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Turing, Gödel and the “Bright Abyss”

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Philosophical Explorations of the Legacy of Alan Turing

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 324))

Abstract

We consider the development of computability in the 1930s from what we have called the formalism free point of view. We employ a dual conceptual framework: confluence together with grounding.

In Gödel’s view, the Turing analysis of computability grounded that notion.

We follow that idea forward, through to Gödel’s introduction, in his 1946 Princeton Bicentennial Lecture, of the concept of ordinal definability in set theory. In particular we trace the influence of Turing’s analysis of computability on the provisional program for definability and, to a lesser extent, provability laid out in that lecture by Gödel.

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Notes

  1. 1.

    Hadamard writes of nonverbal thought, images that resist statement in words, sudden insight, flashes of inspiration. See (Burgess 2015).

  2. 2.

    Some would claim to the contrary that Turing proved a theorem equating human effective calculability with Turing calculability. See below. The faithfulness problem is only a problem for those who take the view that a sharply individuated concept is present, intersubjectively, in the first place. In his (2009) Hodges takes a different view: “Turing’s Thesis is the claim that independent of Turing we have an intuitive notion of an effectively computable function, and Turing’s analysis exactly captures this class of functions... This kind of claim is impossible to verify. Work like Turing’s has a power of creating intuitions. As soon as we read it, we lose our previous innocence.”

  3. 3.

    This is in contrast to the First Incompleteness Theorem, which has nothing to do with the meaning of the undecidable sentence G—all that matters in that case is that G is undecidable. See (Feferman 1960/1961), (Detlefsen 2001), (Franks 2009) and (Pudlak 1996).

  4. 4.

    As given in their (1934/1968).

  5. 5.

    This is “Thesis T” in (Gandy 1988).

  6. 6.

    See (Shapiro 2013).

  7. 7.

    See (Sieg 1997).

  8. 8.

    (Kripke, 2013), p. 81. See also (Burgess 1992), (Rav 2007) and (Shapiro 2013).

  9. 9.

    Turing’s Theorem is defined by Gandy as follows: Any function which is effectively calculable by an abstract human being following a fixed routine is effectively calculable by a Turing machine—or equivalently, effectively calculable in the sense defined by Church—and conversely. (Gandy 1988), p. 83.

  10. 10.

    See (Wimsatt 1994). In computability theory the term “confluence” seems to be preferred over “robustness.” See e.g. (Gandy 1988).

  11. 11.

    In set theory.

  12. 12.

    This is because the concept of “natural number” is usually defined in terms of finiteness. D.A. Martin expressed a similar thought in his recent (2012):

    There are various ways in which we can explain to one another the concept of the sequence of all the natural numbers or, more generally, the concept of an ω-sequence. Often these explanations involve metaphors: counting forever; an endless row of telephone poles (or cellphone towers); etc. If we want to avoid metaphor, we can talk of an unending sequence or of an infinite sequence. If we wish not to pack to pack so much into the word “sequence,” then we can say that that an ω-sequence consists of some objects ordered so that there is no last one and so that each of them has only finitely many predecessors. This explanation makes the word “finite” do the main work. We can shift the main work from one word to another, but somewhere we will use a word that we do not explicitly define or define only in terms of other words in the circle. One might worry—and in the past many did worry—that all these concepts are incoherent or at least vague and perhaps non-objective.

  13. 13.

    See (Harris 2015). According to Harris, Pierre Deligne seems to have been the first to use the term “avatar” in its standard mathematical meaning, i.e. “manifestation,” or “alternative version.”

  14. 14.

    Foundational formalism was coined in (Kennedy 2013) in order to refer to the idea prevalent in foundations of mathematics in the early part of the twentieth century, associated mainly with the Hilbert Program, of embedding the mathematical corpus into a logical calculus consisting of a formal language, an exact proof concept, and (later) an exact semantics, such that the proof concept is sound and complete with respect to the associated semantics as well as syntactically complete in the sense that all propositions that can be written in the formalism are also decided.

  15. 15.

    An example are the Abstract Elementary Classes, classes of models in which the elementary submodel relation is replaced by an abstract relation satisfying certain mathematical properties.

  16. 16.

    We use the term logic in this sense, and the term formalism, or formal system, interchangeably.

  17. 17.

    See (Kennedy 2013) and (Kennedy et al. forthcoming).

  18. 18.

