User blog comment:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal/@comment-5150073-20130813193429

I believe that I can define somewhat simpler definition of psi function up to BHO:

Notes:

1) Start to apply the rules from the bottom of the expression inside $$\psi(\alpha)$$.

2) $$\psi(0)$$ has an alternative form $$\psi$$.

3) # indicates the rest of an expression.

Rule 1. Condition: $$\alpha$$ is 0:

$$\psi(0)[1] = \omega$$

$$\psi(0)[n] = \omega^{\psi(0)[n-1]}$$

Rule 2. Condition: $$\alpha$$ has a successor:

$$\psi(\alpha+1)[1] = \psi(\alpha)+1$$

$$\psi(\alpha+1)[n] = \omega^{\psi(\alpha+1)[n-1])}$$

Rule 3. Condition: $$\alpha$$ is a limit ordinal:

$$\psi(\alpha)[n] = \psi(\alpha[n])$$

Rule 4. Condition: $$\Omega$$ appears somewhere in the expression:

$$\psi(\#+\Omega)[1] = \psi(\#)$$

$$\psi(\#+\Omega)[n] = \psi(\#+\psi(\#+\Omega)[n-1])$$

The key to it was usage of # symbol, which isn't used in the professional math.