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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)Zn(b)m-1(c)@], where Zx = an array of of n's with [@1(a)Zx-1(b)m-1(c)@] entries in each row, [@1(a)Zx-1(b)m-1(c)@] rows in each plane, [@1(a)Zx-1(b)m-1(c)@] planes in each realm etc., with b dimensions. Z0 = n, and @1 contains only 1's and separators
  • R2: n![@1(a)y,@21,m,@] = n![@1(a)y,@2[@1(a)y,@2[@1(a)y,@2[...[@1(a)y,@2[@1(a)y-1,@21,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: [@10@2] = 1, where @1 and @2 contain no set of brackets enclosing the 0.

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