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 , consider the base- writing of the nonnegative integer

where each is an integer between and . *The Cantor base-* writing of is obtained by iteratively applying the base- writing to the exponents as well, until the only values appearing are integers between and . For example, for and , we have

and also

Given a nonnegative integer , consider the *Goodstein sequence* defined for by putting , and by constructing from as follows: Continue reading