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Feferman on Set Theory: Infinity up on Trial

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Feferman on Foundations

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

Abstract

In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear” and “completely definite,” many of the statements of analysis and set theory are “inherently vague” and “indefinite.” I critique his five main arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to this final, remaining point will be that the concept of “being clear enough to secure definiteness” is about as clear a case of an inherently vague and indefinite concept as one might find, and as such it can bear little weight in making a case against the definiteness of analysis and set theory.

“Look into infinity, all you see is trouble."

— Bob Dylan, “Trouble”

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Notes

  1. 1.

    A more thorough account will appear in [28].

  2. 2.

    Throughout this paper I will use ‘analysis’ and ‘second-order number theory’ interchangeably.

  3. 3.

    [17], p. 1.

  4. 4.

    [15], p. 127.

  5. 5.

    The title involves a reference to Feferman’s “Infinity in Mathematics: Is Cantor Necessary?”, which opens with what is perhaps the coolest epigraph ever used in a paper:

    $$\begin{aligned} ``&\text {Infinity is up on trial} \, \dots \text {''} \\&\text {--- Bob Dylan,} \,\, \textit{Visions of Johanna}. \end{aligned}$$
  6. 6.

    I would like to thank Douglas Blue, J.T. Chipman, Solomon Feferman, Gabriel Goldberg, Charles Parsons, Michael Rathjen, Wilfried Sieg, Thomas Strahm, and Hugh Woodin for helpful comments and discussion. I would also like to thank the The Journal of Philosophy for giving me permission to repeat passages that occur [28], which was a high-level summary of the present paper, one that I delivered as a reply to Sol in the conference described below.

  7. 7.

    [19], p. 74.

  8. 8.

    [15], pp. 2–3.

  9. 9.

    [15], pp. 2–3.

  10. 10.

    [4], p. 70.

  11. 11.

    [15], p. 7.

  12. 12.

    [13], p. 314.

  13. 13.

    [13], p. 314.

  14. 14.

    [13], p. 316.

  15. 15.

    [3], p. 1.

  16. 16.

    See [3].

  17. 17.

    [3].

  18. 18.

    [4].

  19. 19.

    [6].

  20. 20.

    [8, 21].

  21. 21.

    The qualification ‘locally at least’ is necessary since the totality \({\mathrm{RA}}_{<\Gamma _0}\) is not itself predicatively characterizable.

  22. 22.

    [13], pp. 313–314.

  23. 23.

    Letter of Jan. 4, 2016.

  24. 24.

    [11], Sect. 6.

  25. 25.

    See [32, 35].

  26. 26.

    [7], pp. 200–201.

  27. 27.

    [12], p. 405.

  28. 28.

    [17], p. 1.

  29. 29.

    [17], p. 1.

  30. 30.

    [20], p. 7.

  31. 31.

    To underscore this point consider a hypothetical scenario involving an arithmetical statement. Suppose that the scientific board were presented with a number-theoretic statement that was equivalent to the consistency of \({\mathrm{ZFC}}+\text {``There}\) is a supercompact cardinal.” They could quite reasonably refrain from putting it on the list because they could quite reasonably think that any positive resolution would require subtle issues surrounding the justification of new axioms. But this would not provide “considerable circumstantial evidence” that the number-theoretic statement is indefinite.

  32. 32.

    [4], p. 70.

  33. 33.

    [9], p. 7, Introduction.

  34. 34.

    [12], p. 405.

  35. 35.

    [12], pp. 410–411.

  36. 36.

    See, for example, [15], p. 130 and [17], p. 2 and p. 21.

  37. 37.

    [17], p. 2, my emphasis.

  38. 38.

    [14], p. 619.

  39. 39.

    [3], pp. 1–2.

  40. 40.

    [13], p. 314.

  41. 41.

    [19], p. 81.

  42. 42.

    [9], Introduction, p. ix.

  43. 43.

    [17], p. 9.

  44. 44.

    [10], p. 107.

  45. 45.

    [17], p. 18. In this quotation I have used ‘P(A)’ in place of Feferman’s ‘S(A)’. As Charles Parsons pointed out to me, the reference should be to Benacerraf’s 1973 paper.

  46. 46.

    [17], p. 18.

  47. 47.

    [19], p. 74.

  48. 48.

