Goodstein Sequence
المؤلف:
Borwein, J. and Bailey, D
المصدر:
Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters
الجزء والصفحة:
...
18-1-2022
1462
Goodstein Sequence
Given a hereditary representation of a number 
in base 
, let ](https://mathworld.wolfram.com/images/equations/GoodsteinSequence/Inline3.svg)
be the nonnegative integer which results if we syntactically replace each 
by 
(i.e., ![B[b]](https://mathworld.wolfram.com/images/equations/GoodsteinSequence/Inline6.svg)
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
=3^(3^(3+1))+3^(3+1)+3.](https://mathworld.wolfram.com/images/equations/GoodsteinSequence/NumberedEquation1.svg) |
(3)
|
Now repeatedly bump the base and subtract 1,
etc.
Starting this procedure at an integer 
gives the Goodstein sequence ![<span style=]()
{G_k(n)}" src="https://mathworld.wolfram.com/images/equations/GoodsteinSequence/Inline43.svg" style="height:24px; width:60px" />. 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).
REFERENCES
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 34-35, 2003.
Goodstein, R. L. "On the Restricted Ordinal Theorem." J. Symb. Logic 9, 33-41, 1944.
Henle, J. M. An Outline of Set Theory. New York: Springer-Verlag, 1986.
Simpson, S. G. "Unprovable Theorems and Fast-Growing Functions." Contemp. Math. 65, 359-394, 1987.
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