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- Goodsteins_theorem abstract "In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Kirby and Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem was a later example.Laurence Kirby and Jeff Paris introduced a graph theoretic hydra game with behavior similar to that of Goodstein sequences: the \"Hydra\" is a rooted tree, and a move consists of cutting off one of its \"heads\" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved 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 long time.".
- Goodsteins_theorem wikiPageExternalLink goodstein.pdf.
- Goodsteins_theorem wikiPageExternalLink ?p=674.
- Goodsteins_theorem wikiPageExternalLink summary?doi=10.1.1.22.3296.
- Goodsteins_theorem wikiPageExternalLink the-hydra-game.
- Goodsteins_theorem wikiPageExternalLink goodstein.
- Goodsteins_theorem wikiPageExternalLink Kaplan_S12_MATH_Thesis_Final_5-8-12.pdf?sequence=1.
- Goodsteins_theorem wikiPageID "150062".
- Goodsteins_theorem wikiPageLength "20319".
- Goodsteins_theorem wikiPageOutDegree "39".
- Goodsteins_theorem wikiPageRevisionID "693327687".
- Goodsteins_theorem wikiPageWikiLink Ackermann_function.
- Goodsteins_theorem wikiPageWikiLink Category:Articles_containing_proofs.
- Goodsteins_theorem wikiPageWikiLink Category:Independence_results.
- Goodsteins_theorem wikiPageWikiLink Category:Large_numbers.
- Goodsteins_theorem wikiPageWikiLink Category:Set_theory.
- Goodsteins_theorem wikiPageWikiLink Category:Theorems_in_the_foundations_of_mathematics.
- Goodsteins_theorem wikiPageWikiLink Computable_function.
- Goodsteins_theorem wikiPageWikiLink Fast-growing_hierarchy.
- Goodsteins_theorem wikiPageWikiLink Gentzens_consistency_proof.
- Goodsteins_theorem wikiPageWikiLink Gerhard_Gentzen.
- Goodsteins_theorem wikiPageWikiLink Grahams_number.
- Goodsteins_theorem wikiPageWikiLink Gxc3xb6dels_incompleteness_theorems.
- Goodsteins_theorem wikiPageWikiLink Hardy_hierarchy.
- Goodsteins_theorem wikiPageWikiLink Independence_(mathematical_logic).
- Goodsteins_theorem wikiPageWikiLink Jeff_Paris.
- Goodsteins_theorem wikiPageWikiLink Journal_of_Symbolic_Logic.
- Goodsteins_theorem wikiPageWikiLink Kanamori–McAloon_theorem.
- Goodsteins_theorem wikiPageWikiLink Kruskals_tree_theorem.
- Goodsteins_theorem wikiPageWikiLink Lernaean_Hydra.
- Goodsteins_theorem wikiPageWikiLink Mathematical_logic.
- Goodsteins_theorem wikiPageWikiLink Natural_number.
- Goodsteins_theorem wikiPageWikiLink Non-standard_model_of_arithmetic.
- Goodsteins_theorem wikiPageWikiLink Ordinal_number.
- Goodsteins_theorem wikiPageWikiLink Paris–Harrington_theorem.
- Goodsteins_theorem wikiPageWikiLink Partial_function.
- Goodsteins_theorem wikiPageWikiLink Peano_axioms.
- Goodsteins_theorem wikiPageWikiLink Reuben_Goodstein.
- Goodsteins_theorem wikiPageWikiLink Second-order_arithmetic.
- Goodsteins_theorem wikiPageWikiLink Turing_machine.
- Goodsteins_theorem wikiPageWikiLink Well-founded_relation.
- Goodsteins_theorem wikiPageWikiLink Woodall_number.
- Goodsteins_theorem wikiPageWikiLinkText "Goodstein function".
- Goodsteins_theorem wikiPageWikiLinkText "Goodstein sequences".
- Goodsteins_theorem wikiPageWikiLinkText "Goodstein's theorem".
- Goodsteins_theorem wikiPageWikiLinkText "Goodstein's theorem#Sequence length as a function of the starting value".
- Goodsteins_theorem wikiPageWikiLinkText "Kirby–Paris theorem".
- Goodsteins_theorem wikiPageWikiLinkText "hydra game".
- Goodsteins_theorem wikiPageUsesTemplate Template:Citation.
- Goodsteins_theorem wikiPageUsesTemplate Template:Citation_needed.
- Goodsteins_theorem wikiPageUsesTemplate Template:Mathworld.
- Goodsteins_theorem wikiPageUsesTemplate Template:OEIS2C.
- Goodsteins_theorem wikiPageUsesTemplate Template:Redirect.
- Goodsteins_theorem wikiPageUsesTemplate Template:Reflist.
- Goodsteins_theorem subject Category:Articles_containing_proofs.
- Goodsteins_theorem subject Category:Independence_results.
- Goodsteins_theorem subject Category:Large_numbers.
- Goodsteins_theorem subject Category:Set_theory.
- Goodsteins_theorem subject Category:Theorems_in_the_foundations_of_mathematics.
- Goodsteins_theorem hypernym Statement.
- Goodsteins_theorem type Proof.
- Goodsteins_theorem type Redirect.
- Goodsteins_theorem type Theorem.
- Goodsteins_theorem comment "In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Kirby and Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second order arithmetic).".
- Goodsteins_theorem label "Goodstein's theorem".
- Goodsteins_theorem sameAs Q1149185.
- Goodsteins_theorem sameAs Goodsteinova_věta.
- Goodsteins_theorem sameAs Goodstein-Folge.
- Goodsteins_theorem sameAs Sucesión_de_Goodstein.
- Goodsteins_theorem sameAs Théorème_de_Goodstein.
- Goodsteins_theorem sameAs Teorema_di_Goodstein.
- Goodsteins_theorem sameAs グッドスタインの定理.
- Goodsteins_theorem sameAs 굿스타인의_정리.
- Goodsteins_theorem sameAs Twierdzenie_Goodsteina.
- Goodsteins_theorem sameAs Teorema_de_Goodstein.
- Goodsteins_theorem sameAs m.0139r1.
- Goodsteins_theorem sameAs Теорема_Гудстейна.
- Goodsteins_theorem sameAs Теорема_Гудштейна.
- Goodsteins_theorem sameAs Q1149185.
- Goodsteins_theorem wasDerivedFrom Goodsteins_theorem?oldid=693327687.
- Goodsteins_theorem isPrimaryTopicOf Goodsteins_theorem.