## Goodstein Sequence

Given a Hereditary Representation of a number in Base, let be the Nonnegative Integer which results if we syntactically replace each by (i.e., is a base change operator that `bumps the base' from up to ). The Hereditary Representation of 266 in base 2 is

so bumping the base from 2 to 3 yields

Now repeatedly bump the base and subtract 1,

etc. Starting this procedure at an Integer gives the Goodstein sequence . Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein's Theorem states that is 0 for any and any sufficiently large .