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