Goodstein's Theorem

Goodstein's Theorem

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical logic, Goodstein's theorem is a statement about the natural numbers, made by Reuben Goodstein, which states that every Goodstein sequence eventually terminates at 0. Kirby & Paris 1982 showed that it is unprovable in Peano arithmetic. This was the third "natural" example of a true statement that is unprovable in Peano arithmetic. Earlier statements of this type had either been, except for Gentzen, extremely complicated, ad-hoc constructions or concerned metamathematics or combinatorial results. Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the "Hydra" is a rooted tree, and a move consists of cutting off one of its "heads," to which the hydra responds by growing a finite number of new heads according to certain rules. The Kirby-Paris interpretation of the theorem says that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very, very long time.show more

Product details

  • Paperback | 64 pages
  • 152 x 229 x 4mm | 104g
  • Fec Publishing
  • United States
  • English
  • 6136617013
  • 9786136617015