    In detail, the probability that a random relational structure on the domain {1, ... , n} satisfies a given first order formula tends to either 0 or 1 as n tends to infinity. But if we allow function symbols as part of the language, even the simple sentence ∃x(f(x) = x) has limit probability 1/e. See Fagin, [14].

  19. 19.

    The slogan is Colin McLarty’s description of Poincaré’s view, which he calls “expansive intuitionism” (1997). In her (2014), p.367 Danielle Macbeth describes Yehuda Rav’s similar thought, of the distinction between “…a mathematician’s proof and a strictly reductive, logical proof, where the former is ‘a conceptual proof of customary mathematical discourse, having irreducible semantic content,’ and the latter, which Rav calls a derivation, is a syntactic object of some formal system.” (subquote in (Rav 1999). She remarks, “Just as Poincaré had argued, to formalise a mathematical proof is to destroy it… ” Gödel would echo the thought in notes to himself in 1944 (unpublished, XI): “Is the difference between living and lifeless perhaps that its laws of action don’t take the form of “mechanical” rules, that is, don’t allow themselves to be “formalized”? That is then a higher level of complication. Those whose intuitions are opposed to vitalism claim then simply: everything living is dead.”

  20. 20.

    (Gandy 1988), p. 93.

  21. 21.

    As Kripke observed in his recent (2013).

  22. 22.

    In (1936) Post gave a similar analysis to that of Turing’s, “with only Church [(1936)] to hand.” See (Kleene 1981a).

  23. 23.

    For a partial list of Gödel remarks on Turing computability see Gödel *193? (in Gödel 1995); see also the Gibbs Lecture, *1951, the 1965 Postscriptum to the (Gödel 1934) Princeton lectures. Gödel remarks to Wang in (Wang 1996), and the Wang-Gödel correspondence (in Gödel 2003b) are also relevant. These are discussed below.

  24. 24.

    The writing of this paper and the particular use in it of the concept of autonomy owes much to the notion of autonomy outlined in Curtis Franks’s (2009).

  25. 25.

    As Gödel wrote to himself in 1943, “A board game is something purely formal, but in order to play well, one must grasp the corresponding content [the opposite of combinatorically]. On the other hand the formalization is necessary for control [note: for only it is objectively exact], therefore knowledge is an interplay between form and content.” (Gödel unpublished IX, quoted in Floyd and Kanamori (2016). Absolute provability and absolute decidability are directly connected. See (van Atten and Kennedy 2009).

  26. 26.

    Or so some have argued. See, e.g. (Rescorla 2007) and the rebuttal to it by Copeland and Proudfoot in their (2010).

  27. 27.

    Gödel used the term “recursive” for what are now called the primitive recursive functions. The primitive recursive functions were known earlier. Ackermann (1928) produced a function which is general recursive, in the terminology of Gödel 1934 Princeton lectures, but not primitive recursive. See also (Péter 1935,1937).

  28. 28.

    (Gandy 1988), section 14.8. As mentioned above, following Gandy we use the term “effectively computable,” or just “effective,” to mean “intuitively computable.”

  29. 29.

    About which Martin Davis would remark, unimprovably, “Not exactly what one dreams of having one’s graduate students do for one” (Gandy 1988, p. 70).

  30. 30.

    See (Kleene and Rosser 1935), which relies on Church’s (1934).

  31. 31.

    See below. The proof of the equivalence developed in stages. See (Davis 1982) and (Sieg 1997).

  32. 32.

    Church, letter to Kleene November 29, 1935. Quoted in (Sieg 1997) and in (Davis 1982).

  33. 33.

    Church, letter to Kleene as quoted in (Sieg 1997).

  34. 34.

    The phrase “Church’s Thesis” was coined by Kleene in his (1943). About the attitude of Church toward the idea of taking the λ-calculus as canonical at this point Davis remarks, “The wording [of Church’s published abstract JK] leaves the impression that in the early spring of 1935 Church was not yet certain that λ-definability and Herbrand-Gödel general recursiveness were equivalent. (This despite Church’s letter of November 1935 in which he reported that in the spring of 1934 he had offered to Gödel to prove that ‘any definition of effective calculability which seemed even partially satisfactory... was included in λ-definability’)” (Davis 1982, p. 10). See Church’s (1935) abstract and (Church 1936).

  35. 35.

    (Church 1936), p. 357.

  36. 36.

    (Kripke 2013), p. 81.

  37. 37.

    (Kripke 2013), p. 80. Kripke’s point in the paper is that such arguments, being valid arguments, can be, via Hilbert’s thesis, stated in a first order language. But then the solution of the Entscheidungsproblem follows almost trivially from the Completeness Theorem for first order logic.

