Following from my post in the watercooling forum[1] , 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


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

\ 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(c/d+e)\\

The last rule is WIP status! I also have an idea for limit ordinals but ill extend on that later on. At the moment f only works for \alpha < \omega .

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