08/23/2009, 01:37 AM
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 \( f(n, m) = \phi(n, m, 3) = {}^{(m+1)}n \) which is an offset from tetration. Because it is an offset of tetration, \( \phi(n, m, 4) \) 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.