  38. 38.

    Conditions (i)–(iii) are Sieg’s formulation of Church’s conditions. See (Sieg 1997) p. 165. For Gandy’s formulation of the step-by-step argument, see his (1988), p. 77.

  39. 39.

    (Church 1936), footnote 21, pp. 357–358.

  40. 40.

    See also Sieg’s discussion of the “semi-circularity” of the step-by-step argument in his (1997).

  41. 41.

    The comparison of Church’s (1936) with Hilbert and Bernays’s (1939/1970) follows that of Sieg’s in his account (2006). See also (Gandy 1988).

  42. 42.

    As Mundici and Sieg wrote in their (1994), “The analysis of Hilbert and Bernays revealed also clearly the ‘stumbling block’ all these analyses encountered: they tried to characterize the elementary nature of steps in calculations, but could not do so without recurring to recursiveness (Church), primitive recursiveness (Hilbert and Bernays), or to very specific rules (Gödel).”

  43. 43.

    (Gandy 1988), p. 71.

  44. 44.

    See (Post 1936). For a penetrating analysis of Post’s work of the 1920s see (de Mol 2006). Post was apparently aware of the work of the Princeton group, but he was unaware of Turing’s. See (Gandy 1988).

  45. 45.

    See (Gandy 1988), p. 72.

  46. 46.

    (Gödel 1986), p. 195. P is a variant of Principia Mathematica.

  47. 47.

    (Gödel 2003b), p. 339.

  48. 48.

    (Gödel 2003b), p. 341.

  49. 49.

    Gödel responded to von Neumann in writing, but his letters seem to have been lost. See (Sieg 2009), p. 548. We know of his response through the minutes of a meeting of the Schlick Circle that took place on 15 January 1931, which are found in the Carnap Archives of the University of Pittsburgh. See Sieg’s introduction to the von Neumann-Gödel correspondence in (Gödel 2003b), p. 331.

  50. 50.

    (Gödel 2003b), p. 23.

  51. 51.

    (Gödel 1995), p. 51. By “system A” Gödel means to refer to systems consisting of finitistically allowable methods. See Feferman’s discussion of this point in his (1995) introduction to 1933o.

  52. 52.

    i.e., standard recursion.

  53. 53.

    The claim, the converse of which is being considered here, is the claim that functions computable by a finite procedure are recursive in the sense given in the lectures (Gödel 1986), p. 348.

  54. 54.

    Gödel’s addenda to the 1934 lectures were published in (Davis 1965/2004), pp. 71–73.

  55. 55.

    (Gödel 1995), p. 47.

  56. 56.

    (Sieg 2005), p. 180.

  57. 57.

    (Sieg 2009), p. 554.

  58. 58.

    (Gödel 1986), p 349. Emphasis added. Such a mechanistic view of the concept of formal system was not a complete novelty at the time. Tarski conceived of “deductive theory” for example, as “something to be performed.” See (Hodges 2008); as Hodges put it, Tarski’s view was “that a deductive theory is a kind of activity.”

  59. 59.

    (Gödel 1986), p. 346.

  60. 60.

    (Gödel 1986), p. 346. This contrasts with Church’s initial presentation of the λ-calculus.

  61. 61.

    Quoted in (Gödel 1986), p. 341. And as Kleene would later write (1981b), “Turing’s computability is intrinsically persuasive but λ-definability is not intrinsically persuasive and general recursiveness scarcely so (its author Gödel being at the time not at all persuaded)”.

  62. 62.

    (Gödel 1986), p. 361.

  63. 63.

    See also (Sieg 2009), p. 551.

  64. 64.

    The theorem has an interesting history: at the Königsberg meeting in 1930, in private

    discussion after the session at which Gödel announced the First Incompleteness Theorem, von Neumann asked Gödel whether the undecidable statement in question could be put in “number-theoretic form,” given the fact of arithmetization. Gödel was initially skeptical but eventually proved the diophantine version of the theorem, to his surprise. See (Wang 1996).

  65. 65.

    Quoted in (Sieg 2006), in turn quoting from an unpublished manuscript of Odifreddi (1990), p. 65.

  66. 66.

    (Gödel 1936), addendum in (Gödel 1986), p. 399. Emphasis added.

  67. 67.

    See Davis (1982). Kleene’s Normal Form Theorem is stated in (Davis 1982) as follows: Every general recursive function can be expressed in the form f(μy(g(x1... xn,y) = 0)), where f and g are primitive recursive functions.

