Abstract
The years up to 1914 were a crucial period of diffusion and recognition for set theory. During the 1890s the new vision of mathematics and the Cantorian ideas spread out, while the 1900s saw fundamental new contributions in the hands of a new generation of cultivators — Zermelo and Hausdorff above all. But this was also a period of heated debates surrounding the notion of arbitrary set and its expressions, the Axiom of Choice and the Well-Ordering Theorem. It was also the time in which the paradoxes emerged, heralded by Russell. Thus, the diffusion of set theory was accompanied by much ambivalence and confusion. The acceptability of abstract mathematics was in question, as were the relations between logic and set theory.
That the word ‘set’ is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new.1
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References
Julius König [1905] as translated in [van Heijenoort 1967, 145].
Hilbert tried to solve it twice unsuccessfully, in [Hilbert 1926] and two years later.
The same happens, e.g., with Dedekind’s chain-condition in his definition of the natural numbers (§VII.3.2).
That can also be presented as a version of Cantor’s definition, according to which a cardinal number is a general concept under which equipollent classes fall, but again not faithfully. We can here observe how by 1900 set theory was not yet completely extensionalized.
This can be found already in [Frege 1879], but one of the novelties in the meantime was precisely the introduction of `courses-of-values.’
He did establish two interesting requirements on manifolds in order that the identical calculus become applicable. They must be “consistent” and “pure” [rein; op.cit., vol. 1, 212–13, 248, 342].
See [Russell 1903; Zermelo 1908]. His work led him to some ideas that played a role in the history of lattice theory, and stimulated Dedekind to important but not very influential contributions in this area [Mehrtens 1979].
On Peano and his school see [Kennedy 1974; Borga, Freguglia & Palladino 1985]. [Rodriguez-Consuegra 1991] discusses carefully his influence on Russell.
kEK means that k is a class, or belongs to the class of all classes; N is the set of natural numbers.
With particular attention to authors such as Weierstrass, Cantor, Dedekind, von Staudt, Pasch, Pieri, Peano, Frege and Schröder.
By the time he wrote this, he was just falling under Frege’s influence; Frege’s work was
On this topic see [Grattan-Guinness 1978; Coffa 1979; Moore & Garciadiego 1981; Garciadiego 1985, 1986, 1992; Moore 1988]. For short summaries see [Garciadiego 1994; Moore, forthcoming].
[Frege 1903, 253]: “Und noch jetzt sehe ich nicht ein, wie die Arithmetik wissenschaftlich begründet werden könne, wie die Zahlen als logische Gegenstände gefasst und in die Betrachtung eingeführt werden können, wenn es nicht — bedingungsweise wenigstens — erlaubt ist, von einem Begriffe zu seinem Umfange überzugehn. Darf ich immer von dem Umfange eines Begriffes, von einer Klasse sprechen? Und wenn nicht, woran erkennt man die Ausnahmefälle? Kann man daraus, dass der Umfang eines Begriffes mit dem eines zweiten zusammenfällt, immer schliessen, dass jeder unter den ersten Begriff fallende Gegenstand auch unter den zweiten falle?”
I thank Alejandro Garciadiego for calling my attention to this topic.
The fact that he found it by 1900 is well-established by statements of his own, of Hilbert and of Husserl: see [Rang & Thomas 1981], also [Zermelo 1908a, 191] and [Peckhaus 1990, 25–26].
[Hilbert 1904], as translated in [van Heijenoort 1967, 131].
See, e.g., [König 1914], which tried to establish a consistent theory of arithmetic and set theory on the basis of a “synthetic logic” based on immediate intuition and with psychologistic overtones [op.cit., iii—iv]. Still, König was deeply acquainted with Cantor’s work and influenced by Dedekind and Hilbert. He distinguished many different senses of the word `set,’ rejecting extensionality, and advanced toward a notion of `Cantorian set’ that allowed him to derive the classical theory.
This conception of the debate was first emphasized by Moore [1978; see also 1982].
[Kowalewski 1950, 202], who also reports that the newspapers mentioned the news of König’s lecture.
Some authors tell the episode differently, following Kowalewski, who in 1950 wrote that Zermelo found the error just the day after the lecture [Moore 1982, 87]. But this is contradicted by another witness who was close to Hilbert and Zermelo, Schoenflies [1922, 100]. Since the only documentary evidence from that early time, Hausdorff’s paper, does not mention Zermelo, it may well be that Kowalewski misremembered.
[Zermelo 1904, 141], as translated by Bauer-Mengelberg, with some changes.
A detailed analysis of implicit uses prior to 1904 is given in [Moore 1982, chap. 1].
König’s final version of the lecture given the year before, and a paper by Georg Hamel on real functions. Only Hamel took the position of openly accepting the axiom.
Zermelo thus relied on the existence of implicitly defined well-ordered subsets of M. This peculiar approach may be quite consistent with Cantor’s conception of sets as given with an ordering (he regarded pure sets as obtained by abstraction from the nature and ordering of the elements of a given set).
Their wrong appreciation of the situation was due to the fact that Bernstein and Schoenflies wished to accept the class S2 of all ordinals as a set, so they had to restrict the `extension’ of sets.
French mathematics had been less prone to abstraction than German mathematics throughout the 19th century.
[Hadamard et al. 1905, 267], as translated in [Moore 1982, appendix, 316] with minor changes.
For biographical data see [Peckhaus 1990, 77ff].
