[FOM] interesting real number

weiermann@math.uu.nl weiermann at math.uu.nl
Tue Apr 18 17:24:13 EDT 2006

```Hereby I would like to thank all contributors
to this thread for their interesting and sensible
contributions.

Besides curiosity in the constant itself
(about which e.g. google or the book by Finch
on constants provide full information)
there might be some further themes which
motivate such alternative definitions
of Ackermannian functions.

1. Dynamical systems. Iterations
of a start function g(a,x) as function of x may be divergent
for certain choices of a. This is a standard theme
in dynamical systems. Perhaps there is interest
in seeing which iterations lead to Ackermannian
growth after suitable extension.
Iterations in complex numbers might
be studied in this context, too,
when one considers the modulus of
g(a,x) or more precisely the
integer part of it, so that
iteration arguments are natural
numbers.

2. Indexed phase transitions.
Given a function g one might consider those
functions h for which the following hierarchy
is Ackermannian.
B_0(x)=g(x)
B_{k+1}(x)=B_k^{h(x)}(x)
B(x)=B_x(x)

That seems to be more interesting than expected at first
sight. The phase transition in h depends on an
additional parameter g.

3. Refined phase transition.
For the original system one might vary the phase
transition problem as follows. Let
g(x)=a_1^{\ldots^{a_n}} where a_i=e^{1/e}+epsilon(i)
where epsilon(i) converges to 0. Again an obvious
question is for what choices of epsilon the resulting
hierarchy is Ackermannian. I am aware
of some non trivial results
on lower and upper bounds on epsilon as a function on i.

4. The ideas of Mani seem sensible to me and might
be rewarding further research.

I hope that this thread indicates that
even nowadays there is still some intriguing stuff
around the Ackermann function.

Best regards,
Andreas Weiermann

> On Sunday 16 April 2006 02:00, Martin Davis wrote:
>> Replying to a query by Bob Solovay, Ron Graham wrote:
>>  > This real is well known to be e^(1/e)

```

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