  68. 68.

    This point was made by Dana Scott in private communication with the author.

  69. 69.

    Or quintuples, in Turing’s original presentation.

  70. 70.

    By a logic, we have meant a combination of a list of symbols, commonly called a

    signature, or vocabulary; rules for building terms and formulas, a list of axioms and rules of proof, and then, usually, a semantics.

  71. 71.

    On the formalism or logic-freeness of the Turing model, of course the concept of a Turing Machine may give rise to a logic; or have a logic embedded within it in some very generalized sense, as when one refers to something like “the logic of American politics.” But a logic it is not. In (Kennedy forthcoming we observed that if one defines a formalism, or alternatively a logic, as we have done here then with very little mathematical work each of the informal notions of computability we have considered so far can be seen as generating formal calculi in this sense, i.e. as an individual, self-standing formalism with its own syntax and rules of proof and so forth. See also Kanamori, “Aspect-Perception and the History of Mathematics” (forthcoming).

  72. 72.

    Rescorla has argued that the model gives rise to other circularities, based on the problem of deviant encodings. See (Rescorla 2007).

  73. 73.

    Remark to Hao Wang, in (Wang 1996), p. 203. Emphasis added.

  74. 74.

    (Wang 1996), p. 205.

  75. 75.

    (Gödel 1986), p. 369.

  76. 76.

    (Floyd 2013).

  77. 77.

    (Gandy 1988), p. 83.

  78. 78.

    I.e. the work on computability of Church and others prior to Turing’s (1936/1937) paper.

  79. 79.

    (Gandy 1988), p. 101.

  80. 80.

    (Gödel 1995), p. 164.

  81. 81.

    (Gödel 1995), p. 168.

  82. 82.

    (Gödel 1990), p.150.

  83. 83.

    See, e.g., (Davis 1958).

  84. 84.

    As Parsons interprets the paragraph in his introduction to (Gödel 1946), Gödel is referring to “the absence of the sort of relativity to a given language that leads to stratification of the notion such as (in the case of definability in a formalized language) into definability in languages of greater and greater expressive power.” The stratification is “driven by diagonal arguments” (Parsons (1990), p. 145).

  85. 85.

    That is, to the version of the lecture published in Davis (ed.) (1965/2004).

  86. 86.

    See also Gödel addendum to his (1936) discussed in Sect. 2.1.

  87. 87.

    A rather weak metatheory is required for this.

  88. 88.

    Emphasis added.

  89. 89.

    See (Woodin 2010).

  90. 90.

    Gödel is actually referring to the hereditarily ordinal definable sets in the lecture, denoted HOD. A set belongs to HOD if its transitive closure is ordinal definable.

  91. 91.

    Gödel must have known this, judging from these remarks, though modern proofs of this depend on the Levy Reflection Principle, which was only proved in 1960.

  92. 92.

    The principle says that for every n there are arbitrarily large ordinals α such that Vα ≺ n V.

  93. 93.

    Some of the material in this section was drawn from our (2013). That confluence led to grounding is only accurate from the point of view of the logicians in Princeton. Turing gave the grounding example outright, without confluence in the background.

  94. 94.

    And to some degree, HOD, taking the concepts identified by Gödel in the lecture.

  95. 95.

    Interesting intermediate models, i.e. lying strictly between L and HOD, can also be obtained in this way. An analysis of some of these models is given in Kennedy, Magidor and Väänänen’s [33]. For example, if the Magidor-Malitz quantifier \( {Q}_{\alpha}^{M\ M, n}{x}_1,\dots, {x}_n\varphi \left({x}_1,,\dots,, {x}_n\right) \) is defined as follows:

    $$ \begin{array}{c}\mathcal{M}\models {Q}_{\alpha}^{M\ M, n}{x}_1,\dots, {x}_n\phi \left({x}_1,\dots, {x}_n\right)\\ {}\iff \exists X\subseteq \mathcal{M}\ (\left| X\right|\ge {\aleph}_{\alpha}\wedge \forall {a}_1,\dots, {a}_n\in X:\mathcal{M}\models \phi \left(\forall {a}_1,\dots, {a}_n\right)),\end{array} $$

    then by a result of Magidor, if \( {\mathcal{L}}^{\ast } \) is taken to be first order logic with the Magidor-Malitz quantifiers adjoined to it, then the version of L built with Magidor-Malitz logic and with first order logic are the same, assuming 0♯ exists. See [33] for details.

