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Predicativity and Feferman

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Feferman on Foundations

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

Abstract

Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell’s type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predicativity have since taken a life of their own, but have also produced a deeper understanding of the notion of predicativity, therefore witnessing the “light logic throws on problems in the foundations of mathematics.” [30, p. vii] Predicativity has been at the center of a considerable part of Feferman’s work: over the years he has explored alternative ways of explicating and analyzing this notion and has shown that predicative mathematics extends much further than expected within ordinary mathematics. The aim of this note is to outline the principal features of predicativity, from its original motivations at the start of the past century to its logical analysis in the 1950–1960s. The hope is to convey why predicativity is a fascinating subject, which has attracted Feferman’s attention over the years.

Dedicated to Professor Solomon Feferman

I thank Andrea Cantini for helpful comments on a draft of this article. I also profited from remarks by Gerhard Jäger and Wilfried Sieg, as well as two anonymous referees. I gratefully acknowledge funding from the School of Philosophy, Religion and History of Science of the University of Leeds and from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement n. 312938.

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Notes

  1. 1.

    See, for example, [69,70,71,72,73, 80,81,82].

  2. 2.

    See [79, 82, 97, 99].

  3. 3.

    See [16, 54, 85, 86].

  4. 4.

    See e.g. [26, 33, 36, 88, 89].

  5. 5.

    See, for example, [34], and [16, 24,25,26,27,28,29,30,31, 33].

  6. 6.

    Feferman [34] writes: “Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and so I shall give them special attention.”

  7. 7.

    In his introduction to a chapter on the ordinal analysis of predicativity, Pohlers [68, p. 134] writes: “The notion of predicativity is still controversial. Therefore we define and discuss here predicativity in a pure mathematical – and perhaps oversimplified – setting.” See [22, 50, 55, 56, 96] for discussions pertaining to the logical analysis of predicativity. See also the discussion on “metapredicativity” in [52]. As to the philosophical and historical aspects of predicativity, see, e.g., [32, 37, 48, 59, 65,66,67].

  8. 8.

    Predicativity-related themes have appeared in different forms over the years, both in classical and constructive settings. In fact, predicativity is gaining renewed prominence today especially in the constructive context. I shall postpone to another occasion a discussion of other forms of predicativity, as the constructive predicativity which characterizes Martin-Löf type theory see, e.g., [60,61,62] and forms of “strict predicativity” [64, 65, 67]. See also [11, 12].

  9. 9.

    In the following, I shall also write “predicativity” to denote predicativity given the natural numbers. See Sect. 3.4 for some remarks on the notion of predicativity given the natural numbers.

  10. 10.

    See e.g. [92, 95].

  11. 11.

    See, in particular, [73, 97]. See also [2, 42, 58] for a discussion of arbitrary sets and [66] for an analysis of Weyl’s conception of predicative set and its reception by Weyl’s contemporaries.

  12. 12.

    See [69,70,71, 81, 82]. Note that the term “class” is used here as in Russell and Poincaré’s texts, that is, to refer to a generic collection. Hence it should be carefully distinguished from the notion of proper class that is found in contemporary set theory. In the original literature one frequently finds also the word “totality”. In this section I shall try to avoid the use of the term “set”, since the latter has in the meantime acquired additional connotations (as set in e.g. ZFC) that should not be presupposed in this discussion.

  13. 13.

    See [69,70,71, 80, 81]. See e.g. [47, p. 455] for discussion. Note also that today the distinction between predicative and impredicative definitions is typically framed as relating to sets. However, Russell and Poincaré’s discussions are concerned with definitions of different kinds of entities, including propositions, properties, etc.

  14. 14.

    The issue of how we establish whether an entity is impredicative (and in which context) is more complex than this coarse characterization of impredicativity may suggest. This complexity was further addressed by the development of Russell’s type theory, Weyl’s [97] and the logical analysis of predicativity to be discussed below.

  15. 15.

    Here I shall follow [7, p 48].

  16. 16.

    Carnap [7, p. 48] concludes that this definition of natural number is “circular and useless”. It is worth recalling that Carnap in [7] also hints at a form of platonism, attributed to Ramsey (but not endorsed by Carnap), which finds no fault with impredicative definitions. See also [47, 73] for further discussion.

  17. 17.

    Poincaré’s texts make use of other examples, more directly drawn from the mathematical practice. See also [13, 14] for a similar reading of Russell’s paradox. Cantini [4] proposes a detailed analysis of Poincaré’s ideas.

