Goodstein Sequences

Given a hereditary representation of a number n in base b, let B[b](n) be the nonnegative integer which results if we syntactically replace each b by b+1 (i.e., B[b] is a base change operator that ‘bumps the base’ from b up to b+1). 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 n gives the Goodstein sequence {G_k(n)}. Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein’s theorem states that G_k(n) is 0 for any n 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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