  96. 96.

    Finding absolute notions of provability and definability was an explicit concern of Post. See (Post and Davis (ed.) (1994)).

  97. 97.

    One preserving the so-called human profile.

References

  • Ackermann, W. 1928. Zum Hilbertschen Aufbau der reellen Zahlen. Mathematische Annalen 99 (1): 118–133.

    Google Scholar 

  • Burgess, J.P. 1992. Proofs About Proofs: A Defense of Classical Logic. In Proof, Logic and Formalization, 8–23. London: Routledge.

    Google Scholar 

  • ———. 2015. Rigor and Structure. Oxford: Oxford University Press.

    Google Scholar 

  • Church, A. 1932. A Set of Postulates for the Foundation of Logic. I, II. Annals of Mathematics (2) 33: 346–366.

    Google Scholar 

  • ———. 1934. The Richard Paradox. American Mathematical Monthly 41 (6): 356–361.

    Google Scholar 

  • ———. 1935. An Unsolvable Problem of Elementary Number Theory. Bulletin of the American Mathematical Society 41: 332–333.

    Google Scholar 

  • ———. 1936. A Note on the Entscheidungsproblem. The Journal of Symbolic Logic 1: 40–41. (Correction 1:101–102).

    Google Scholar 

  • Copeland, B.J., and D. Proudfoot. 2010. Deviant Encodings and Turing’s Analysis of Computability. The Journal of Symbolic Logic 41 (3): 247–252.

    Google Scholar 

  • Davis, M. 1958. Computability and Unsolvability. New York: McGraw-Hill Book Co..

    Google Scholar 

  • ———. 1965/2004. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. New York: Raven Press/Dover Publications.

    Google Scholar 

  • ———. 1982. Why Gödel Didn’t Have Church’s Thesis. Information and Control 54 (1–2): 3–24.

    Google Scholar 

  • De Mol, L. 2006. Closing the Circle: An Analysis of Emil Post’s Early Work. The Bulletin of Symbolic Logic 12 (2): 267–289.

    Google Scholar 

  • Detlefsen, M. 2001. What Does Gödel’s Second Theorem Say? Philosophia Mathematica. Philosophy of Mathematics, Its Learning, and Its Application. Series III 9 (1): 37–71.

    Google Scholar 

  • Fagin, R. 1976. Probabilities on Finite Models. The Journal of Symbolic Logic 41 (1): 50–58.

    Google Scholar 

  • Feferman, S. 1960. Arithmetization of Metamathematics in a General Setting. Polska Akademia Nauk. Fundamenta Mathematicae 49: 35–92.

    Google Scholar 

  • ———. 1995. Introductory Note to Gödel *1933. In Kurt Gödel Collected Works Volume III: Unpublished Essays and Lectures, ed. S. Feferman et al., 36–44. New York: Oxford University Press.

    Google Scholar 

  • Floyd, J. 2013. The Varieties of Rigorous Experience. In The Oxford Handbook of the History of Analytic Philosophy, ed. M. Beaney, 1003–1042. Oxford: Oxford Univ. Press.

    Google Scholar 

  • Floyd, J., and A. Kanamori. 2016. Gödel vis-à-vis Russell: Logic and Set Theory to Philosophy. In Kurt Gödel: Philosopher-Scientist, ed. G. Crocco and E.-M. Engelen, 243–326. Aix-en-Provence: Presses Universitaires de Provence.

    Google Scholar 

  • Franks, C. 2009. The Autonomy of Mathematical Knowledge. Cambridge: Cambridge University Press.

    Google Scholar 

  • Gandy, R.O. 1988. The Confluence of Ideas in 1936. In The Universal Turing Machine: A Half-Century Survey, ed. R. Herken, 55–112. New York: Oxford University Press.

    Google Scholar 

  • Gödel, K. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik 38: 173–198. Reprinted with English translation in (Gödel 1986), 144–195.

    Google Scholar 

  • ———. *1933o. The Present Situation in the Foundations of Mathematics. Lecture to Meeting of the Mathematical Association of America and the American Mathematical Society, Cambridge, MA, 29–30 December 1933, in (Gödel 1995), 45–53.

    Google Scholar 

  • ———. 1934. On Undecidable Propostions of Formal Mathematical Systems. Lectures at the Institute for Advanced Study, Feb.-May 1934, notes taken by S.C. Kleene and J. B. Rosser. Published in (Gödel 1986), 346–371, which includes the Postscriptum (Gödel 1964).