  18. 18.

    See [55] for discussion of this characterization from a modern logical perspective. See also [18, 39].

  19. 19.

    My translation; italics by Poincaré. The word “disordered” translates the French “bouleverseé”.

  20. 20.

    See [5] for a rich discussion of the impact of the paradoxes on mathematical logic.

  21. 21.

    Russell’s ideas on type theory appeared first in an appendix to [79], and were further developed (with ramification) in [82] and then in [99].

  22. 22.

    In the present context we may follow [34], and identify the notion of propositional function with that of open formula, i.e. a formula with a free variable, say \(\varphi (x)\). Note, however, that the interpretation of the notions of proposition and propositional function in Russell is complex. See e.g. [57].

  23. 23.

    See [9, 74].

  24. 24.

    See [49] for a discussion of the reasons that might support Russell’s (and Whitehead’s) choice of a ramified type theory over a simple type theory.

  25. 25.

    See [82]. See also [63]. See [10, 31] for introductory expositions of the ideas underlying ramified type theory and the difficulties it encounters.

  26. 26.

    Russell [82, p. 243] also writes: “Thus a predicative function of an individual is a first-order function; and for higher types of arguments, predicative functions take the place that first-order functions take in respect of individuals. We assume, then, that every function is equivalent, for all its values, to some predicative function of the same argument.”.

  27. 27.

    Wilfried Sieg has informed me about perceptive discussions by Hilbert and Bernays on predicativity and Russell’s logicism, including the axiom of reducibility. See e.g. Hilbert’s lecture notes from 1917/18 entitled “Prinzipien der Mathematik” and those from 1920 entitled “Probleme der mathematischen Logik” published in [15]. See also [87] for discussion. Sieg [87] also draws important correlations between [97] and Hilbert and Bernays’ work around 1920. As suggested by Sieg, the relations between Hilbert and Bernays ’ writings and Weyl’s [97] deserve more thorough investigations.

  28. 28.

    The (LUB) states that every bounded, non–empty subset M of the real numbers has a least upper bound. See [16, 26] for discussion.

  29. 29.

    Weyl, in particular, made use of sequential rather than Dedekind completeness, the first being amenable to predicative treatment. See also [26].

  30. 30.

    In addition to properties, Weyl [97] also considers relations, here omitted for simplicity.

  31. 31.

    The notion of “judgment” is so clarified by Weyl [97, p. 5]: a “judgment affirms a state of affairs”.

  32. 32.

    Weyl also considers a principle of substitution [97, p. 10]. In addition, in the paradigmatic case of the natural numbers as basic domain, one also applies a principle of iteration, as further discussed below.

  33. 33.

    As remarked by Feferman [26] see also [31], it is not completely clear how strong is the system Weyl sketches in [97]. Feferman has, however, verified that system W of [26], which is inspired by [97], suffices to carry out all of Weyl’s constructions in “Das Kontinuum”. System W is a conservative extension of Peano Arithmetic, PA [38]. As clarified in Sect. 3.3, Feferman has also shown that W allows for the development of a more extensive portion of contemporary analysis, compared with [97].

  34. 34.

    Poincaré see, e.g., [71] states that mathematical induction is synthetic a priori. Note also that Weyl expresses mathematical induction by appeal to a principle of iteration. See [30, p. 264-5] for discussion. Poincaré and Weyl fully realized the significance of the assumption of unrestricted mathematical induction. This is further clarified by a comparison with approaches to predicativity which instead introduce restrictions on induction [64, 65].

  35. 35.

    See [9, 74].

  36. 36.

    See also [34] for additional thoughts on what “pushed predicativity to the sidelines.”.

  37. 37.

    Gödel [47] also mentions a prominent example of the fruitfulness of predicativity: the constructible hierarchy [45, 46], inspired by Russell’s idea of ramification.

  38. 38.

    See also Sect. 3.4 for more on the notion of predicativity given the natural numbers.

  39. 39.

    The use of classical logic marks a crucial difference with the form of predicativity that is to be found in e.g. Martin-Löf type theory [60].

  40. 40.

    The hyperarithmetical hierarchy has a central place in the development of mathematical logic because of its prominence within a number of fundamental areas in mathematical logic: definability theory, recursion theory and admissible set theory. This witnesses the centrality within logic of themes that pertain to the predicativity debate, and further explains the interest of this notion form a logical point of view.

  41. 41.

