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

			(1)
			(2)

so bumping the base from 2 to 3 yields

(3)

Now repeatedly bump the base and subtract 1,

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

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 . 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).

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