High Quality Content by WIKIPEDIA articles! 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 & Paris 1982 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 "natural" example of a true statement that is unprovable in Peano arithmetic (after Gerhard Gentzen's 1943 direct proof of the unprovability of ?0-induction in Peano arithmetic and the Paris–Harrington theorem). Earlier statements of this type had either been, except for Gentzen, extremely complicated, ad-hoc constructions (such as the statements generated by the construction given in Godel's incompleteness theorem) or concerned metamathematics or combinatorial results (Kirby & Paris 1982). Данное издание представляет собой компиляцию сведений, находящихся в свободном доступе в среде Интернет...

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