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The Operational Perspective: Three Routes

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Advances in Proof Theory

Part of the book series: Progress in Computer Science and Applied Logic ((PCS,volume 28))

Abstract

Let me begin with a few personal words of appreciation, since Gerhard Jäger is one of my most valued friends and long time collaborators.

For Gerhard Jäger, in honor of his 60th birthday.

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Notes

  1. 1.

    Let me also take this opportunity to thank Thomas Strahm and Thomas Studer for organizing the December 2014 meeting in honor of Gerhard Jäger, and for arranging for me to participate via Skype since I was unable to attend in person.

  2. 2.

    There are several possible formulations of the definition by cases operator. In the one originally taken in [23], sometimes called definition by cases on V, this takes the form d xyuv = (x if \(u=v\), else y). However, when added to the axioms for k and s, extensionality is inconsistent for operations. More restrictive versions have subsequently been used, mainly definition by cases on the natural numbers, allowing both extensionality and totality of operations; cf. [59].

  3. 3.

    First steps in that direction had already been made in [19] via the system IR. For subsequent explorations cf. [20, 22, 26].

  4. 4.

    In [25] I also examined T\(_{0}\) within intuitionistic logic.

  5. 5.

    The scheme ECA can be finitely axiomatized by adding constants for the identity relation, the first-order logical operations for negation, conjunction, existential quantification, and inverse image of a class under an operation.

  6. 6.

    There is a difference in terminology, though: Jäger used ‘types’ for our classes.

  7. 7.

    In certain subsystems of T\(_{1}\) with restricted induction we need to add to the (\(\upmu )\) axiom that if \({\upmu }f \in \) N then\( f \in \) (N \(\rightarrow \) N).

  8. 8.

    Parts of T\(_{0}\) relate to Aczel’s Frege structures and Martin-Löf’s constructive theory of types; cf. for example, Beeson [5], Chaps. XI and XVII. But neither of these approaches goes on to the adjunction of non-constructive functional operators like \(\mu \) (or E\(_{0})\) and E\(_{1}\).

  9. 9.

    For the proof theory of systems of explicit mathematics with \({\mathbf {E}}_{1}\) see Jäger and Strahm [60] and Jäger and Probst [58].

  10. 10.

    Then Jäger [47] and Avigad [3] showed that they are of the same proof-theoretical strength, by proof-theoretical methods, the first via theories of iterated admissible sets without foundation and the second via fixed point theories.

  11. 11.

    In Sect. 4 below I conjecture that the unfolding of a suitable subsystem of T\(_{0}\) is equivalent in strength to predicative analysis.

  12. 12.

    Recently, Sato [74] has shown how to establish the reduction of \(\Delta ^{1}_{2}\) − CA + BI to T\(_{0}\) without going through the ordinal notation system for \(\kappa \).

  13. 13.

    Another interesting group of questions concerns the strength over T\(_{0}\) (or its restricted version \(\text {T}_0{\upharpoonright }\)) of the principle MID that I introduced in [27]. That expresses that if f is any monotone operation from classes to classes then f has a least fixed point. Takahashi [80] showed that \(\text {T}_0\) + MID is interpretable in \(\Pi ^1_2\) − CA + BI, and then Rathjen [69] showed that it is much stronger than T\(_{0}\). Next, exact strength of \(\text {T}_0{\upharpoonright }\) + MID was determined by Glass et al. [42]. A series of further results by Rathjen for the strength of \(\text {T}_0\) + MID and \(\text {T}_0\) + UMID, where UMID is a natural uniform version of the principle, are surveyed in the paper Rathjen [70].

  14. 14.

    To explain some anomalies of the dates of subsequent work on this subject, it should be noted that my 2009 paper was submitted to the journal Information and Control in December 2006 and in revised form in April 2008. In the meantime, Jäger [50] had appeared and so I could refer to it in that revised version.

  15. 15.

    Further important work is contained in Zumbrunnen [87] and Sato and Zumbrunnen [75]. Constructive operational theories of sets have been treated by Cantini and Crosilla [12, 13] and Cantini [11].

  16. 16.

    Rathjen [71] uses (Pow(P)) for our formulation in the language of KP as well, in order to distinguish it from the usual power set axiom formulated without the additional constant. It would have been better to do that in [34], but not having done so I here follow the notation from there.

  17. 17.

    An earlier such result for the system with a restricted form of set induction is due to Mathias [66].

  18. 18.

    However, I did say that I had not checked the details. In fact, I hadn’t thought them through at all.

  19. 19.

    Cf. Footnote 2.

  20. 20.

    Since referred to as KF in the literature.

  21. 21.

    This follows the proposed formulation of U(NFA) via a truth predicate in Feferman [31, p. 14].

  22. 22.

    Ulrik Buchholtz originally thought that \(\uppsi (\Gamma _{\Omega +1})\) is the same as the ordinal H(1) of Bachmann [4]. This seemed to be supported by Aczel [1] who wrote (p. 36) that H(1) may have proof theoretical significance related to those of the ordinals \(\varepsilon \), \(\Gamma _{0}\) and \(\upvarphi \varepsilon _{\Omega +1}\)0. And Miller [67, p. 451] had conjectured that “H(1) [is] the proof-theoretic ordinal of ID\(_{1}\)* which is related to ID\(_{1}\) as predicative analysis ID\(_{0}\)* is to first-order arithmetic ID\(_{0}\).” However, Wilfried Buchholz recently found that the above representation of H(1) in terms of the \(\uppsi \) function is incorrect. This suggests one should revisit the bases of Aczel’s and Miller’s conjectures.

  23. 23.

    Alternatively, one can of course work with the formulation of explicit mathematics in terms of the representation relation \(\mathfrak {R}(x\), X).

  24. 24.

    This would provide another answer to the question of finding a system of explicit mathematics of the same strength as predicative analysis.

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Acknowledgments

I wish to thank Ulrik Buchholtz, Gerhard Jäger, Dieter Probst, Michael Rathjen, and Thomas Strahm for their helpful comments on a draft of this article.

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    Solomon Feferman

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    Reinhard Kahle

  2. Inst.of Computer Sci. applied math., University of Bern, Bern, Switzerland

    Thomas Strahm

  3. Institute of Computer Science and A, University of Berne, Berne, Switzerland

    Thomas Studer

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Feferman, S. (2016). The Operational Perspective: Three Routes. In: Kahle, R., Strahm, T., Studer, T. (eds) Advances in Proof Theory. Progress in Computer Science and Applied Logic, vol 28. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29198-7_7

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