Rathjen, M (2015) Goodstein's theorem revisited. In: Kahle, R and Rathjen, M, (eds.) Gentzen's Centenary: The Quest for Consistency. Springer International Publishing , London , pp. 229-242. ISBN 3319101021
Abstract
Prompted by Gentzen’s 1936 consistency proof, Goodstein found a close fit between descending sequences of ordinals <ε0 and sequences of integers, now known as Goodstein sequences. This chapter revisits Goodstein’s 1944 paper. 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.e. those induced by the shift function, is not provable in PA. This was first proved by Kirby and Paris in 1982, using techniques from the model theory of arithmetic. The proof presented here arguably only uses tools that would have been available in the 1940s or 1950s. Thus we ponder the question whether striking independence results could have been proved much earlier? In the same vein we also wonder whether the search for strictly mathematical examples of an incompleteness in PA really attained its “holy grail” status before the late 1970s. 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.
Metadata
Item Type: | Book Section |
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Authors/Creators: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 May 2015 11:45 |
Last Modified: | 20 Apr 2016 19:00 |
Published Version: | http://dx.doi.org/10.1007/978-3-319-10103-3_9 |
Status: | Published |
Publisher: | Springer International Publishing |
Identification Number: | 10.1007/978-3-319-10103-3_9 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:85842 |