    Google Scholar 

  • ———. 1936. Über die Länge von Beweisen. Abstract in Ergebnisse eines mathematischen Kolloquiums; published with English translation, including an addendum, as “On the Length of Proofs” in Gödel (1986), 397–398.

    Google Scholar 

  • ———. 1946. Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics. In (Gödel 1990), 150–153.

    Google Scholar 

  • ———. *1951o. Some Basic Theorems on the Foundations of Mathematics and Their Implications. Josiah Willard Gibbs Lecture, Brown University, 26 December 1951, in (Gödel 1995), 304–232.

    Google Scholar 

  • ———. 1964. Postscriptum to the 1934 Princeton Lectures. In (Gödel 1986), 369–370. Originally Published in Davis (ed.) (1965/2004), 71–73.

    Google Scholar 

  • ———. 1986. Kurt Gödel Collected Works Vol. I: Publications 1929–1936. New York: Oxford University Press.

    Google Scholar 

  • ———. 1990. Kurt Gödel Collected Works Volume II: Publications 1938–1974. New York: Oxford University Press.

    Google Scholar 

  • ———. 1995. Kurt Gödel Collected Works Volume III: Unpublished Essays and Lectures. New York: Oxford University Press.

    Google Scholar 

  • ———. 2003a. Collected Works. IV: Correspondence A-G. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2003b. Kurt Gödel Collected Works Volume V: Correspondence H-Z. New York: Oxford University Press.

    Google Scholar 

  • ———. unpublished. Max Phil Notebooks. Available in Princeton, New Jersey, Firestone Library and the Library of the Institute for Advanced Study, Princeton University.

    Google Scholar 

  • Harris, M. 2015. Mathematics Without Apologies: A Portrait of a Problematic Vocation. Princeton: Princeton University Press.

    Google Scholar 

  • Hilbert, D., and P. Bernays. 1934/1968. Grundlagen der Mathematik. I, Die Grundlehren der Mathematischen Wissenschaften 40. Berlin, New York: Springer Verlag.

    Google Scholar 

  • ———. 1939/1970. Grundlagen der Mathematik. II, Die Grundlehren der Mathematischen Wissenschaften 50. Berlin, New York: Springer Verlag.

    Google Scholar 

  • Hodges, W. 2008. Tarski’s Theory of Definition. In New Essays on Tarski and Philosophy, 94–132. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2009. Set Theory, Model Theory, and Computability Theory. In The Development of Modern Logic, 471–498. Oxford: Oxford University Press.

    Google Scholar 

  • Kanamori, A. forthcoming. Aspect Perception and the History of Mathematics. In Seeing-as and Novelty, ed. M. Beaney and D. Shaw. New York: Routledge.

    Google Scholar 

  • Kennedy, J. 2013. On Formalism Freeness: Implementing Gödel’s 1946 Princeton Bicentennial Lecture. The Bulletin of Symbolic Logic 19 (3): 351–393.

    Google Scholar 

  • Kennedy, J. forthcoming. Gödel’s 1946 Princeton Bicentennial Lecture: An Appreciation. In J. Kennedy, (ed.), Interpreting Gödel, 109–132. Cambridge: Cambridge University Press.

    Google Scholar 

  • Kennedy, J., M. Magidor, and J. Väänänen (forthcoming). Inner Models from Extended Logics.

    Google Scholar 

  • Kleene, S.C. 1936. A Note on Recursive Functions. Bulletin of the American Mathematical Society 42 (8): 544–546.

    Google Scholar 

  • ———. 1943. Recursive Predicates and Quantifiers. Transactions of the American Mathematical Society 53: 41–73.

    Google Scholar 

  • ———. 1981a. Origins of Recursive Function Theory. Annals of the History of Computing 3 (1): 52–67.

    Google Scholar 

  • ———. 1981b. The Theory of Recursive Functions, Approaching Its Centennial. American Mathematical Society. Bulletin. New Series 5 (1): 43–61.

    Google Scholar 

  • Kleene, S.C., and J.B. Rosser. 1935. The Inconsistency of Certain Formal Logics. Annals of Mathematics (2) 36 (3): 630–636.

    Google Scholar 

  • Kripke, S. 2013. The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem. In Computability: Gödel, Church, and Beyond, ed. B.J. Copeland, C.J. Posy, and O. Shagrir. Cambridge, MA: MIT Press.

    Google Scholar 

  • Macbeth, D. 2014. Realizing Reason. Oxford: Oxford University Press.

