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Volker Peckhaus†and Reinhard Kahle‡ In diesem Aufsatz wird erstmals die Hilbertsche Antinomie vollst¨andigpubliziert. David Hilbert hat sie w¨ahrend seiner Auseinandersetzun-gen mit der Cantorschen Mengenlehre gefunden. Seinen Angaben zu-folge wurde Ernst Zermelo durch sie zu seiner Version der Zermelo-Russellschen Antinomie angeregt. Es handelt sich um die Antinomieder Menge aller durch Addition (Vereinigung) und Selbstbelegung er-zeugbaren Mengen. Sie ¨ahnelt der Cantorschen Antinomie der Mengealler Kardinalzahlen, ist aber, so Hilbert, "rein mathematisch", da inihr ein offensichtlicher Bezug zur Cantorschen Kardinal- und Ordinal-zahlarithmetik vermieden wird.
In this paper Hilbert's paradox is for the first time published com-pletely. It was discovered by David Hilbert while struggling with Can-tor's set theory. According to Hilbert, it initiated Ernst Zermelo'sversion of the Zermelo-Russell paradox. It is the paradox of all setsderived from addition (union) and self-mapping. It is similar to Can-tor's paradox of the set of all cardinals, but, following Hilbert, of"purely mathematical nature", because an open reference to Cantor'scardinal and ordinal arithmetic is avoided.
Mathematics Subject Classification 2000 : 01A55 01A60 03-03 03A05 03E30Key Words: Paradoxes, set theory, union of sets, mapping In 1903 Gottlob Frege published the second volume of his Grundgesetze derArithmetik [Frege 1903] containing the admission that the logical system ∗Paper submitted to Historia Mathematica on 16 March 2001.
†Institut f¨ur Philosophie der Universit¨at Erlangen-N¨urnberg, Bismarckstr. 1, D-91054 ‡Wilhelm-Schickard-Institut, Universit¨at T¨ubingen, Sand 13, D-72076 T¨ubingen, e- Volker Peckhaus and Reinhard Kahle used there for the foundation of arithmetic had proved to be inconsistent.
He sent a copy of this volume to David Hilbert, who thanked him in a letterdated 7 November 1903. In this letter Hilbert referred to Frege's descriptionof Russell's paradox in the postscript, and wrote that "this example" wasalready known in G¨ottingen. In a footnote he added "I believe Dr Zermelodiscovered it three or four years ago after I had communicated my examplesto him" and continued I found other even more convincing contradictions as long as four orfive years ago; they led me to the conviction that traditional logicis inadequate and that the theory of concept formation needs to besharpened and refined.1 Hence, Hilbert maintained that he had formulated logical paradoxes around1898 or 1899 which he communicated to Zermelo, thereby initiating Zermelo'sindependent discovery of Russell's paradox which took place around 1899 or1900.
Zermelo's part in this story is well-known, Hilbert's role, however, remains almost completely obscure. Hilbert never published a new paradox. There isno paradox associated to Hilbert in standard catalogues of paradoxes. Whatcould it be? What could be more convincing than Russell's paradox? In this paper we present a candidate for Hilbert's paradox. In the first part we give evidence for our suggestion and provide the historical context. Inthe second part Hilbert's paradox is described and its systematic significanceis discussed.
Throughout the paper we use the term "paradox", bearing in mind, how- ever, that as early as 1907 Ernst Zermelo had suggested to use "antinomy"instead. After having read the proof sheets of the paper "Bemerkungen zuden Paradoxieen von Russell und Burali-Forti" co-authored by his studentKurt Grelling and his philosophical colleague in G¨ottingen Leonard Nelson[Grelling/Nelson 1908], he criticized in a comment to Nelson the use of theterm "paradox", "antinomy" being much more precise. "Paradox" means,he wrote, "a statement contradicting the common opinion, it doesn't containanything of an inner contradiction (as is the case for the paradoxes of Russelland Burali-Forti, and expressed by the term "antinomy").2 1[Frege 1980, p. 51]. German original [Frege 1976, pp. 79–80].
