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Feferman’s Skepticism About Set Theory

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Feferman on Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 13))

Abstract

Solomon Feferman has expressed skepticism or reserve about set theory, especially higher set theory, in many writings, and in his mathematical work he has largely stayed away from set theory. The paper undertakes to describe and diagnose Feferman’s attitude toward set theory, especially higher set theory. Section 1 discusses his opposition to Platonism in relation to some understandings of what Platonism is. Section 2 discusses his interest in predicativity, his analysis of predicative provability, and his sympathy for “predicativism,” although he denies being an adherent and has used impredicative methods in his own metamathematical work. It also notes his reconstruction of Hermann Weyl’s attempt to construct the elements of analysis on a predicative basis. Section 3 concerns his attitude toward proof theory. Section 4 turns more to philosophy. His anti-platonism is compared with the well known platonism of Gödel. Unlike Gödel, Feferman views concepts as human creations. He notes that basic mathematical concepts can differ in clarity and argues that less clear concepts, in particular those of set theory, can give rise to questions that do not have definite answers. It is questioned whether Feferman’s conceptual structuralism gives mathematics the degree of objectivity that its application in science requires. In Sect. 5 remarks are made about Hilary Putnam’s criticism of Feferman’s claim that a predicative system conservative over PA is adequate for the mathematics applied in science. The difference is seen to turn on Putnam’s scientific realism.

This paper was in virtually final form before Sol Feferman’s unfortunate death in July 2016. That event implies that this volume will not contain the reply by him that was originally envisaged. However, Feferman [17], presented in a symposium at Columbia University in April 2016, contains ideas that he would probably have expressed in his reply.

Thanks to Wilfried Sieg for comments on earlier versions and to him and Peter Koellner for discussions of Feferman’s views over an extended period.

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Notes

  1. 1.

    Consideration of some large cardinal notions seems to have been forced on him by developments in proof theory.

  2. 2.

    For further discussion of Tait on realism, see Sect. 3 of Parsons [15].

  3. 3.

    It is likely that he would see a difficulty of principle in establishing such a limit with mathematical precision; cf. the remarks in Sect. 2 below on his analysis of predicativity.

  4. 4.

    The earliest in date of publication of any of the essays in In the Light of Logic (Feferman [10]) is 1979.

  5. 5.

    See his introductory note to Gödel’s letters to Cohen in Gödel [25]. Cohen did not allow his letters to Gödel to be published, although some information about their content can be gleaned from Feferman’s introduction.

  6. 6.

    Feferman’s own papers will be cited only by the number in the list of references.

  7. 7.

    One such person is Hilary Putnam in Putnam [37], p. 137. In fairness to Putnam, it should be noted that he has primarily in mind those essays in which Feferman argues that the mathematics applied in natural science can be developed in theories that are conservative extensions of PA and predicative by his own lights. (Cf. the remarks on Putnam at the end of Sect. 5 below.)

  8. 8.

    Usually translated as, “God created the natural numbers; everything else is the work of man.” As we shall see in Sect. 4, Feferman could accept the theological framework only as metaphor.

  9. 9.

    Throughout it is predicativity relative to the natural numbers that is under consideration. That is usually called predicativity simpliciter. Clearly that is what is at issue in the quotation just given.

  10. 10.

    The results concerning ramified systems were obtained independently by Kurt Schütte, who also brought out the significance of the ordinal \(\Gamma \) \(_{0}\). See Schütte [38, 39]. I myself was fascinated by Feferman’s and Schütte’s work. In 1970 I obtained a Gödel functional interpretation (in Spector’s extensional version) of IR and some related systems. See the abstract Parsons [31].

  11. 11.

    I don’t believe that in the earlier history of foundations predicativism was a very well-defined position. Cf. the comments on this issue in Parsons [34], pp. 63–65.

  12. 12.

    Reference [3], p. 486. Some years later, William Tait said virtually the same thing about his own analysis of fintism. See Tait [42], pp. 527, 533 of original.

  13. 13.

    One could ask whether this is still true of work in the 1980s and later, where work on stronger classical systems aimed at proof-theoretic reduction to Per Martin-Löf’s intuitionistic theory of types, a much more powerful theory than had been envisaged in traditional intuitionism.

  14. 14.

    Feferman indicates that this is a reason in the Preface to [10], p. ix.

  15. 15.

    See Parsons [35]. I identify as advocates of the view Brouwer, Weyl, and Hilbert.

  16. 16.

    While working on this paper I was not able to find my copy of that manuscript, although I was present at the talk. But see the comments below on the theses as published in [15].

  17. 17.

    See Parsons [33], pp. 97–99 of reprint. For some aspects of this view, particularly (1) (which is close to but not quite the same as Tait’s default realism), ‘realism’ is a more appropriate term. In his discussions with Hao Wang Gödel preferred the term ‘objectivism’, which has not been taken up by others.

  18. 18.

    Gödel [24], p. 320.

  19. 19.

    Ibid., p. 325.

  20. 20.

    Reference [15], p. 170. This is the first of the ten “theses of conceptual structuralism.”

  21. 21.

    It follows, I believe, that he would reject Michael Dummett’s view that asserting bivalence of statements independently of whether we have any means of deciding their truth is a criterion of realism.

  22. 22.

    Donald A. Martin is at least inclined to that view; see Martin [28]. However, Martin holds that if CH does not have a determinate truth-value, then a structure that would be a standard model of third-order arithmetic does not exist. In his language, the concept would not be instantiated. It is still possible to investigate mathematically what is implied by the concept. See further Martin [29].

  23. 23.

    In the case of institution-dependent facts such as my example of home ownership, it is not independent of what anyone thinks. Would it still be true that we own the house in Springfield if it came to be widely believed, in particular by those in authority, that the records in the Registry of Deeds of Sullivan County, New Hampshire, are meaningless pieces of paper or records of a hopelessly distant past? (We bought the house in 1973.)

  24. 24.

    Gödel [23], p. 260, from the 1964 version of “What is Cantor’s continuum problem?”.

  25. 25.

    An even more radical view in this direction is expressed in Hamkins [26].

  26. 26.

    Gödel [23], p. 128 (from “Russell’s mathematical logic”).

  27. 27.

    Of course if the ascent is into the transfinite, the issues arise that are addressed by the autonomy condition on ordinal levels in Feferman’s own analysis.

  28. 28.

    This is quite in accord with Gödel’s classic analysis of Russell’s vicious circle principle.

  29. 29.

    I am not confident enough in my understanding of Feferman’s systems of explicit mathematics to say whether they are naturally interpreted so that all sets are definable. Evidently this could be so only with the help of inductively defined predicates.

  30. 30.

    See Simpson [41].

  31. 31.

    Of great importance was Feferman’s work with Leon Henkin in 1955–56, when Tarski was away. See his [9, 16], as well as his autobiography in this volume.

  32. 32.

    See p. 423 of his contribution to Feferman et al. [13].

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  1. Department of Philosophy, Harvard University, Emerson Hall 25 Quincy St, Cambridge, MA, 02138, USA

    Charles Parsons

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  1. Institut für Informatik, Universität Bern, Bern, Switzerland

    Gerhard Jäger

  2. Department of Philosophy, Carnegie Mellon University Dept. Philosophy, Pittsburgh, Pennsylvania, USA

    Wilfried Sieg

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Parsons, C. (2017). Feferman’s Skepticism About Set Theory. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_20

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