A concrete piece of evidence for incompleteness

On Thursday, the 25th of March 2015, Venanzio Capretta gave a Theory Lunch talk about Goodstein’s theorem. Later, on the 9th of March, Wolfgang Jeltsch talked about ordinal numbers, which are at the base of Goodstein’s proof. Here, I am writing down a small recollection of their arguments.

Given a base b \geq 2, consider the base-b writing of the nonnegative integer

n = b^m \cdot a_m + b^{m-1} \cdot a_{m-1} + \ldots + b \cdot a_1 + a_0

where each a_i is an integer between 0 and b-1. The Cantor base-b writing of n is obtained by iteratively applying the base-b writing to the exponents as well, until the only values appearing are integers between 0 and b. For example, for b = 2 and n = 49, we have

49 = 32 + 16 + 1 = 2^{2^2 + 1} + 2^{2^2} + 1

and also

49 = 27 + 9 \cdot 2 + 3 + 1 = 3^3 + 3^2 \cdot 2 + 3 + 1

Given a nonnegative integer n, consider the Goodstein sequence defined for i \geq 2 by putting x_2 = n, and by constructing x_{i+1} from x_i as follows: Continue reading