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Thanks to everyone who commented on my previous blog, it helped me spot all the errors, typos and false claims in it, and these have all now been corrected. This post will contain an extension to my hyperfactorial array notation (as the title suggests), which (I think) will take it up to order type \omega^\omega (please correct me if i'm wrong, it may be \omega^3 or possibly lower (but hopefully not)). This extension is also different to anything in Bowers' work (or at least more different than it was before). Anyway, this is the extension:

Before, the notation was in the form a![b,c,d,e...]. This worked in a way that meant its order type is \omega^2. To reach \omega^\omega, we need to add another level of expansion to the notation, a new way to recursively expand the notation. This is done by adding more exclamation marks, so making it in the form a!!...!![b,c,d,e...] (shorthand a!^n[b,c,d,e\dots] for n !'s). This could be evaluated by working out the value with one exclamation mark and putting that back in the brackets, but that would not get the level of recursion we need. Instead, we define it with a new rule to the three usual ones, taking precedence over the others for the order:

  • If n > 1, a!^n[x_1,x_2,\dots,x_{k-1},x_k] = a!^{n-1}[x_1,x_2,\dots,x_{k-1},x_k!^{n-1}[x_1,x_2,\dots,x_{k-1},(x_k-1)!^{n-1}[x_1,x_2,\dots,x_{k-1},(x_k-2)!^{n-1}[\dots\dots!^{n-1}[x_1,x_2,\dots,x_{k-1}]\dots\dots]]]]

This is a pretty sizable extension to the previous notation, and it certainly ramps up the numbers generated by this notation. Here are two examples of how it works:

  • 3!![3,2] = 3![3,2![3]] = 3![3,2] = 3![3![3]] ~ 3![7.6x10^12]. Not that impressive , but it gets better:
  • 3!!![3,3] = 3!![3,3!![3,2!![3]]] = 3!![3,3!![3,2]] = 3!![3,3![3,2![3]]] = 3!![3,3![3,2]] = 3!![3,3![3]] ~ 3!![3,7.6x10^12] = 3![3,(7.6x10^12)![3,((7.6x10^12)-1)![3,((7.6x10^12)-1)...![3,2![3]]...]]]. That's more like it! Just remember it will only get a lot worse from there on.

Again, there is a list of named numbers coming soon, and there is another expansion i'm working on to get to the next major order level, but it is still a work in progress, as this is, so may take a while to actually get there. Any and all constructive comments are appreciated. If there are any problems please point them out, and I will make changes where necessary.

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