# 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