Given a hereditary representation of a number in base , let be the nonnegative integer which results if we syntactically replace each by i. The hereditary representation of in base 2 is. Starting this procedure at an integer gives the Goodstein sequence. Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein's theorem states that is 0 for any and any sufficiently large.
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Gentzen's Centenary pp Cite as. In light of new historical details found in a correspondence between Bernays and Goodstein, we address the question of how close Goodstein came to proving an independence result for PA. We also present an elementary proof of the fact that already the termination of all special Goodstein sequences, i. This was first proved by Kirby and Paris in , using techniques from the model theory of arithmetic.
The proof presented here arguably only uses tools that would have been available in the s or s. Thus we ponder the question whether striking independence results could have been proved much earlier?
Almost no direct moral is ever given; rather, the paper strives to lay out evidence for the reader to consider and have the reader form their own conclusions. However, in relation to independence results, we think that both Gentzen and Goodstein are deserving of more credit.
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Cichon, A short proof of two recently discovered independence results using recursion theoretic methods. Gentzen, Die Widerspruchsfreiheit der reinen Zahlentheorie. Goodstein, On the restricted ordinal theorem. Grzegorczyk, Some Classes of Recursive Functions.
Rozprawy Mate No. IV Warsaw, Google Scholar. Kirby, J. Paris, Accessible independence results for Peano arithmetic. Kreisel, On the interpretation of non-finitist proofs II. Paris, L. Harrington, A mathematical incompleteness in Peano arithmetic, in Handbook of Mathematical Logic , ed.
Barwise North-Holland, Amsterdam, , pp. Simpson, Subsystems of Second Order Arithmetic , 2nd edn. Wainer, A classification of the ordinal recursive functions. Personalised recommendations. Cite chapter How to cite? ENW EndNote. Buy options.
For all , there exists a such that the th term of the Goodstein sequence. In other words, every Goodstein sequence converges to 0. The secret underlying Goodstein's theorem is that the hereditary representation of in base mimics an ordinal notation for ordinals less than some number. For such ordinals, the base bumping operation leaves the ordinal fixed whereas the subtraction of one decreases the ordinal. But these ordinals are well ordered, and this allows us to conclude that a Goodstein sequence eventually converges to zero. Amazingly, Paris and Kirby showed in that Goodstein's theorem is not provable in ordinary Peano arithmetic Borwein and Bailey , p.
In mathematical logic , Goodstein's theorem is a statement about the natural numbers , proved by Reuben Goodstein in , 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. 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" named for the mythological multi-headed Hydra of Lerna 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.
Gentzen's Centenary pp Cite as. In light of new historical details found in a correspondence between Bernays and Goodstein, we address the question of how close Goodstein came to proving an independence result for PA. We also present an elementary proof of the fact that already the termination of all special Goodstein sequences, i. This was first proved by Kirby and Paris in , using techniques from the model theory of arithmetic. The proof presented here arguably only uses tools that would have been available in the s or s.