Goodstein’s Theorem Revisited


Prompted by Gentzen’s 1936 consistency proof, Goodstein found a close fit between descending sequences of ordinals \(<\varepsilon _{0}\) 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.


Recursive Function Elementary Proof Sequent Calculus Peano Arithmetic Independence Result 
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The author acknowledges support by the EPSRC of the UK through Grant No. EP/G029520/1.

He is also indebted to Jan von Plato for showing him letters from the Goodstein-Bernays correspondence.


  1. 1.
    J. Barwise (ed.), Handbook of Mathematical Logic (North-Holland, Amsterdam, 1977)Google Scholar
  2. 2.
    P. Bernays, Letter to Goodstein, dated September 1st, 1942, Bernays collection of the ETH ZürichGoogle Scholar
  3. 3.
    P. Bernays, Letter to Goodstein, dated September 29th, 1943, Bernays collection of the ETH ZürichGoogle Scholar
  4. 4.
    A. Cichon, A short proof of two recently discovered independence results using recursion theoretic methods. Proc. Am. Math. Soc. 87, 704–706 (1983)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    G. Gentzen, Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann. 112, 493–565 (1936)MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Gentzen, Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exacten Wissenschaften, Neue Folge 4 (Hirzel, Leipzig, 1938), pp. 19–44Google Scholar
  7. 7.
    R.L. Goodstein, On the restricted ordinal theorem. J. Symb. Log. 9, 33–41 (1944)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Grzegorczyk, Some Classes of Recursive Functions. Rozprawy Mate No. IV (Warsaw, 1953)Google Scholar
  9. 9.
    L. Kirby, J. Paris, Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 14, 285–293 (1982)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Kreisel, On the interpretation of non-finitist proofs II. J. Symb. Log. 17, 43–58 (1952)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Paris, L. Harrington, A mathematical incompleteness in Peano arithmetic, in Handbook of Mathematical Logic, ed. by J. Barwise (North-Holland, Amsterdam, 1977), pp. 1133–1142CrossRefGoogle Scholar
  12. 12.
    D. Schmidt, Well-Partial Orderings and Their Maximal Order Types (Habilitationsschrift, Universität Heidelberg, 1979)Google Scholar
  13. 13.
    H. Schwichtenberg, Eine Klassifikation der \(\varepsilon _{0}\)-rekursiven Funktionen. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 17, 61–74 (1971)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    S.G. Simpson, Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher Bäume. Archiv für mathematische Logik 25, 45–65 (1985)MATHCrossRefGoogle Scholar
  15. 15.
    S.G. Simpson, Subsystems of Second Order Arithmetic, 2nd edn. (Cambridge University Press, Cambridge, 2009)MATHCrossRefGoogle Scholar
  16. 16.
    S.S. Wainer, A classification of the ordinal recursive functions. Archiv für Mathematische Logik und Grundlagenforschung 13, 136–153 (1970)MATHMathSciNetCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsEngland

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