I have put formal definitions of multidimensional arrays on my website here. There have also been a couple of slight changes to my notation, but nothing that will affect anything on the wiki other than a few of the pages on the larger numbers, which have had very slight definition changes, to make them more formal. Also, a couple of the numbers in the gigantixul group no longer exist.

Another thing I have done is to put more comparisons to the FGH out on my website, and (I believe) that the limit of multidimensional arrays is indeed the Bachmann-Howard ordinal.

This is the definition of multidimensional arrays (just copied and pasted over):

R means rule. To use these, first check if rule 1 applies. If it does use it. If not, check if rule 2 applies. If it does, use it. If not, check the rules for the first row. S means sub-rule. These can just be used whenever they apply.

- R1: n![@
_{1}(a)1(b)m(c)@] = n![@_{1}(a)Z_{n}(b)m-1(c)@], where Z_{x}= an array of of n's with [@_{1}(a)Z_{x-1}(b)m-1(c)@] entries in each row, [@_{1}(a)Z_{x-1}(b)m-1(c)@] rows in each plane, [@_{1}(a)Z_{x-1}(b)m-1(c)@] planes in each realm etc., with b dimensions. Z_{0}= n, and @_{1}contains only 1's and separators - R2: n![@
_{1}(a)y,@_{2}1,m,@] = n![@_{1}(a)y,@_{2}[@_{1}(a)y,@_{2}[@_{1}(a)y,@_{2}[**...**[@_{1}(a)y,@_{2}[@_{1}(a)y-1,@_{2}1,m,@],m-1,@]**...**],m-1,@],m-1,@],m-1,@] with n [@_{1}(a)y,@_{2}@,m-1,@]'s, where @_{1}contains only 1's and separators and @_{2}only 1's. - S1: n![@
_{1}(a)1(b)1(c)@] = n![@_{1}(a)1(c)@] iff a ≥ b < c - S2: [@
_{1}0@_{2}] = 1, where @_{1}and @_{2}contain no set of brackets enclosing the 0.