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On the Performance of Axiom Systems

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Axiomatic Thinking II

Abstract

One of the aims of proof theory is to calibrate the strength of axiom systems by invariants. According to Gödel’s discoveries these invariants will in general not be finite but rather transfinite objects. Pioneering work in this direction had been done by Gerhard Gentzen who characterized the axiom system for Peano arithmetic by the transfinite ordinal \({\varepsilon _0}\). In this paper we try to develop a general framework for characterizing ordinals of axiom systems and study to what extend these ordinals embody a measure for their performance.

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Notes

  1. 1.

    By the elementary language of a structure we understand the basis language of \(\mathfrak {M}\). This is in principle always a first-order language. However, it does not exclude many sorted first-order languages, e.g. weak second-order languages.

  2. 2.

    The properties of the ordinal \(\pi ^{\mathfrak M}\) are studied in [31]. In case that the structure \({\mathfrak M}\) is the structure \({\mathfrak N}\) of arithmetic, we get \(\pi ^{\mathfrak N}=\omega _1^{\hbox {\tiny CK}}\).

  3. 3.

    This theorem, in a different formulation, already appears in [15].

  4. 4.

    By a pseudo \(\Pi _{1}^{1}\)-sentence we understand a formula in the elementary language of \({\mathfrak M}\), which must not contain free first-order variables but may well contain free second-order variables. Semantically pseudo \(\Pi _{1}^{1}\)-sentences are treated as \(\Pi _{1}^{1}\)-sentences.

  5. 5.

    That similarly \(\pi ^\mathfrak {M}\) is characteristic rather for an universe above \(\mathfrak {M}\) than for \({\mathfrak M}\) itself follows from Sect. 5 in [31].

  6. 6.

    Therefore many of the mathematical results presented here are not new. Most of them are already contained in [33] but are presented here in a different context.

  7. 7.

    Cf. Definition 3.25 and Note 3.30.

  8. 8.

    \(\pi ^\mathfrak {M}\) corresponds to \(o(\Gamma )\) if \(\Gamma \) is the least Spector class above \(\mathfrak {M}\) (cf. Theorems 3.32 and 3.34).

  9. 9.

    Therefore, in some sense, we are selling “old wine in new skins”. Nevertheless we decided to include—at least extended sketches of—many of the old proofs. First, of course, to make the paper more self-contained and thus retain the survey character of the paper. Secondly because in many cases the central ideas of the proofs are helpful (or even needed) to make our approach transparent.

  10. 10.

    Here one should observe that the spectrum of an axiomatization \(\textsf{ID}_{\nu }({\textsf{T}})\) for \({\mathfrak M}_{\nu }\) also comprises the ordinal \(\pi ^{{\mathfrak M}_{\nu }}(\textsf{ID}_{\nu }({\textsf{T}}))\).

  11. 11.

    Introduced in Sect. 3.3.3.

  12. 12.

    A more elaborated sketch is in [33].

  13. 13.

    For a complete proof cf. e.g. [31] Theorem 4.4.

  14. 14.

    By an axiomatization of \({\mathfrak M}\) we understand a set \(\textsf{T}\) of \({\mathscr {L}}(\mathfrak {M})\)-sentences such that \(\mathfrak {M}\models \textsf{T}\).

  15. 15.

    This is just because any axiom system \(\textsf{T}\) is consistent iff there is a formula F such that \(\textsf{T}\not \vdash F\).

  16. 16.

    A full proof is given in [1].

  17. 17.

    This implies that the well-ordering on the countable domain of \({\mathfrak M}\) is elementarily definable.

  18. 18.

    Cf. Sects. 5.3 and 5.4 in [31].

  19. 19.

    To what extend the ordinal also has a meaning for the ground structure \(\mathfrak {M}\) will be briefly touched in Sect. 3.5.

  20. 20.

    Rumor has it that this observation is due to Kreisel. Unfortunately I know no reference.

  21. 21.

    This is familiar from “Reverse Mathematics” as, e.g. presented in [38]. The increasing strength of the axiom systems treated there is due to additional set-existence axioms.

  22. 22.

    To distinguish Analysis in the meaning of “Real Analysis” from analysis in concepts such as “ordinal analysis” we capitalize Analysis, whenever we mean it in the former sense.

  23. 23.

    Spector classes are introduced in [24] Chap. 9.

  24. 24.

    In abuse of notation we will often use \({\vec x}\in R\) and \(R({\vec x})\) synonymously.

  25. 25.

    For strictly acceptable structures \(\mathfrak {M}\) and acceptable axiom systems \(\textsf{T}\) which prove Weak König’s Lemma we always have \(\delta ^{\mathfrak M}(\textsf{T})=\kappa ^{\mathfrak M}(\textsf{T})\) (cf. [30] Thm. 6.7).

  26. 26.

    The lack of direct access to ordinals in \({\mathscr {L}}^2(\mathfrak {M})\) is one of the reasons why ordinal analysis shifted from the study of analytical universes to the study of set-theoretic universes in which ordinal numbers occur naturally (cf. [32] for a brief overview).

  27. 27.

