User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5150073-20130415141127/@comment-5150073-20130416080906

DrCeasium, to make it clearer, look at the FGH: tetrational arrays has limit ordinal e_0, and multidimensional arrays has limit ordinal w^(w^w). So f_w^(w^w)(n) ~ {n,n (0,1) 2}, which means n-D array with size n. So X^^4 & 3[3] should be {3,7625597484987 (0,1) 2}? Well, 7625597484987 << f_w(3), and X^^4 & 3[3] ~ f_w^(w^w)(f_w(3)) << f_w^(w^w)+1(n). So, by you tetrational arrays doesn't add even one more level of recursion past dimensional arrays, and grow even slower than {n,n,2 (0,1) 2}, not so speak about {n,n (1,1) 2}, {n,n (0,2) 2}, {n,n (0,0,1) 2}, {n,n ((1) 1) 2} and higher tetrational structures.

I can give an example to how solve X^(X+1) & 3[2]. It is {3,2 (1,1) 2} in Bowers' notation.

Then:

{3,2 (1,1) 2} = {X^X & 3[2] (0,1) X^X & 3[2]} = {3,3 (1) 3,3 (0,1) 3,3 (1) 3,3}. Now imagine when we reduce it in something in the form {3,N (0,1) 3,3 (1) 3,3} = {X^N & 3[N] (0,1) 3,3 (1) 3,3}, N will be some immensely large number itself. Then X^N & 3[N] means the N-D hypercube with size N. Tetrational arrays really can't be solved simply and intuitively, you should have imagination that X^^X & n[m] structure makes numbers derived from dimensional arrays smaller than subatomic dust, even in the recursive sence.