2Zermelo's postcard to Leonard Nelson, Glion (Switzerland), no date (postmark 22 December 1907): "Wollen Sie nicht auch lieber ‘Antinomie' sagen, statt ‘Paradoxie', dader erstere Ausdruck sehr viel pr¨aziser ist." A month later Zermelo wrote to Nelson in apostcard, Glion, no date (postmark 20 January 1908): "Das Wort ‘Paradoxie' scheint mirvon Hessenberg [Gerhard Hessenberg, co-editor of the new series of the Abhandlungen derFries'schen Schule, where the joint paper was published] weil es eben etwas ganz anderesbedeutet, n¨amlich eine Aussage, welche der herk¨ommlichen Meinung widerstreitet; von Hilbert's Paradox Historical Context Zermelo's Paradox We turn to Zermelo's part in this story. Zermelo came to G¨ottingen in 1897in order to work for his Habilitation. His special fields of competence were thecalculus of variations and mathematical physics, such as thermodynamics andhydrodynamics.3 Under the influence of Hilbert he changed his focus of inter-ests to set theory and foundations. He became Hilbert's collaborator in thefoundations of mathematics, a first member of Hilbert's school before it waseven established. Zermelo's first set-theoretical publication on the additionof transfinite cardinals dates from 1902 [Zermelo 1902], but as early as thewinter term 1900/1901 he gave a lecture course on set theory in G¨ottingen.
It is possible that he found the paradox while preparing this course. He re-ferred to it in the famous polemical paper "A New Proof of the Possibilityof a Well-Ordering" of 1908 [Zermelo 1908a]. There Zermelo noted that hehad found the paradox independently of Russell, and that he had mentionedit to Hilbert and other people already before 1903, the year when it was firstpublished by Frege and Russell ([Frege 1903], [Russell 1903]). And indeed,among the papers of Edmund Husserl, until 1916 professor of philosophy inG¨ottingen, a note in Husserl's hand was found, partially written in Gabels-berger shorthand, saying that Zermelo had informed him on 16 April 1902that the assumption of a set M that contains all of its subsets m, m0, . .
as elements, is an inconsistent set, i. e., a set which, if treated as a set atall, leads to contradictions.4 Zermelo's message was a comment on a reviewthat Husserl had written on the first volume of Ernst Schr¨oder's Vorlesungen¨uber die Algebra der Logik [Schr¨oder 1890]. Schr¨oder had criticized GeorgeBoole's interpretation of the symbol 1 as the class of everything that can be asubject of discourse (the universe of discourse, universal class).5 Husserl haddismissed Schr¨oder's argumentation as sophistical [Husserl 1891, p. 272], andwas now advised by Zermelo that Schr¨oder was right concerning the matter, einem inneren Widerspruch enth¨alt es gar nichts," Archiv der sozialen Demokratie, Bonn,Nelson papers.
3On Zermelo's activities in G¨ottingen cf. esp. [Moore 1982], [Peckhaus 1990a, pp. 76– 122], [Peckhaus 1990b].
4Critical edition in Husserliana XXII [Husserl 1979, p. 399]: "Zermelo teilt mit (16.
April 1902) [ . . ] Eine Menge M, welche jede ihrer Teilmengen m, m0 . . als Elemententh¨alt, ist eine inkonsistente Menge, d. h. eine solche Menge, wenn sie ¨ Menge behandelt wird, f¨ uhrt zu Widerspr¨ uchen." English translation in [Rang/Thomas 5[Schr¨oder 1890, p. 245]: "Es [ist] in der That unzul¨assig [ . . ], unter 1 eine so um- fassende, sozusagen ganz offene Klasse, wie das oben geschilderte ‘Universum des Diskus-sionsf¨ahigen' (von Boole) zu verstehen." Schr¨oder referred to Boole's definition of theuniverse of discourse and his interpretation of the symbol 1, cf. [Boole 1854, pp. 42–43].
Volker Peckhaus and Reinhard Kahle but not in his proof.