    The theory of fixed points is studied in [25]. We will, however, by far not exhaust the very general approach in [25].

  28. 28.

    See [24, 25] for details.

  29. 29.

    Cf. [25] Sect. 8.

  30. 30.

    Cf. [36].

  31. 31.

    Although some proof-theoretic ordinals are known via embeddings into set-theoretic universes.

  32. 32.

    Hence the notation \(\kappa ^\mathfrak {M}_{\mu +1}\).

  33. 33.

    Cf. e.g. [27] Sect. 7.2.

  34. 34.

    The proof of the modification follows in principle the same pattern as the proof of Theorem 3.14. Because of the more blurred estimate \(2^\alpha \) it is even a bit easier and does not need the notion of co-enumeration. However, the extra 2-power is indispensable.

  35. 35.

    An assumption that will be substantiated later.

  36. 36.

    This axiom system is essentially the axiom system for iterated inductive definitions as introduced by Sol Feferman in [8].

  37. 37.

    Cf. [22].

  38. 38.

    Cf. e.g. [8, 11, 10] and the bibliographic notes given there.

  39. 39.

    Observe that \(\mu \) in the derived formula stands for the ordinal notation and \(\mu \) in the derivation height for its order-type in \(\prec \).

  40. 40.

    The idea to control semi-formal derivations by ordinal operators has been introduced by Wilfried Buchholz in [5] as a simplification of local predictivity which has been used in [26].

  41. 41.

    By \(\aleph \) we denote the enumerating function of the infinite cardinals. As usual we write \(\aleph _\mu \) instead of \(\aleph (\mu )\).

  42. 42.

    This is the familiar diagonal argument.

  43. 43.

    Cf. e.g. [24] Theorem 2B.1.

  44. 44.

    Cf. [5] for a survey and further citations.

  45. 45.

    The assumption that \(\nu +1\) is among the generators of \(\mathcal{H}\) is usually omitted. For example, I omitted it in [33]. This is harmless since it suffices to know that (3.24) is provable with an upper bound \(\Omega _{\nu }+\Omega _{1}\), which is operator controlled.

  46. 46.

    Cf. Sect. 3.4.2.

  47. 47.

    More details are in [33] Theorem 8.4.

  48. 48.

    One of the “simplest” examples for a theory which is not simple is \({\textsc {ID}}_1\) (i.e. \(\textsf{ID}_{1}({\mathord {\textsc {PA}}})\)). Its spectrum consists of two points \(\{\Psi _{\Omega _1}({\varepsilon _{\Omega _{1}+1}}),\varepsilon _{\Omega _{1}+1}\} \).

  49. 49.

    In case that the notation for \(\alpha \) needs n-ary Veblen functions it is wise to count also these functions among the generating functions of \(\mathcal{H}\). This is, however, not absolutely necessary. Even in our definition of \(\mathcal{H}\) the Veblen functions \(\varphi _{\xi }\) for \(\xi >0\) are dispensable in principle. These Veblen functions can be expressed in terms of the \(\Psi \)-functions. However, the then obtained results look somewhat weird, very unfamiliar and thus much less attractive (and are therefore also more difficult to read and to memorize).

  50. 50.

    Examples could be systems in weak second-order logic, e.g. (\(\Delta ^1_1\)-CA).

  51. 51.

    Cf. [19].

  52. 52.

    As an example the Peano axioms suffice to prove the properties of \(\mathcal{H}^{\Gamma _{\Omega _{\nu }+1}}(0)\).

  53. 53.

    Recall that an axiom system is simple, if its spectrum is a singleton.

  54. 54.

    Recall that in general we have no ordinals in \({\mathfrak M}_{\mu }\) but have to represent them by elements in \(O_\mu \) as discussed in Sect. 3.3.3.1.

  55. 55.

    Cf. Sect. 3.3.2.2.

  56. 56.

    Without having checked that we conjecture that also the “No Enhancement Theorem” can be generalized to this situation.

  57. 57.

    A more detailed proof sketch can be found in [33].

  58. 58.

    E.g. \(\Phi (n):=3^{n+2}\) is sufficiently increasing for the basis structure \({\mathfrak N}\).

  59. 59.

    Cf. [7].

  60. 60.

    Such a result has been first obtained by Andreas Weiermann. (Cf. [40, 2]).

  61. 61.

    The intersection with \(\Omega _{1}\) is of course superfluous and should only illustrate the bridge to the previous results.

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Acknowledgements

I want to thank the anonymous referee for her or his criticism and the many extremely helpful comments and corrections.

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  1. Institut für Mathematische Logik und Grundlagenforschung, Einsteinstraße 62, 48149, Münster, Germany

    Wolfram Pohlers

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  1. Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal

    Fernando Ferreira

  2. Carl Friedrich von Weizsäcker-Zentrum, Universität Tübingen, Tübingen, Germany

    Reinhard Kahle

  3. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Giovanni Sommaruga

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Pohlers, W. (2022). On the Performance of Axiom Systems. In: Ferreira, F., Kahle, R., Sommaruga, G. (eds) Axiomatic Thinking II. Springer, Cham. https://doi.org/10.1007/978-3-030-77799-9_3

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