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Gödel’s Cantorianism

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Abstract

Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.

Originally published in C. Ternullo, Goedel’s cantorianism, in Kurt Gödel: Philosopher-Scientist, ed. by G. Crocco, E.-M. Engelen (Presses Universitaires de Provence, Aix en Provence, 2015), pp. 413–442.

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Notes

  1. 1.

    Wang [48, p. 164].

  2. 2.

    Gödel [16], revised and extended version, [20].

  3. 3.

    Further information on biographical and philosophical aspects of Gödel’s life can be found in Feferman’s introduction to [21, pp. 1–34]. A precise and exhaustive reconstruction of the development of Gödel’s thought is also carried out by van Atten and Kennedy in [45]. None of these works mentions specific connections between Cantor and Gödel.

  4. 4.

    I have only found two passages in Wang [48], where Gödel says something directly about Cantor. The first, on p. 175, concerns the philosophy of physics: “5.4.16 The heuristics of Einstein and Bohr are stated in their correspondence. Cantor might also be classified together with Einstein and me. Heisenberg and Bohr are on the other side. Bohr [even] drew metaphysical conclusions from the uncertainty principle.” The second, on p. 276, is about the distinction between set and class (for whose relevance see Sect. 6): “8.6.13 Since concepts can sometimes apply to themselves, their extensions (their corresponding classes) can belong to themselves; that is, a class can belong to itself. Frege did not distinguish sets from proper classes, but Cantor did this first.” Both remarks show at least some familiarity with Cantor’s writings.

  5. 5.

    Among these few exceptions should be counted some remarks by Wang and van Atten, in, respectively, [48] and [44], concerning the so-called Cantor-von Neumann axiom, for which also see Sect. 6.

  6. 6.

    See Sect. 5.

  7. 7.

    Throughout this article, all quotations from Gödel’s published and unpublished works reproduce the established text in the II and III volume of his Collected Works [22, 23].

  8. 8.

    However, Gödel’s interest in theology is noticeable in the Max-Phil Notebooks.

  9. 9.

    See Parsons [38], Martin [35] and Crocco [7] for a careful examination of the issues related to this position.

  10. 10.

    However, as late as 1933, Gödel stated ([14], in [23, p. 50]): “The result of the preceding discussion is that our axioms, if interpreted as meaningful statements, necessarily presuppose a form of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent.” See Feferman’s comments on this in his Introduction to [14]. But, apart from that, it seems very plausible that Gödel embraced Platonism, in at least some of its forms, at a very early stage in his career.

  11. 11.

    Purely mathematical sets, in ZFC, or in alternative systems, with or without urelements, are sets formed through the iteration of the power-set operation at successor-stages and the union of all previous stages at limit-stages, starting from Ø or ur-elements.

  12. 12.

    However, it is not wholly uncontroversial what Gödel thought to be the objects of “perception”, whether mathematical objects or concepts or both. For instance, in Wang [48, p. 253], Gödel is reported to have said: “7.3.12 Sets are objects but concepts are not objects. We perceive objects and understand concepts. Understanding is a different kind of perception: it is a step in the direction of reduction to the last cause.” I thank Eva-Maria Engelen for pointing me to this quotation and to the subtle difference between these two forms of “perception” in Gödel’s thought.

  13. 13.

    No doubt, this conception has a Leibnizian ancestry, but Gödel may have also deliberately wanted to refer to the logicist standpoint. For instance, see Frege [13, p. 85]. “Thus, arithmetic becomes only a further developed logic, every arithmetical proposition a logical law, albeit a derivative one.” However, Frege never affirmed that arithmetical truths are tautologies. In any case, as we have seen, in the passage quoted, Gödel fosters a different notion of “analytic”, meaning: “owing to the meaning of the terms occurring in it”.

  14. 14.

    The English translation of all Cantor’s quotations from [1] comes from Ewald [10, pp. 878–920]. In reproducing it, I have also kept Ewald’s annotations in square brackets.

  15. 15.

    See Hallett [24, p. 17].