The document from the Husserl papers provides convincing evidence for Hilbert's assertion concerning Zermelo. It is furthermore confirmed by Zer-melo's own recollections. In 1936, Heinrich Scholz was working on the papersof Gottlob Frege which he had acquired for his department at the Universityof M¨unster. He had found Hilbert's letter to Frege, mentioned above, andnow asked Zermelo what paradoxes Hilbert referred to in this letter.6 Zer-melo answered that the set-theoretic paradoxes were often discussed in theHilbert circle around 1900, and he himself had given at that time a preciseformulation of the paradox which was later named after Russell.7 Traces of Hilbert's Paradox But what about Hilbert's own paradox? It left some traces in history. Themost prominent one is Otto Blumenthal's hint in his biography of Hilbertpublished in the third volume of Hilbert's Collected Works [Blumenthal1935]. There Blumenthal mentions the paradoxes of set theory and relatesthem to the second of Hilbert's problems presented in the famous Paris prob-lems lecture in 1900 [Hilbert 1900a], i. e., the problem of proving the consis-tency of the axioms of arithmetic. According to Blumenthal the paradoxesshowed that certain operations with the infinite, which everyone thought tobe allowed, led unquestionably to contradictions. Blumenthal reports thatHilbert convinced himself of this fact by constructing the example of an in-consistent set of all sets resulting from union and self-mapping, i. e., purelymathematical operations [Blumenthal 1935, pp. 421–422].
Another trace can be found in the year 1907. The G¨ottingen philoso- pher Leonard Nelson and the student of mathematics and philosophy KurtGrelling were working on one of the first philosophical papers to discuss theparadoxes, here especially the ones of Russell and Burali-Forti [Grelling/Nel-son 1908]. The joint paper contained a general formulation suitable for severalparadoxes, among them the semantical "heterological paradox" or "Grelling'sparadox" (cf. [Peckhaus 1990a, pp. 168–195], [Peckhaus 1995]). From a letterof the G¨ottingen mathematician Ernst Hellinger to Leonard Nelson, dated 28December 1907,8 we learn that Hellinger had read a manuscript version of the 6Heinrich Scholz to Zermelo, dated M¨unster, 5 April 1936, University Archive Freiburg i. Br., Zermelo papers, C 129/106.
7Zermelo to Scholz, dated Freiburg i. Br., 10 April 1936, Institut f¨ur mathematische Logik und Grundlagenforschung, M¨ unster, Scholz papers: " ¨ Uber die mengentheoretischen Antinomien wurde um 1900 herum im Hilbert'schen Kreise viel diskutiert, und damalshabe ich auch der Antinomie von der gr¨oßten M¨achtigkeit die sp¨ater nach Russell benanntepr¨azise Form (von der ‘Menge aller Mengen, die sich nicht selbst enthalten') gegeben. BeimErscheinen des Russellschen Werkes [ . . ] war uns das schon gel¨aufig." 8Hellinger to Nelson, dated Breslau, 28 December 1907, Archiv der sozialen Demokratie, Hilbert's Paradox paper. He suggested to add a note on Hilbert's paradox, because its appear-ance was more mathematical and perhaps more suitable for mathematiciansnot working in set theory. In the end Hilbert's paradox was not included, be-cause Grelling failed to reduce it to the general formulation. Nevertheless wecan state that, at least in G¨ottingen, Hilbert's paradox was generally knownin that time.
Hilbert and Cantor Given the time period referred to by Hilbert, it can be assumed that Hilbertformulated the paradox during his discussions with Georg Cantor, docu-mented in their correspondence between 1897 and 1900.9 The main topicswere Cantor's problems with the assumption of a set of all cardinals. Alreadyin the first of Cantor's letters to Hilbert, dated 26 September 1897 [Cantor1991, no. 156, pp. 388–389], Cantor proves that the totality of alephs does notexist, i. e., that this totality is not a well-defined, finished set [fertige Menge].
If it is taken to be a finished set, a certain larger aleph would follow on thistotality. So this new aleph would at the same time belong to the totalityof all alephs, and not belong to it, because of being larger than all alephs(ibid., p. 388). Cantor consequently distinguished sets from other kinds ofmultiplicities, i. e., "finished" sets from multiplicities which are not sets, likethe totality of all cardinals. The latter multiplicities are "absolutely infinite",unlike the former ones, the "transfinite" sets. In a later letter Cantor gave thefollowing characterization of a finished set: A set can be imagined as finishedif it is consistently possible to imagine all of its elements as being gathered,the set itself therefore as one compound thing, i. e., if it is possible to imaginethe totality of its elements as existing.10 This is, however, impossible for theabsolute infinite which he identifies with God. The absolute infinite doesn'tallow any determination [Cantor 1883, p. 556]. Realized in its highest per-fection in God it has to be strictly opposed to the actual infinite which hecalls the transfinite [Cantor 1887, pp. 81–82].