    [20], pp. 2–3.

  49. 49.

    [17], p. 11.

  50. 50.

    [17], p. 13.

  51. 51.

    See [38, 39], Chap. 4.

  52. 52.

    [17], p. 12.

  53. 53.

    For a very clear discussion of this distinction see [2].

  54. 54.

    [10], p. 107.

  55. 55.

    [11], p. 72.

  56. 56.

    See also [17], p. 19, [12], p. 411, and [19], p. 79.

  57. 57.

    Personal communication: Letter of March 14, 2016.

  58. 58.

    [15], p. 127.

  59. 59.

    [5], pp. 72–73, [12], pp. 404–405, and [15].

  60. 60.

    [3], p. 1.

  61. 61.

    [17], p. 15.

  62. 62.

    It is often pointed out that statements like \(\varphi \) are not “natural.” That may be true but it is not relevant to our present discussion, which concerns the question of definiteness. Feferman is maintaining that all of the statements of number theory have a determinate truth value, not just the “natural” ones. And he is providing arguments which apply to statements regardless of whether they are “natural” or not. (It would be far-fetched to say “not-(currently)-being-settled-by-remotely-plausible-assumptions secures indefiniteness \(\dots \)” and then add “but only for “natural” statements.”

  63. 63.

    See the last section of [24] for further discussion.

  64. 64.

    [33]. See also [34].

  65. 65.

    [30, 31].

  66. 66.

    [30].

  67. 67.

    For example, this argument is put forward by [22].

  68. 68.

    One can choose theories other than \({\mathrm{PA}}\) and one can arrange that \(\varphi \) is quite simple—it can be \(\Delta ^0_2\) (provably over \({\mathrm{PA}}\) (or over whichever background theory one is working with)), as is demonstrated in the first appendix.

  69. 69.

    ‘PU’ is short hand for the statement that all projective sets have the uniformization property. This is a statement of (schematic) second-order number theory, and so concerns the concept of subsets of natural numbers. CH, in contrast, is a statement of third-order number theory, one that concerns the concept of subsets of the set of subsets of natural numbers.

  70. 70.

    [17], p. 2. See [16] for the technical work on these systems. See [17,18,19], for an informal, philosophical discussion.

  71. 71.

    It would take us too far afield, and it would not be especially illuminating, to pause to lay out the axioms of these two systems. For a precise account of these systems the interested reader can consult [36]. (I am using ‘\({\mathrm{SCS}}\)’ for the system Rathjen also calls ‘\({\mathrm{SCS}}\)’ and I am using ‘\({\mathrm{SCS}}^+\)’ for the system he calls T, that is, the system \({\mathrm{SCS}}+{}\)“\(\mathbb R\) is a set”.).

  72. 72.

    [36].

  73. 73.

    [18], p. 23.

  74. 74.

    This result and the remaining results in this section were proved jointly with Hugh Woodin. It seems likely that the hypothesis can be weakened to just \({\mathrm{ZFC}}\) but the technical hurdles are difficult and it doesn’t affect the point I wish to make. I am confident that the hypothesis is true, and hence that the conclusion follows. If one has an issue with the hypothesis we have a whole other debate \(\dots \) In any case, the skeptical reader can take the theorem as it stands.

  75. 75.

    The former follows from the existence of infinitely many Woodin cardinals. The latter follows from the existence of infinitely many Woodin cardinals with a measurable above them all. The assumption of Woodin cardinals is much stronger than the assumption of measurable cardinals.

  76. 76.

    See for example [29, 37, 40].

  77. 77.

    See [23, 29].

  78. 78.

    [12], p. 405.

  79. 79.

    It is unclear to me that he should say this. For open determinacy is logically equivalent over \({\mathrm{RCA}}_0\) (a system Feferman accepts) to the statement that all countable well-orderings are comparable, and it does not seem that Feferman can say that the latter is definite since it involves what he regards as a bankrupt notion, namely, the notion of a well-ordering.

  80. 80.

    [20], p. 3.

  81. 81.

    For some of the elements that go into making a positive case see the expository articles [25,26,27], and the references therein.

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  1. Department of Philosophy, Harvard University, Cambridge, MA, 02138, USA

    Peter Koellner

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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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Koellner, P. (2017). Feferman on Set Theory: Infinity up on Trial. 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_19

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