  16. 16.

    Frege’s “objectivistic” views about concepts can be found, in particular, in Frege [12]. It should be noticed that Frege thought that the main value of his work had consisted, among other things, precisely in the clarification of the essence of concepts (see the letter quoted by Ricketts in [40, p. 149]).

  17. 17.

    The bulk of Hill’s careful work on the relationships between Cantor and Husserl can be found in [36]. See also [37]. In [36], Hill identifies three stages of influence of Cantor’s thought on Husserl. But she clearly acknowledges, although only conjecturally, that there might be a further, fourth stage, that she does not examine, which “…would consist of the assimilation of certain of Cantor’s ideas into Husserl’s phenomenology and extends far beyond the compass of this study. Here it would be a matter of studying the relationship between Cantor’s theories and, for example, Husserl’s Mannigfaltigkeitslehre, his theories about eidetic intuition, the phenomenological reductions, noemata, horizons, infinity, whole and part…” (p. 166).

  18. 18.

    On this point, see, in particular, Wang’s mentioned books, Kennedy and van Atten [45], Tieszen [41,42,43], Hauser [25] and Crocco [7]. Føllesdal’s introduction to [19, pp. 364–373] also provides interesting insights.

  19. 19.

    The Continuum Problem is the problem of determining the cardinality of \(\mathbb {R}\) (denoted \(\mathfrak {c}\)). The Continuum Hypothesis (CH) is Cantor’s conjecture that \(\mathfrak {c}=\aleph _1\). See footnote 27 below.

  20. 20.

    The problem with Gödel’s claim that set-theoretic statements might be shown to have a determinate and unique truth-value as a result of conceptual refinements is explained very neatly by Hauser in [25, pp. 539–540]: “On this view, the meaning of the continuum problem is tied to an unfolding of concepts through successive refinements of mathematical intuition. One difficulty is why it should lead to a unique resolution of CH, for our intuitions could conceivably evolve into different directions inducing us to formulate axioms with opposite outcomes of CH.”

  21. 21.

    See footnote 12 above.

  22. 22.

    The bulk of Husserl’s phenomenological ideas can be found in the three volumes of the Ideen, [26, 27] and [29]. See also [28]. A quick review of the main concepts of phenomenology can be found in one of the articles/books I mentioned above, footnote 16, or, for instance, in Christian Beyer’s entry “Husserl” in the Stanford Philosophical Encyclopedia, 2013, which also includes an up-to-date bibliography.

  23. 23.

    See, in particular, Parsons [38, pp. 56–70], concerning the difficulties with Gödel’s notion of intuition. Parsons’ interpretation, especially of Gödel’s quotations from [20], seems inclined to explain away the presence of phenomenological elements in Gödel’s thought. For instance, with regard to the notion of “immediately given”, he says: “The picture resembles Kant’s, for whom knowledge of objects has as “components” a priori intuition and concepts. It is un-Kantian to think of pure concepts as given, immediately or otherwise. But Gödel’s picture seems clearly to be that our conceptions of physical objects have to be constructed from elements, call them primitives, that are given, and that some of them (whether or not they are much like Kant’s categories) must be abstract and conceptual.” (p. 68).

  24. 24.

    See, in particular, Hallett [24, pp. 128–142].

  25. 25.

