User blog comment:Wythagoras/All my stuff/@comment-7484840-20130708083606/@comment-5529393-20130708142718

I don't think the ordinals need a lot more rules - yes you need rules for fundamental sequences, but that just mirrors the rules you need for any recursive notation. You need one rule for 0, but you need a base case for any notation.

The FGH up to omega needs just two rules:

1. F_0 (b) = b+1

2. F_{a+1} (b) = (F_a)^b (b)

I guess you need three rules to define F_omega(n), but can you define a function of that growth or greater using just two?

Here's a 4 rule notation for H_epsilon_0:

1. (X+1)[n] = X

2. (X + omega^(Y+1))[n] = (X + omega^Y) * n

3. (X + omega^Y)[n] = X + omega^(Y[n])

4. H_X(n) is the smallest k such that X[n][n+1]...[k] = 0

I'm pretty sure one can define $$F_{\theta(\Omega_{\Omega_{\Omega_\ldots}},0)}$$ using fewer than 10 rules.