User blog:Fejfo/SAH: Super Ackermann Hierachy

When I made my aray ordinals I noticed linear BEAF grows faster than it's array ordinal (array without the prime and base) in the fgh. This is beacause BEAF always iterates in the most important value, in the limit case to. This is why I define:
 * 1) \( A_0(x)=x+1 \)
 * 2) \( A_{\alpha+1}(x+1)=A_\alpha(A_{\alpha+1}(x)) \)
 * 3) \( A_{\alpha+1}(0)=A_\alpha(1) \)
 * 4) \( A_\alpha(x+1)=A_{\alpha[A_\alpha(x)]}(x+1) \)
 * 5) \( A_\alpha(0)=A_{\alpha[0]}(1) \)

Analysis
For natural numbers \( A_\alpha(x)=Ack(\alpha,x) \).

For ordinals this iterates in the collapsing argument so \( A_\omega(x)\approx f_{\omega+1}(x) \)

Succesor ordinals iterate the normal argument again so \( A_{\omega+n}(x)\approx f_{\omega+n+1}(x) \)

We can conclude that \( A_\alpha\approx f_{\alpha+1} \) so it's not that much more powerfull and it's defintion is to complex to justify the extra growth, the fgh is better.