User blog:Chronolegends/Chronolegend's Shrinking Hierarchy

Following from my post in the watercooling forum, and thanks to Littlepeng9 for pointing out an error.

Using normal indexes instead of negative ones (also using alpha now since i'm not really defining it for any limit ordinals) $$f_0(n)=n/2$$

$$f_{\alpha+1}(n)=f^n_{\alpha}(n)$$

Then, its just a matter of Extending f^x(b) to allow x and b to be in $$\mathbb{Q}$$

Express x in the form a/b+c, a and b are minimal and positive integers, and c is a non-negative integer.

For example, The natural numbers n are the cases n/1+0

$$ \begin{cases} \ c>0 \rightarrow f^{a/b+c}_\alpha(n)=f^{c}_{\alpha}(f^{a/b}_\alpha(n)) \\ \ c=0,a=b=1 \rightarrow f^{a/b}_\alpha(n) = f_\alpha(n) \\ \ f^{a/b}_0(c/d+e) = (ac+ade)/(db*2) \\ \ f^{a/b}_{\alpha+1}(c/d+e)=f^{(a/b)*(c/d+e)}_\alpha(n)\\ \end{cases} $$

I also have an idea for limit ordinals but ill extend on that later on. At the moment f only works for $$\alpha < \omega$$.