Logic seminar

A Walk with Goodstein

The classical Goodstein process is based on writing numbers in "normal form" in terms of addition and exponentiation with some base k. By iteratively changing base and subtracting one, one obtains very long sequences of natural numbers which eventually terminate. The latter is proven by comparing base-k normal forms with Cantor normal forms for ordinals, and in fact this proof relies heavily on the notion of normal form. The question then naturally arises: what if we write natural numbers in an arbitrary fashion, not necessarily using normal forms? What if we allow not only addition and exponentiation, but also multiplication for writing numbers?

A "Goodstein walk" is any sequence obtained by following the standard Goodstein process but arbitrarily choosing how each element of the sequence is represented. As it turns out, any Goodstein walk is finite, and indeed the longest possible Goodstein walk is given by the standard normal forms. In this talk we sketch a proof of this fact and discuss how a similar idea can be applied to other variants of the Goodstein process, including those based on the Ackermann function.

Joint work with A. Weiermann