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
Now repeatedly bump the base and subtract 1,
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 .
See also Goodstein's Theorem, Hereditary Representation
© 1996-9 Eric W. Weisstein
1999-05-25