It is well-known that Cantor later changed his terminology. In May 1899 Bonn, Nelson papers: "Es w¨are vielleicht nicht unzweckm¨aßig, es [Hilbert's paradox] zuerw¨ahnen, da es mathematischer aussieht als die andern, und vielleicht auch dem nicht-mengentheoretischen Mathematiker sympathischer aussieht, als das W-Paradoxon [i. e.,Burali-Forti's paradox of the set W of all ordinals]." 9For a comprehensive discussion of this correspondence cf. [Purkert/Ilgauds 1987, 147– 166]. Extracts are published in [Cantor 1991]. For Cantor's reaction to the paradox seealso [Ferreir´os 1999, 290–296].
10Cantor's letter to Hilbert, dated 2 October 1897 [Cantor 1991, p. 390], also published in [Purkert/Ilgauds 1987, no. 44, pp. 226–227]. A similar definition can be found in Cantor'sletter to Hilbert, Halle, 10 October 1898 [Cantor 1991, no. 158, 396–397, definition on p.
396].
Volker Peckhaus and Reinhard Kahle he wrote to Hilbert that he had become accustomed to replace what heformerly had called "finished" by the expression "consistent". The notion"sets" stood now for "consistent multiplicities".11 Cantor disproves the existence of the totality of all cardinals by show- ing that the assumption of its existence contradicts his definition of a setas a comprehension of certain well distinguished objects of our intuition orour thinking in a whole.12 The totality of all cardinals (and of all ordinals)cannot be thought of as one such thing, contrary to actual infinite objectslike transfinite sets. He is therefore not really concerned with paradoxes andtheir solution, but with non-existence proofs using reductio-ad-absurdum ar-guments.13 From these passages we learn that Hilbert was concerned with what was later called "Cantor's paradox", i. e., the paradox of the greatest cardinal,or of the set of all cardinals. It is clear, however, that the contradiction dis-cussed by Cantor served only as a paradigmatic example for other inconsis-tent multiplicities, i. e., totalities resulting from unrestricted comprehension.
Nevertheless, there is no evidence that Cantor and Hilbert discussed the con-tradiction resulting from the assumption of a greatest ordinal, today knownas "Burali-Forti's paradox", although this has been claimed by several au-thors.14 Usually Cantor's letter to Philip E. B. Jourdain of 4 November 1903is taken as evidence for Cantor having known the paradox of the greatestordinal before its publication by Cesare Burali-Forti [Burali-Forti 1897], andthat he had communicated this paradox to Hilbert as early as 1896.15 In factCantor showed in this letter to Jourdain that the assumption of a system ofall ordinals leads to a contradiction. In his communication with Hilbert of9 May 1899, however, he only referred to the assumption of a greatest car-dinal.16 Purkert and Ilgauds made it furthermore plausible [Purkert/Ilgauds 11Cantor's letter to Hilbert, Halle, 9 May 1899 [Cantor 1991, no. 160, p. 399].
12[Cantor 1895/97], quoted in [Cantor 1932, p. 282]: "Unter einer ‘Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unsrerAnschauung oder unseres Denkens (welche die ‘Elemente' von M genannt werden) zueinem Ganzen." 13We follow in this evaluation [Moore/Garciadiego 1981], [Garciadiego Dantan 1992].
14E. g., [Fraenkel 1930, pp. 261], [Meschkowski 1983, p. 144].
15The letter was quoted by Jourdain [Jourdain 1904] and mentioned by Felix Bernstein [Bernstein 1905, 187]. Gerhard Hessenberg referred to Bernstein when maintaining Can-tor's priority [Hessenberg 1906, § 98, p. 631]. From there it became standard folklore. Cf.
[Grattan-Guinness 2000, pp. 117–119].