    In [4], reprinted in [5, p. 380], Cantor also uses the Greek word \(\upmu {\mathrm{{o}}}\upnu \acute {\upalpha } \upvarsigma \) [monás] to refer to number-sets, a term which is borrowed from a definition in Euclid’s Elements he mentions in that work. He says (my translation): “Cardinal numbers as well as order-types are simple conceptual formations; each of them is a true Unity \((\upmu o\upnu \acute {\upalpha } \upvarsigma )\), as in them a plurality and multiplicity of Ones is unitarily bound together [Die Kardinalzahlen sowohl, wie die Ordinungstypen sind einfache Begriffsbildungen; jede von ihnen ist eine wahre Einheit \((\upmu {\mathrm{{o}}}\upnu \acute {\upalpha } \upvarsigma )\), weil in ihr eine Vielheit und Mannigfaltigkeit von Einsen einheitlich verbunden ist].” He also reports instances of the notion of \(\upmu {\mathrm{{o}}}\upnu \acute {\upalpha } \upvarsigma \) as can be found in Nicomachus’ Institutio Arithmetica and Leibniz. Nicomachus’ neo-Pythagorean view about numbers also implied that they are \(\upsigma \upupsilon \upsigma \uptau \acute {\upeta } \upmu \upalpha \uptau \upalpha \quad \upmu {\mathrm{{o}}} \upnu \acute {\upalpha } \updelta \upomega \upnu \) [systémata monádon], that is, aggregations of unities (monads). In Cantor’s quoted passage from Leibniz’s De arte combinatoria, Leibniz says: “Abstractum autem ab uno est unitas, ipsumque totum abstractum ex unitatibus, seu totalitas, dicitur numerus”.

  26. 26.

    Cantor [2], reprinted in [5, p. 372].

  27. 27.

    At the time of composition of [16], and of its revision in [20], it was not known that large cardinal axioms do not fix the power of the continuum, as Gödel had conjectured that they might. This was first shown by Solovay and Lévy in [32] using measurables, but the result generalises to all known large cardinals. An analogous result for smaller large cardinals had already been proved by Cohen in [6]. See Kanamori [31, p. 126].

  28. 28.

    Wang discusses them in both [46] and [48], but this latter contains a more detailed account.

  29. 29.

    Wang [48, p. 261]: “[..] Cantor called multitudes “like” V inconsistent multitudes, and introduced a general principle to distinguish them from sets.”

  30. 30.

    English translation in [10, pp. 931–935].

  31. 31.

    However, as we have seen (footnote 4), Gödel was fully aware of the fact that Cantor, not von Neumann, had first introduced the distinction between sets and classes. On this point, see also Van Atten [44].

  32. 32.

    See [24], and particularly, p. 164–194. With regard to von Neumann’s re-statement of Cantor’s principle (what Gödel calls “von Neumann axiom”), see, in particular, pp. 286–298.

  33. 33.

    See, in particular, Hallett [24, pp. 41–48 and 165–176]. Wang discusses Cantor’s conception in connection with Gödel’s criteria for introducing new axioms especially in Wang [46, pp. 188–190]. Jané addresses Cantor’s conception in full, in his [30]. Jané lays strong emphasis on the tension between the idea that the Absolute cannot be measured (and determined), and the fact that it can still be seen as a sort of “quantitative maximum” for the actual infinite, a tension which was perceived and addressed by Cantor in different ways over his career. Jané thinks that, in the end, God’s absoluteness and mathematical absoluteness fell apart, as Cantor was forced to accept, mathematically, that the absolute infinite is not a form of the actual infinite.

  34. 34.

    However, this form of reflection principle does not justify very large large cardinal hypotheses. Gödel was maybe already aware of this fact, when, in a footnote added in 1966 to his [20], referring to the axiom asserting the existence of a measurable cardinal, he stated (pp. 260–261): “That these axioms are implied by the general concept of set in the same sense as Mahlo’s has not been made clear yet […]”, whereas he is aware of the fact that small large cardinals such as Mahlo’s can be connected to reflection successfully: “Mahlo’s axioms have been derived from a general principle about the totality of sets which was first introduced by Levy (1960). It gives rise to a hierarchy of precise formulations”. For details about this hierarchy, see Kanamori [31, pp. 57–59].

  35. 35.

    This claim can be made precise by saying that Gödel’s First Incompleteness Theorem shows that any theory of arithmetic of the same strength as PA (Peano Arithmetic) is incomplete, namely, that it does not prove all arithmetical truths.

  36. 36.

    However, it could actually turn out that “subjective” mathematics is larger than “mechanical” mathematics, should human minds prove to be stronger than machines, something Gödel had already cast as a conjecture in the Gibbs lecture and which is again reported by Wang in [48, p. 186]. I am indebted to Gabriella Crocco for pointing me to the subtle difference between “subjective” and “mechanical” in Gödel’s formulations.

