INSANITY GROWING HIERARCHY

$$\S_0(a) = a+1$$

Define a thing to use on functions:

$$f_{\alpha}^{f_\alpha}(a) = f_{\alpha}^{f_\alpha(a)}(a)$$

continue...

$$f_{\alpha}\uparrow\uparrow b(a)$$is obvious.

Now use BEAF...

$$\S_{\alpha+1}(a) = \{\S_{\alpha},\S_{\alpha}(\S_{\alpha})\S_{\alpha}\}(a)$$

(using BEAF)