16Cantor's letter to Philip E. B. Jourdain, dated Halle 4 November 1903 [Cantor 1991, no. 172, pp. 433–434, quote p. 433]: "Den unzweifelhaft richtigen Satz, daß es außer denAlephs keine anderen transfiniten Cardinalzahlen giebt, habe ich vor ¨ uber 20 Jahren (bei der Entdeckung der Alephs selbst) intuitiv erkannt. [ . . ] Schon vor 7 Jahren machte ichHerrn Hilbert, vor 4 Jahren Herrn Dedekind darauf bez¨ ugliche briefliche Mitteilung." The extensive correspondence between Cantor and Jourdain is published in [Grattan-Guinness Hilbert's Paradox 1987, p. 151] that Cantor's recollections were erroneous. He most probablyreferred to his letter to Hilbert of 26 September 1897, mentioned above. Thenotion of the greatest ordinal was also the topic of a letter Cantor wroteto Dedekind on 3 August 1899. There he proved that the system Ω of allnumbers is an inconsistent, absolutely infinite multiplicity.17 In this letterCantor also referred to the totality of everything imaginable ("Inbegriff allesDenkbaren"), i. e., Dedekind's own assumption in Was sind und was sollendie Zahlen? [Dedekind 1888], needed to prove that there are infinite sys-tems (sets).18 Cantor showed that his non-existence proofs also hold withthis assumption.
Hilbert's responses in correspondence have not been preserved,19 but he published his opinion at prominent places. In the paper "On the Concept ofNumber" from 1900 [Hilbert 1900b], Hilbert's first paper on the foundationsof arithmetic, he gave a set of axioms for arithmetic, and claimed that onlya suitable modification of known methods of inference would be needed forproving the consistency of the axioms. If this proof were successful, the ex-istence of the totality of real numbers would be shown at the same time. Inthis context he referred to Cantor's problem of whether the system of realnumbers is a consistent, or finished, set. He stressed: Under the conception above, the doubts which have been raised againstthe existence of the totality of all real numbers (and against the ex-istence of infinite sets in general) lose all justification; for by the setof real numbers we do not have to imagine the totality of all possiblelaws according to which the elements of a fundamental sequence canproceed, but rather—as just described—a system of things whose in-ternal relations are given by a finite and closed set of axioms [ . . ],and about which new statements are valid only if one can derive themfrom the axioms by means of a finite number of logical inferences.20 He also claimed that the existence of the totality of all powers or of allCantorian alephs could be disproved, i. e., in Cantor's terminology, that thesystem of all powers is an inconsistent (not finished) set (ibid.).
17Cantor to Dedekind, dated Halle, 3 August 1899, [Cantor 1991, no. 163, pp. 407–411].
It is one of the best known of Cantor's letters, published already in Zermelo's edition ofCantor's collected works [Cantor 1932, pp. 443–447]. Ivor Grattan-Guinness has shown,however, that Zermelo combined this letter with the one of 28 July 1899 and even changedthe original wording at some places [Grattan-Guinness 1974–75]. The correct text of theletter of 28 July 1899 is found in [Cantor 1991, no. 162, p. 405].
18[Dedekind 1888, p. 14]: "Meine Gedankenwelt, d. h. die Gesamtheit S aller Dinge, welche Gegenstand meines Denkens sein k¨onnen, ist unendlich." 19In his letter to Hilbert of 2 October 1897 Cantor referred to some of Hilbert's objec- tions, quoted in [Purkert/Ilgauds 1987, pp. 226–227].
20[Hilbert 1996b, p. 1095]. German original [Hilbert 1900b, p. 184].
Volker Peckhaus and Reinhard Kahle Hilbert took up this topic again in his famous Paris lecture on "Mathe- matical Problems".21 In the context of his commentary on the second problemconcerning the consistency of the arithmetical axioms he used the same exam-ples from Cantorian set theory and the continuum problem as in the earlierlecture. "If contradictory attributes be assigned to a concept," he wrote, "Isay, that mathematically the concept does not exist" [Hilbert 1996a, p. 1105].
