Goodstein's theorem [SEP] named after
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" 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
well, meaning that it must reach 0.
While this proof of Goodstein's theorem is fairly easy, the "Kirby–Paris theorem", which shows that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countable nonstandard models of Peano arithmetic.
Extended Goodstein's theorem.
Suppose the definition Goodstein sequence is changed so that instead of
replacing each occurrence of the base "b" with 
it replaces it with . Would the sequence still