User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5150073-20130415141127/@comment-5529393-20130415214812

Yes, you have misunderstood how BEAF works. 3^3^3^3 & 3 is not a 762559748498D array. First of all the numbers don't match up - 3^3^n & 3 is an n-dimensional array, so perhaps you meant a 27D array. But 3^3^3^3 & 3 is much larger than a 27D array or a 7625597484987D array. Let's look at the much smaller 3^(3^3 + 1) & 3:  this is an X^X + 1 structure, which diagonalizes over X^X structures, i.e. multidimensional arrays. Let q be the X^X + 1st entry, which is 3 in 3^(3^3 + 1) & 3. If we set q to 1, we get 3^3^3 & 3, which is a 3D array. When we set q to 2, we first evaluate the array 3^3^3 & 3, then set the dimension to the huge value we get from that, which of course will be much larger than 7625597484987;  so we get an mD array where m is very large. If we set q to 3, then we take the very large number that results from the mD array, and define an array with that many dimensions. The final result is 3^(3^3 + 1) & 3. So even 3^(3^3 + 1) & 3 is much larger than what you think 3^3^3^3 & 3 is, or even pentational or higher arrays.

If BEAF actually worked like you think, it would not go much higher than omega^omega^omega in the FGH, since you would just evaluate any array down to dimensional arrays.

I will go over you notation soon.