Common
Goodstein's theorem
Theorem about natural numbers
Goodstein's theorem
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 (as defined below) eventually terminates at 0. Laurence Kirby and Jeff Paris showed in 1982 that Goodstein's theorem is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo–Fraenkel set theory). This was the third known example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example. Kirby and Paris also introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move by "Hercules" consists of cutting off one...
Open boosters, collect cards and trade them. Free to play.
Play for free