Goodstein’s theorem, proved by Reuben Goodstein in 1944, states that every Goodstein sequence eventually terminates at 0. Laurence Kirby and Jeff Paris showed that the theorem is unprovable in Peano arithmetic but can be proven in stronger systems. Goodstein sequences are defined using a concept called “hereditary base-n notation,” which is similar to positional notation but with some differences. The Hydra game, introduced by Kirby and Paris, has behavior similar to that of Goodstein sequences and also cannot be proven in Peano arithmetic alone. Despite the rapid growth of Goodstein sequences, they always reach 0. The proof of Goodstein’s theorem is fairly easy, while the proof that it is unprovable in Peano arithmetic is more difficult and technical. The extended Goodstein’s theorem explores sequences with different base changes. The Goodstein function, which calculates the length of the Goodstein sequence that starts with a number, has an extremely high growth rate. Goodstein’s theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total.
https://en.wikipedia.org/wiki/Goodstein%27s_theorem