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Date: 28-5-2020
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Date: 17-11-2019
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Date: 25-10-2020
680
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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
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so bumping the base from 2 to 3 yields
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Now repeatedly bump the base and subtract 1,
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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).
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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