    Google Scholar 

  • Martin, D. A. 2012. Completeness or Incompleteness of Basic Mathematical Concepts. In Exploring the Frontiers of Infinity, ed. P. Koellner.

    Google Scholar 

  • McLarty, C. 1997. Poincaré: Mathematics & Logic & Intuition. Philosophia Mathematica. Philosophy of Mathematics, Its Learning, and Its Applications. Series III 5 (2): 97–115.

    Google Scholar 

  • Mundici, D., and W. Sieg. 1994. Computability Theory. Annals of the History of Computing.

    Google Scholar 

  • Myhill, J., and D. Scott. 1971. Ordinal Definability. Axiomatic Set Theory. Proceedings of the Symposium of Pure Mathematics, Vol. XIII, Part I, University of California, Los Angeles, (1967)). Providence: American Mathematical Society, 271–278.

    Google Scholar 

  • Odifreddi, P. (ed.). 1990. About Logics and Logicians—A Palimpsest of Essays by Georg Kreisel, Vol II: Mathematics. Manuscript.

    Google Scholar 

  • Parsons, C. 1990. Introductory Note to Gödel (1946). In (Gödel 1990), 144–150.

    Google Scholar 

  • Péter, R. 1935. Konstruktion nichtrekursiver Funktionen. Mathematische Annalen 111 (1): 42–60.

    Google Scholar 

  • ———. 1937. Über die mehrfache Rekursion. Mathematische Annalen 113 (1): 489–527.

    Google Scholar 

  • Post, E.L. 1936. Finite Combinatory Processes, Formulation 1. Journal of Symbolic Logic 1:103–105. Reprinted in Davis (1965/2004), 288–291 and in Post, E. L., and Davis, M. (ed.) (1994), 103–105.

    Google Scholar 

  • ———. 1944. Recursively Enumerable Sets of Positive Integers and Their Decision Problems. Bulletin of the American Mathematical Society 50: 284–316. Reprinted in Davis (1965/2004), 304–337 and in Post, E. L., and Davis, M. (ed.) (1994), 461–494.

    Google Scholar 

  • ———. 1947. Recursive Unsolvability of a Problem of Thue. Journal of Symbolic Logic 12(1): 1–11. Reprinted in Davis (1965/2004), 292–303 and in Post, E. L., and Davis, M. (eds.) (1994), 503–513.

    Google Scholar 

  • Post, E.L., and M. Davis (eds.). 1994. Solvability, Provability, Definability: The Collected Works of Emil L. Post. Boston/Basel/Berlin: Birkhäuser.

    Google Scholar 

  • Pudlák, P. 1996. On the Lengths of Proofs of Consistency: A Survey of Results. In Collegium Logicum, vol. 2, 65–86. Vienna: Springer.

    Google Scholar 

  • Rav, Y. 1999. Why Do We Prove Theorems? Philosophia Mathematica. Philosophy of Mathematics, Its Learning, and Its Application. Series III 7 (1): 5–41.

    Google Scholar 

  • ———. 2007. A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians’ Proof Practices. Philosophia Mathematica. Series III 15 (3): 291–320.

    Google Scholar 

  • Rescorla, M. 2007. Church’s Thesis and the Conceptual Analysis of Computability. Notre Dame Journal of Formal Logic 48 (2): 253–280. (electronic).

    Google Scholar 

  • Shapiro, S. 1982. Acceptable Notation. Notre Dame Journal of Formal Logic 23 (1): 14–20.

    Google Scholar 

  • ———. 1990. Review: Stephen C. Kleene, Origins of Recursive Function Theory; Martin Davis, Why Godel Didn’t Have Church’s Thesis; Stephen C. Kleene, Reflections on Church’s Thesis. Journal of Symbolic Logic 55 (1): 348–350.

    Google Scholar 

  • ———. 2013. The Open Texture of Computability. In Computability—Turing, Gödel, Church, and Beyond, 153–181. Cambridge, MA: MIT Press.

    Google Scholar 

  • Sieg, W. 1997. Step by Recursive Step: Church’s Analysis of Effective Calculability. The Bulletin of Symbolic Logic 3 (2): 154–180.

    Google Scholar 

  • ———. 2005. Only Two Letters: The Correspondence Between Herbrand and Gödel. The Bulletin of Symbolic Logic 11 (2): 172–184.

    Google Scholar 

  • ———. 2006. Gödel on Computability. Philosophia Mathematica 14: 189–207.

