlers. It is defined as follows:

\(C_0 (\alpha) = \lbrace 0, \Omega \rbrace \\

C_{n+1} (\alpha) = \lbrace \beta + \gamma, \omega^{\beta}, \psi(\delta) | \beta, \gamma, \delta \subset C_n (\alpha); \delta < \alpha \rbrace \\

C (\alpha) = \bigcup_{n = 1}^{\infty} C_n (\alpha) \\

\psi (\alpha) = \min \lbrace \beta | \beta \subset C(\alpha) \rbrace \)

(more to come)