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I wanted to say something about the first part, "1 Introduction". In this part of the paper, you equate the original Ackermann function with a[n]b, which (stictly speaking) is not true. Robert Munafo has discussed this on his website (http://www.mrob.com/pub/math/largenum.html), and I have also verified this for myself by reading the original paper Ackermann wrote. Ackermann's tetration is a function which is an offset from tetration. Because it is an offset of tetration, is not pentation at all. It wasn't until Reuben Goodstein that the "offset" disappeared. I think we should take this into account when discussing the original Ackermann function, because its evolution is way more complicated than any introduction can really summarize.
@Andrew: Good! Thats a very attentive observation.


While verifying myself I found that the deviation (up to a difference of 1 in the rank) from our operator sequence comes from forming an unnecessary odd initial condition. I dont know why he does, perhaps it is more suitable for his proof of non-primitive recursiveness.

In his article
Ackermann, W. (1928 ). Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann., 99, 118–133.

Ackermann defines:

.

Where in our notation ( is a free variable here to determine the argument of the function to be applied).

So everything would be good if the initial value was for . Then would . But instead Ackermann defines:

where and in today notation with the Kronecker-.
This definition is hence equivalent to:
for
for
for
as he also mentiones in his paper.

So he introduces a third initial value besides 0 and 1 which causes the deviation from our operator sequence:
.



PS: Munafo gives a very detailed description of the different versions of the Ackermann-function here. It is a very good reference to show to someone for explaining about different versions of the Ackermann-function. All glory to Andrew for digging out such references.
Also, about a month ago, I redesigned the Hyperoperation page, to try and explain these differences.
(08/23/2009, 09:45 AM)bo198214 Wrote: [ -> ]While verifying myself I found that the deviation (up to a difference of 1 in the rank) from our operator sequence comes from forming an unnecessary odd initial condition. I dont know why he does, perhaps it is more suitable for his proof of non-primitive recursiveness.

Oh now I found out where this odd initial conditions comes from!
I assert that Ackermann originally wanted to define left-braced hyperoperations!
Then this initial condition for makes sense!

Left-braced hyperoperations would similarly be defined by:



here again we have and .
But the forth operation is not as one would obtain with the initial condition , but it is due to the initial value !

So this initial condition makes left-braced hyperoperations look simpler, while it makes right-braced hyperoperations looking odd.
I think he started with the left-braced hyperoperations and then switched to the faster growing right-braced hyperoperations, perhaps it was more suitable for his proof of non-primitive recursiveness of