According to Hilbert a suitable axiomatization would be able to avoid the contradictions resulting from the attempt to comprehend absolute infinitemultiplicities as units, because only those concepts had to be accepted whichcould be derived from an axiomatic base.
Although it is evident that Hilbert was at that time deeply concerned withthe problems of set theory, we have found no direct evidence that Hilberthad formulated contradictions in this context, or even a paradox of his own.
Indirect evidence can be found, however, in documents dating from a fewyears later.
Only after the publication of the paradoxes by Russell and Frege, and especially through Frege's reaction, the logical significance of this kind ofcontradiction became evident.22 Now mathematicians understood that theseparadoxes were not the simple contradictions that they were familiar with intheir everyday reductio ad absurdum arguments. As logical paradoxes theyseriously affected Hilbert's axiomatic programme, especially the proposedconsistency proof for arithmetic. It is a matter of course that a consistencyproof, based on a logic proved to be inconsistent, could not be given. Hilbertfirst expressed this new insight in a talk delivered at the Third InternationalCongress of Mathematicians in Heidelberg in August 1904 [Hilbert 1905c].
In this lecture "On the Foundations of Logic and Arithmetic" he demandeda "partly simultaneous development of the laws of logic and arithmetic"[Hilbert 1905c, p. 176]. According to Blumenthal [Blumenthal 1935, p. 422],this lecture remained completely misunderstood and several of Hilbert's ideasproved to be defective. Nevertheless it was the first step in the constructionof a foundational system of mathematics avoiding the paradoxes.
The next step was taken in a lecture course on the "Logical Principles of Mathematical Thinking" which Hilbert gave in G¨ottingen in the summerterm of 1905. Two sets of notes of this lecture course were preserved. The"official" notes are from Ernst Hellinger, then a student of mathematics.
They contain marginal notes in Hilbert's hand [Hilbert 1905a]. Another set 21[Hilbert 1900a], English translations [Hilbert 1902], [Hilbert 1996a].
22Cf. [Moore 1978], [Moore 1980, pp. 104–105], [Moore/Garciadiego 1981], [Garciadiego Dantan 1992].
Hilbert's Paradox was produced by the student of mathematics and physics Max Born [Hilbert1905b]. Part B of these notes, on "The Logical Foundations", starts witha comprehensive discussion of the paradoxes of set theory. It begins withmetaphorical considerations on the general development of science: It was, indeed, usual practice in the historical development of sciencethat we began cultivating a discipline without many scruples, pressingonwards as far as possible, that we thereby, however, then ran into dif-ficulties (often only after a long time) that forced us to turn back andreflect on the foundations of the discipline. The house of knowledge isnot erected like a dwelling where the foundation is first well laid-outbefore the erection of the living quarters begins. Science prefers toobtain comfortable rooms as quickly as possible in which it can rule,and only subsequently, when it becomes clear that, here and there, theloosely joined foundations are unable to support the completion of therooms, science proceeds in propping up and securing them. This is noshortcoming but rather a correct and healthy development.23 Although contradictions are quite common in science, Hilbert continued, inthe case of set theory they seem to be different, because there they have atendency towards the side of theoretical philosophy. In set theory the commonAristotelian logic and its standard methods of concept formation were usedwithout hesitation. And these standard tools of purely logical operations,especially the subsumption of concepts under a general concept, proved tobe responsible for the new contradictions.
Hilbert elucidated these considerations by presenting three examples. The first paradox discussed is the Liar paradox. The third one is "Zermelo's para-dox," as the Russell-Zermelo paradox was called in G¨ottingen at that time.
Hilbert described this paradox as purely logical, assuming that it might bemore convincing for non-mathematicians. He stressed, however, that it wasderived from his own paradox, the second one in his list of examples, and thissecond paradox was, according to Hilbert, of purely mathematical nature.24Hilbert expressed his opinion that this paradox appears to be especially important; when I found it, I thought inthe beginning that it causes invincible problems for set theory thatwould finally lead to the latter's eventual failure; now I firmly believe,however, that everything essential can be kept after a revision of thefoundations, as always in science up to now. I have not published this 23[Hilbert 1905b, p. 122], published in [Peckhaus 1990, p. 51].
