# [FOM] a real number for the Ackermann function

Robert M. Solovay solovay at Math.Berkeley.EDU
Fri Apr 14 21:22:59 EDT 2006

Do you have some other, more transparent, characterization of this real?

--Bob Solovay

On Thu, 13 Apr 2006, Andreas Weiermann wrote:

> There might be some interest in the
> following sharp phase transition result
> for the Ackermann function
> in terms of a specific (possibly transcendental) real number.
>
> Theorem: Let a>1 be a real number. Define a hierarchy of unary functions
> a_k as follows. Let a_0(x) be a tower of a's of hight x where x is a natural
> number argument. Let a_{k+1}(x) be the x-th iterate of a_k applied to x.
> Finally define the diagonal function a(x)=a_x(x).
> Then x\mapsto a(x) is Ackermannian iff a>1.44466781...
>
> Best regards,
> Andreas Weiermann
>
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