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In the last post, I claimed that I had a challenger to Jonathan Bowers' origional linear array notation. Well, here it is:

EDIT: not it isn't, see next blog (when it comes out) for the challenge.

There has been one change since my last post, and that is that now [n] means \uarr^n, not \uarr^{n-2}

It is another array notation (as can be guessed from the name). Similarly to the origional linear array notation by Jonathan Bowers, it has several elements seperated by commas. It is written as a![b,c,d,...], with as many letters inside the bracket as you want. It is defined by three rules. If rule 1 applies, follow it, if not, if rule 2 applies do it, and if neither of them apply do rule 3. The part of the equation you work on at any one time is the part inside the most sets of brackets (for example in 10![4,[7,[9],4],3,[4,[7,[7,6]],3]], you would expand the [7,6] as this is the part inside the most sets of brackets. If there are two parts drawing over which is inside the most brackets, just choose one, as it should make no difference to the outcome. The rules are:

For x_k\in(N) and x_k\ne0 for k\in(N) and k\ne0,

  1. If x_k = 1, remove it. (Remove all trailing 1's)
  2. If k = 2, x_1![x_2] = x_1\uarr^{x_2}(x_1-1)\uarr^{x_2}(x_1-2)\uarr^{x_2}\dots\uarr^{x_2}2\uarr^{x_2}1
  3. Else, x_1![x_2,x_3,x_4,\dots,x_{k-1},x_k] = x_1![x_2,x_3,x_4,\dots,x_{k-2},(x_{k-1}![x_2,x_3,x_4,\dots,x_{k-1},x_k-1]![x_2,x_3,x_4,\dots,x_{k-1},x_k-2]!\dots![x_2,x_3,x_4,\dots,x_{k-1},1]),x_k-1]

Note 1: if a 1 occurs anywhere inside a set of square brackets, everything after and including the 1 in that set of brackets is irrelivent and can be removed, (for example 5![6,1,4,7] = 5![6]), because 1![x]![y]!... = 1.

Note 2: 2![x,y,z...] = 2, because 2\uarr^x1 = 1

The definitions of these hyperfactorial arrays may seem a little daunting, but is not that difficult, these are a few examples to demonstrate it:

  • 3![3,2] = 3![3![3,1],1] (using rule 3) = 3![3![3]] (using rule 1) = 3![3^{2^1}] (using rule 2) = 3![9] = 3\uarr^92\uarr^91 (using rule 2) = 3\uarr^83, which is pretty big.
  • 4![3,3] (this not seem much different, but really is) = 4![3![3,2]![3],2] = 4![3![3![3]]![3],2] = 4![3![9]![3],2] = 4![3\uarr^83![3],2] = 4![(3\uarr^83![3])![3\uarr^83![3]]] = 4\uarr^{((3\uarr^83![3])\uarr^{(3\uarr^83![3])}((3\uarr^83![3])-1)\uarr^{(3\uarr^83![3])}\dots3\uarr^{(3\uarr^83![3])}2)}3\uarr^{((3\uarr^83![3])\uarr^{(3\uarr^83![3])}((3\uarr^83![3])-1)\uarr^{(3\uarr^83![3])}\dots3\uarr^{(3\uarr^83![3])}2)}2 ... that's.... quite big. On closer inspection, this is easily far larger than g(2) in the Graham's number sequence.
  • 5![5,5,5,5] = 5![5,5,(5![5,5,5,4]![5,5,5,3]![5,5,5,2]![5,5,5]),4] = 5![5,5,(5![5,5,(5![5,5,5,3]![5,5,5,2]![5,5,5]),3]![5,5,(5![5,5,5,2]![5,5,5]),2]![5,5,5![5,5,5]]![5,(5![5,5,4]![5,5,3]![5,5,2]![5,5]),4]),4] =.... I think you probably get the picture.

The major problem with these, other than their complexity, is that it is in a relatively new direction, so it can be difficult to make direct comparisons with other numbers, but certainly not impossible:

A common benchmark for big numbers is Graham's number. I have already refered to some of the terms in its sequence as comparison. These are some comparisons i was able to draw that because 3\uarr^n2 = 3\uarr^{n-1}3, and 3![n] = 3\uarr^n2, and g(n) = 3\uarr^{g(n-1)}3, it follows that 3![g(n)+1] = g(n+1). Therefore, Graham's number itself, or g(64) = 3![3![\dots3![3\uarr^43]+1]\dots+1]+1], with 63 sets of brackets. This could be greatly overpowered by 3![3,64], as this would contain the necessary brackets and a lot, lot more. In array notation, Graham's number is less than {3,66,1,2}, therefore, because of the relative sizes of the representations (or overpowerings) of Graham's number, Hyperfactorial Array Notation can be more powerful than standard linear array notation with a few small inputs, but will be overtaken with more, or larger, inputs and it lacks the ability to be hugely expanded, which BEAF,  another one of Bowers' systems, does very well (or at least it lacks it at the moment ;) ). Don't think for a second this is the end though, as I have Extended Hyperfactorial Array Notation building up right now (or built if its been a long time since I published this). Soon I will also post a list of named numbers formed from this notation.

EDIT: Standard hyperfactorial array notation has order type \omega^2, which is less than standard linear array notation, which has order type \omega^\omega. Yes they are reasonably similar in appearance, but they expand in different ways (which leads to the huge difference in order type). The continuations to it should differentiate it from array notation completely.

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