[Moore 1980, 130; Peckhaus 1990, 82]: “Schon vor 30 Jahren, als ich Privatdozent in Göttingen war, begann ich unter dem Einflusse D. Hilberts, dem ich überhaupt das meiste in meiner wissenschaftlichen Entwickelung zu verdanken habe, mich mit den Grundlagenfragen der Mathematik zu beschäftigen, insbesondere aber mit den grundlegenden Problemen der Cantorschen Mengenlehre, die mir in der damals so fruchtbaren Zusammenarbeit der Göttinger Mathematiker erst in ihrer vollen Bedeutung zum Bewusstsein kamen.”
For further details on this point, see [Peckhaus 1991, 90–97], which offers numerous quotations from Zermelo’s manuscripts and letters to Hilbert.
See his 1905 letter to Hurwitz, in [Dugac 1976, 271] or [Moore 1982, 109].
For other cases in different areas, like Hamel and Vitali, see [Moore 1982, 100–01, 112].
[Steinitz 1910, 170–71], as translated in [Moore 1982, 172].
[Zermelo 1908], as translated by Stefan Bauer-Mengelberg in [van Heijenoort 1967, 200].
The expression `logical foundations’ is ambivalent, since it can be taken (or not) to mean that the foundations of mathematics are purely logical.
[Zermelo 1908], as translated in [van Heijenoort 1967, 200].
As we have seen (§VIII.8) Cantor’s intention had been to exclude `inconsistent sets’ by emphasizing the `collection into a whole’ of the elements. But in the absence of a more detailed explanation and development, his attempt was not even noticed by Zermelo and others.
[Zermelo 1908, 201–04], as translated by S. Bauer-Mengelberg in [van Heijenoort-1967].
This is the principle of extensionality, which had been indicated by Dedekind and, perhaps not so clearly, by Cantor.
By `class-statement’ Zermelo means a logical condition in one variable, i.e., what Russell was calling a propositional function.
The axiom is formulated for a family of disjoint sets in order to make it simple and more intuitive.
Zermelo indicates that this axiom is esentially due to Dedekind. Indeed, he simply postulates a set that is infinite according to Dedekind’s definition, the relevant mapping being a -3 {a}.
[Zermelo 1908], as translated in [van Heijenoort 1967, 201], with a small change to accommodate the word `Klassenaussage.’
This is not quite true, for No, (not to mention large cardinals) cannot yet be reached, see §Xl.l.
Around 1915, Zermelo himself was also working on an axiomatic definition of the ordinal numbers, see §X1.2.
He was thus forced to prove that, given sets M and N, there exists another set M’ equivalent to Mand disjoint from N [op.cit., 206].
This [op.cit., 212] was a generalization of König’s inequality, as Zermelo went on to notice. Of course, he formulated it for abstract sets, not for cardinal numbers.
Thus, Boolean algebra should apply to the whole universe of classes, not just to the subsets of a given set.
Perhaps his unwillingness to do so was one of the reasons why he abandoned the theory of types for a few years.
Some authors have argued that Quine’s systems NF and ML are closest to being a zig-zag theory (see [Fraenkel, Bar-Hillel & Levy 1973; Ullian 1986; Wang 1986]. For a discussion of Quine’s systems in the historical context of set theory and logicism, see [Ferreirós 1997].
The ZF system seems to be compatible with postulating sizes as big as one wishes. The view that the system implements a `limitation of size’ will only appeal, it seems, to those who regard the universal class as natural, and who are foreign to the notion of the cumulative hierarchy.
On this topic, see [Goldfarb 1988].
As Weyl did in Das Kontinuum [1918], developing a predicative alternative to set theory.
It was a great contribution to the axiomatization of logic, corroborated by the painfully detailed, explicit derivation of hundreds of propositions. This seems to be the main reason why Hilbert, for instance, greatly admired the book.
Part IV presented a generalization of ordinal arithmetic, the so-called theory of `relation numbers.’
Introduction to the second edition of [1903].
A noteworthy example can be found in [Russell 1908, 170].
The strange features of the famous Tractatus by his student Wittgenstein [1921] are thus more a symptom than a deviation.
Likewise, systems of propositional and predicate logic started to be presented in the now customary way during the 1920s. Elements of the distinction between syntax and semantics can be found in Peano and Schröder, and also in Frege, insofar as he differentiates clearly between a name and its referent, between use and mention of an expression
Fora brief review, written from a modem standpoint, see [Kanamori 1996].
On this topic, see [Kanamori 1994].
[Hausdorff 1914, 1–2]. His confidence in the Zermelo system is even clearer in the second, very abridged edition [1927, 34].
The bibliography only cites [Dedekind 1872]; see also chapters 2 and 9, where he fails to mention Dedekind in connection with functions and mappings.
The emergence of topology was a very complex, many-sided process. Here we pay attention to developments in set-theoretic topology leading up to the fundamental notion of topological space; for further details see [Manheim 1964, chap. 6; Johnson 1979; 1981]. Aspects of the rise of combinatorial and algebraic topology are studied in [Bollinger 1972; vanden Eynde 1992; Epple 1995].
Weyl had made a similar contribution previously [1913], also stimulated by Hilbert’s proposal. Hausdorff claims that his work was independent and that he presented it in 1912 at the University of Bonn [1914, 456–57].
The argument emphasized the similarity between the theory of order types and that of topological spaces: an order relation can be taken to be a two-valued function of two arguments, and topology can be developed on the basis of a distance function.
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Ferreirós, J. (1999). Diffusion, Crisis, and Bifurcation: 1890 to 1914. In: Labyrinth of Thought. Science Networks. Historical Studies, vol 23. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5049-0_9
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