[FOM] a real number for the Ackermann function

Andreas Weiermann weiermann at math.uu.nl
Thu Apr 13 14:11:16 EDT 2006

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