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,

￼We repeatedly bump the base and subtract 1.

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 . Even more amazingly, Paris and Kirby showed in 1982 that Goodstein’s theorem is not provable in ordinary Peano arithmetic.

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