Given a hereditary representation of a number n in base b, let be the nonnegative integer which results if we syntactically replace each b by (i.e., is a base change operator that `bumps the base' from b 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 n 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 n and any sufficiently large k. Even more amazingly, Paris and Kirby showed in 1982 that Goodstein's theorem is not provable in ordinary Peano arithmetic (Borwein and Bailey 2003, p. 35).

 

Goodstein's Theorem, Hereditary Representation, Paris-Harrington Theorem