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:
...
24-11-2019
1184
Goodstein Sequence
Given a hereditary representation of a number 
in base 
, let ](http://mathworld.wolfram.com/images/equations/GoodsteinSequence/Inline3.gif)
be the nonnegative integer which results if we syntactically replace each 
by 
(i.e., ![B[b]](http://mathworld.wolfram.com/images/equations/GoodsteinSequence/Inline6.gif)
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.](http://mathworld.wolfram.com/images/equations/GoodsteinSequence/NumberedEquation1.gif) |
(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="http://mathworld.wolfram.com/images/equations/GoodsteinSequence/Inline43.gif" style="height:14px; width:44px" />. 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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