    In the language of second order arithmetic, a \(\Sigma ^1_1\) formula is one of the form: \(\exists X \, \varphi (X)\), with \(\varphi \) an arithmetical formula, that is, a formula that does not quantify over sets (but may quantify over natural numbers). Note that here the upper case letter X denotes a second order variable, standing for a set of natural numbers. A \(\Pi ^1_1\) formula is one of the form \(\forall X \, \psi (X)\), with \(\psi \) an arithmetical formula.

  42. 42.

    See [53], see also [84]. See [16, 55] for further clarification of why this may be seen as offering a predicative justification for this kind of second order quantification. Kreisel [55] offers additional considerations that directly relate to Poincaré’s notion of invariance discussed above.

  43. 43.

    See also [95].

  44. 44.

    See also the review by Gandy [44].

  45. 45.

    Here ordinals are not to be considered set-theoretically, rather as notations from a suitable ordinal notation system. See [68] for details on ordinal notation systems.

  46. 46.

    See e.g. [68, Ch. 1] for details. In the branch of proof theory known as ordinal analysis, suitable (countable) ordinals, termed “proof-theoretic ordinals”, are assigned to theories as a way of measuring their consistency strength and computational power. The “proof-theoretic strength” of a theory is then expressed in terms of such ordinals. The countable ordinal \(\Gamma _0\) is the proof-theoretic ordinal assigned to the progression of ramified systems mentioned above. It is relatively small in proof-theoretic terms. As a way of comparison, it is well above the ordinal \(\epsilon _0\) which encapsulates the proof-theoretic strength of Peano Arithmetic, but it is much smaller than the ordinal assigned to a well–known theory, called \({ ID}_1\), of one inductive definition. The latter ordinal is known in the literature as the Bachmann–Howard ordinal [3]. The strength of \({ ID}_1\) is well below that of second order arithmetic, which is in turn much weaker than full set theory. For surveys on proof theory and ordinal analysis see, for example, [76,77,78].

  47. 47.

    See [50, p. 283] for discussion.

  48. 48.

    Further approaches to predicativity were explored, for example, in [17, 19, 20, 22, 23]. See also [21]. More recently, Feferman has developed the notion of “unfolding”; see [40, 41] and [93]. See also [6] for relations between Feferman’s work on predicativity and theories of truth.

  49. 49.

    See e.g. [28] for discussion of this notion, and [34] for an informal account of its application to an analysis of predicativity.

  50. 50.

    The expression “ordinary mathematics” refers to mainstream mathematics, and has been so characterized, for example, by Simpson [89, p. 1]: “that body of mathematics which is prior to or independent of the introduction of abstract set-theoretic concepts”. That is: “geometry, number theory, calculus, differential equations, real and complex analysis, countable algebra, the topology of complete separable metric spaces, mathematical logic and computability theory”.

  51. 51.

    See [89] for details and [90] for independence results.

  52. 52.

    See [26, 34, 88] for informal discussions and further references.

  53. 53.

    Note that Feferman draws different conclusions on the impact of the logical research on indispensability arguments in the philosophy of mathematics [28].

  54. 54.

    In the following I shall often omit explicit reference to the unrestricted principle of mathematical induction, and simply write “the natural numbers”; however, I shall presuppose that in the case of predicativity given the natural numbers full induction is also assumed.

  55. 55.

    Feferman [16, 34] also describes the predicativist position as one that takes the natural numbers as a “completed totality”, and views the rest in potentialist terms. However, I could find no further elucidation of the notion of complete totality, beyond the claim that we can use classical logic to reason about it. In [32, 35], Feferman proposes to read the “giveness” of the natural numbers in terms of realism in truth value (restricted to the natural numbers). A fundamental theme that emerges within Feferman’s discussions on predicativity is an opposition, analogous to Weyl’s, to arbitrary sets, and in particular to the powerset of an infinite set (see e.g., [33]).

  56. 56.

    See e.g. [29, 54].

  57. 57.

    Wilfried Sieg has drawn my attention to a passage in Hilbert’s 1920s lectures [15, p. 363-4] which suggests that this view of predicativity is in agreement with a Hilbertian perspective. It is also worth observing that with the shift of the logical analysis of predicativity to proof-theoretic considerations this enterprise gained a clear Hilbertian character.

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  1. School of Philosophy, Religion and History of Science, University of Leeds, LS2 9JT, Leeds, UK

    Laura Crosilla

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    Gerhard Jäger

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

    Wilfried Sieg

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Crosilla, L. (2017). Predicativity and Feferman. 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_15

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