  37. 37.

    Using the subsequent ℵ-notation, the powers of the first and the second class are, respectively, ℵ 0 and ℵ 1. Under CH, the power of ethereal matter is \(\mathfrak {c}\). For further details on this claim, and its connection to Cantor’s set-theoretic work, see Dauben [8, p. 126] and Ferreirós [11], in particular, pp. 75–77.

  38. 38.

    An articulated review of extra-mathematical themes in Cantor’s thought is the aforementioned Ferreirós [11]. Cantor’s ideas related to organicism were not altogether foreign to the scientific debate, as Ferreirós shows (see p. 77). On this point, see also Purkert-Ilgauds [39, pp. 67–68].

  39. 39.

    Cf., in particular, Tieszen [43, p. 38]. According to Husserl, “Monads are transcendental egos in their full concreteness. Transcendental egos in their full concreteness are not “mere poles of identity”, but are rather egos with all the predicates that attach to these poles of identity, so that each monad is distinct from every other monad. We know that Leibniz has a range of different kinds of monads, but Husserl’s focus is much narrower. It is on the kind of ‘monads’ that we are”. Tieszen also observes that we do not know to what extent Gödel wanted to use Leibniz’s monadological conception in a way which would conform to Husserl’s.

  40. 40.

    Van Atten notices (p. 4), that the sole fact that “Leibniz denies the existence of infinite wholes of any kind” would doom Gödel’s attempt to failure.

  41. 41.

    Cf. Wang [48, p. 112]: “Gödel gave his own religion as “baptized Lutheran” (though not a member of any religious congregation) and noted that his belief was theistic, not pantheistic, following Leibniz, rather than Spinoza. […] Gödel was not satisfied with Spinoza’s impersonal God.”

  42. 42.

    Generating principles are the three principles Cantor uses in the Grundlagen to “construct” the whole series of transfinite ordinals (see [1], in [10, pp. 907–909]), viz., the successor, the limit and the restriction principle, whereby one can build, respectively, successor-ordinals (ω + 1, ω + 2, …), limit-ordinals (ω, ω + ω, …) and initial ordinals (ω0, ω1, …).

  43. 43.

    In the Cantor paper, Gödel’s optimism is very robust, but elsewhere (the Gibbs lecture and some other remarks in Wang [48]), it is mitigated by the observation that the intuition of mathematical objects may be fallible and, what is more important, incomplete, thus leaving it open whether we are able to find solutions to all set-theoretic problems.

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Acknowledgements

The writing of this article has been supported by the JTF Grant ID35216 (research project “The Hyperuniverse. Laboratory of the Infinite”). A preliminary version was presented and discussed at the conference Kurt Gödel Philosopher: from Logic to Cosmology held in Aix-en-Provence, July, 9–11, 2013. I wish to thank Richard Tieszen for insightful comments on my presentation and Mark van Atten for pointing me to bibliographical material which has proved of fundamental importance for the subsequent writing of this article. Gabriella Crocco and Eva-Maria Engelen read earlier drafts of this work, providing me with extremely helpful comments and suggestions. Finally, I owe a debt of gratitude to Mary Leng, whose advice and help in the final stage of revision have been invaluable.

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    Claudio Ternullo

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  1. Zukunftskolleg, University of Konstanz, Konstanz, Germany

    Carolin Antos

  2. Kurt Gödel Research Center, University of Vienna, Vienna, Austria

    Sy-David Friedman

  3. Department of Logic, Charles University, Prague, Czech Republic

    Radek Honzik

  4. Kurt Gödel Research Center, University Wien, Wien, Austria

    Claudio Ternullo

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Ternullo, C. (2018). Gödel’s Cantorianism. In: Antos, C., Friedman, SD., Honzik, R., Ternullo, C. (eds) The Hyperuniverse Project and Maximality. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-62935-3_11

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