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
 = \phi(n, m, 3) = {}^{(m+1)}n)
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:
=a+b)
=\varrho_c(\varphi(a,c,n),\alpha(a,n),b))
.
Where
,a,n)=f^{[n]}(a))
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
=1)
for
. Then would
=a [n+1] b)
. But instead Ackermann defines:
 = \iota(n,1)\cdot \iota(n,0) \cdot a + \lambda(n,1))
where
=1-\delta_{a,b})
and
=\delta_{a,b})
in today notation with the Kronecker-
.
This definition is hence equivalent to:
 = 0)
for
 = 1)
for
 = a)
for
as he also mentiones in his paper.
So he introduces a third initial value
=a)
besides 0 and 1 which causes the deviation from our operator sequence:
=(c\mapsto\varphi(a,c,0)^{[b]}(\alpha(a,0))=(c\mapsto a+c)^{[b]}(0)=a\cdot b)
.
=(c\mapsto \varphi(a,c,1))^{[b]}(\alpha(a,1))=(c\mapsto a\cdot c)^{[b]}(1)=a^{b})
=(c\mapsto \varphi(a,c,2))^{[b]}(\alpha(a,2))=(c\mapsto a^c)^{[b]}(a)=a [4] (b+1))
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:
=a+b)
=(x\mapsto \psi(x,a,n))^{[b]}(\alpha(a,n)))
here again we have
=ab)
and
=a^b)
.
But the forth operation is not
as one would obtain with the initial condition
=1)
, but it is
=a^{a^b})
due to the initial value
=a)
!
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
)