User blog comment:Alejandro Magno/Make an extension of the Notation! (my style)/@comment-5982810-20140925002326

(Note: This version is much stronger)

Let (a,b,c,...,z) be defined for ordinals a,b,c,...,z. To define this set begin with the set {0,1,2,...}.

If A and B are ordinals AB is an ordinal. If a,b,c,...,z are ordinals then (a,b,c,...,z) is an ordinal. Let @,& be an ordinals, L be a limit ordinal, and $ be an array.

1. 0#n = n+n

2. @k+1#n = @k#( @k#(... @k#(n) ... )) w/n @k#s

3. If we have a limit ordinal then ... L#n = L[n]#n

Defining L[n]

Note: @(@0) = @(@) and @($,0) = @($)

I. &(0)[n] = &n

II. &(@k+1)[n] = &(@k) (@k) (@k)...(@k) w/n (@k)s

III. &(@L$)[n] = &(@L[n]$)

IV. &(0,0,...,0,L$)[n] = &(0,0,...,0,L[n]$)

V. &(0,0,...,0,0,k+1$)[n] = %%...%%n ,k$) ,k$) ... ,k$) ,k$) w/n %s where % = " &(0,0,...,0, "

VI. &(k+1$)[0] = &(&(k$)1$)

VII. &(k+1$)[n+1] =  &(&(k+1$)[n]$)