    Google Scholar 

  • ———. 2009. On Computability. In Philosophy of Mathematics, ed. A.D. Irvine, 535–630. Amsterdam/Boston/Oxford: Elsevier.

    Google Scholar 

  • Soare, R.I. 1996. Computability and Recursion. The Bulletin of Symbolic Logic 2 (3): 284–321.

    Google Scholar 

  • Soare, R. 2015. Why Turing’s Thesis Is Not a Thesis. In eds. Sommaruga and Strahm, 297–310.

    Google Scholar 

  • Sommaruga, G., and T. Strahm, eds. 2015. Turing’s Revolution: The Impact of His Ideas About Computability, 297–310. Basel: Birhäuser/Springer.

    Google Scholar 

  • Turing, A. 1936. On Computable Numbers, with an Application to the Entscheidungsproblsm. Proceedings of the London Mathematical Society 2 (42): 230–265.

    Google Scholar 

  • van Atten, M., and J. Kennedy. 2009. Gödel’s Modernism: On Set-Theoretic Incompleteness, Revisited. In Logicism, Intuitionism, and Formalism, What has Become of Them? ed. Sten Lindström, Erik Palmgren, Krister Segerberg, and H. Stoltenberg. New York: Springer.

    Google Scholar 

  • Wang, H. 1996. A Logical Journey. Cambridge, MA: MIT Press.

    Google Scholar 

  • Wimsatt, W. 1994. The Ontology of Complex Systems: Levels of Organization, Perspectives, and Causal Thickets. Canadian Journal of Philosophy 20: 207–274.

    Google Scholar 

  • Woodin, W.H. 2010. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. Berlin: Walter de Gruyter GmbH & Co. KG.

    Google Scholar 

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Acknowledgments

For discussions about definability as well as many other issues connected to this paper I wish to thank especially Menachem Magidor. For very helpful comments and correspondence I am also grateful to John Baldwin, Patricia Blanchette, Juliet Floyd, Curtis Franks, Wilfried Sieg and Jouko Väänänen. This paper was finished while I was a guest of the mathematics department at Hebrew University during November of 2014. I am very grateful for the support of my visit, which enabled me to benefit from the extremely stimulating atmosphere of the logic group in Jerusalem.

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  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Juliette Kennedy

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    Juliet Floyd

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    Alisa Bokulich

Appendix: Deviant Encodings

Appendix: Deviant Encodings

In a strange twist of fate the essentially philosophical problem of deviant encodings, as it is now called, cannot be avoided. That is, Gödel’s condition in his 1934 Princeton lectures, that “the relation between a positive integer and the expression which stands for it must be recursive” will always introduce circularity, insofar as the condition is a constituent of an all-embracing, or in Gödel’s other terminology, absolute notion of computability.

In brief: As a general rule , Turing Machines require an input, and at the end of the computation they generate an output. Inputs and outputs must be encoded, as, after all, Turing Machines operate on finite strings, not natural numbers.

There are, of course, more or less obvious ways to do this. The obvious choice, “unary” encoding, involves counting the number of consecutive ones in the beginning of the tape and declaring that as the input, and, respectively, counting at the end of the computation the number of consecutive ones in the beginning of the tape, and declaring that as the output. As far as one can see, this approach is completely unproblematic! But one has to accept the effectivity of unary encoding by faith.

In fact, there are also other, equally obvious encodings. For example, one could represent the input number as a binary string which would then be written on the tape. This should be as computable as unary encoding, and in fact it is even more effective because one gets logarithmic compression. But again the effectivity of the (binary) encoding must be taken on faith.

The problem of “deviant encodings”, originally pointed out by Shapiro in his (1982), is that the encoding of input and output data on the tape should be itself computable, devoid of unintended elements which may give the machine non-computable power. For example, using Copeland and Proudfoot’s deviant “mirabilis” output encoding (2010), one can design a Turing Machine that computes the Halting Problem.

Who would question the effectivity of unary encoding ? Nobody we know! The problem is that one would like to have a general method for separating computable from noncomputable encodings, rather than a list of acceptable ones. But such a general criterion cannot be given.

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Kennedy, J. (2017). Turing, Gödel and the “Bright Abyss”. In: Floyd, J., Bokulich, A. (eds) Philosophical Explorations of the Legacy of Alan Turing. Boston Studies in the Philosophy and History of Science, vol 324. Springer, Cham. https://doi.org/10.1007/978-3-319-53280-6_3

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