FAST-GROWING HIERARCHY

FAST-GROWING HIERARCHY IS A SET OF FUNCTIONS. EACH FUNCTION IS DENOTED $$f_\alpha$$ FOR COUNTABLE ORDINAL $$\alpha$$. IT IS DEFINED AS FOLLOWS:

$$f_\alpha(n)=\begin{cases}n+1\text{ if }\alpha=0 \\ f_{\alpha-1}^n(n)\text{ if }\alpha\text{ is successor} \\ f_{\alpha[n]}(n)\text{ if }\alpha\text{ is limit}\end{cases}$$

USUALLY, A FUNCTION WILL VASTLY OUTGROW ALL PREVIOUS FUNCTIONS, however it depends on the system of fundamental sequences used to define the 3rd case