Three-Dimensional Geometry and Topology, Volume 1

is a recursive definition!) The first term of the 'Goodstein sequence' of a number is the number itself. The nth term of the Goodstein sequence is obtained by expressing the previous term in 'pure base n', replacing all the bases n in the expression with n + 1, then subtracting one. Goodstein's theorem is that every Goodstein sequence eventually terminates at zero. This is rather surprising, since the effect of incrementing the bases in the pure base n expansion would seem to be much greater than the effect of subtracting one. The proof requires an understanding of notation for infinite ordinals.