24[Hilbert 1905a, p. 210]: "Als drittes Beispiel dieser Widerspr¨uche stelle ich neben diesen meinen rein mathematischen noch einen rein logischen, den Dr. Zermelo aus jenemherausgezogen hat [ . . ]." Volker Peckhaus and Reinhard Kahle contradiction, but it is known to set theorists, especially to G. Can-tor.25 This paradox, arising from uniting sets and mapping them to themselves, isexactly the one Blumenthal referred to in his biography. It is most likely theone Hilbert himself referred to in his letter to Frege.
Hilbert's Paradox Hilbert's Presentation The full text of Hilbert's paradox is given in the appendices, both in Englishtranslation (appendix I) and in the German original (appendix II). Here, wereconstruct the main steps of Hilbert's argument.
The paradox is based on a special notion of set which Hilbert introduces by means of two set formation principles starting from the natural numbers.
The first principle is the addition principle. In analogy to the finite case,Hilbert argued that the principle can be used for uniting two sets together"into a new conceptual unit [ . . ], a new set that contains each element ofeither sets." This operation can be extended: "In the same way, we are ableto unite several sets and even infinitely many into a union." The secondprinciple is called the mapping principle. Given a set M, he introduces theset MM of self-mappings of M to itself.26 A self-mapping is just a totalfunction which maps the elements of M to elements of M.27 Now, he considers all sets which result from the natural numbers "by applying the operations of addition and self-mapping an arbitrary number oftimes." By use of the addition principle which allows to build the union ofarbitrary sets one can "unite them all into a sum set U which is well-defined."In the next step the mapping principle is applied to U, and we get F = UU asthe set of all self-mappings of U. Since F was built from the natural numbersby using the two principles only, Hilbert concludes that it has to be containedin U. From this fact he derives a contradiction.
Since "there are ‘not more' elements" in F than in U there is an assign- ment of the elements ui of U to elements fi of F such that all elements offi are used. Now one can define a self-mapping g of U which differs from all 25[Hilbert 1905a, p. 204], published in [Peckhaus 1990, p. 52].
26Hilbert used the German term "Selbstbelegung" which is translated here by "self- mapping". The term "Belegung" was already used by Cantor [Cantor 1895/97, § 4, p. 486(1895)], cf. also [Cantor 1932, p. 287]. In his edition of Georg Cantor's collected worksZermelo explained Belegung as a function with explicitly given domain and (potential)range [Cantor 1932, footnote [3], p. 352].
27In classical logic, MM is isomorphic to 2M, and the set of all mappings from M to {0, 1} is isomorphic to P(M), the power set of M.
Hilbert's Paradox fi. Thus, g is not contained in F. Since F was assumed to contain all self-mappings we have a contradiction. In order to define g Hilbert used Cantor'sdiagonalization method. If fi is a mapping ui to fi(ui) = u f (i) element ug(i) different from u as the image of u f (i) i under g. Thus, we have g(ui) = ug(i) 6= u and g "is distinct from any mapping f f (i) k of F in at least one assignment."28 Hilbert finishes his argument with the following observation: We could also formulate this contradiction so that, according to thelast consideration, the set UU is always bigger [of greater cardinality]29than U but, according to the former, is an element of U.
Brief Reconstruction In order to make the argument more comprehensible, the paradox can bepresented in the following way. First we define a notion of set: Definition 1 We define inductively: 1. The natural numbers as a whole are a set.30 2. Addition principle: If we have an arbitrary, possibly infinite collection of sets, the union of all these sets is a set. 3. Mapping principle: The totality of all total functions from a given set into itself is a set. Now we take the closure of all sets introduced according to the followingdefinition (this union is well defined according to the addition principle): Definition 2 Let U be the union of all sets defined according to definition1. Now we can apply the mapping principle to it.
Definition 3 Let F be the set UU . Obviously F is built according to our definition of sets. We have used theaddition principle to define U and then the mapping principle to define F.
But that means, F has to be contained in U because U was the union of allsets built according to the definition of sets. Thus, we get the following 28Hilbert's notation ug(i) is somewhat clumsy. In fact, it is enough to say that g(ui) = vi for an element vi of U with vi 6= fi(ui).
29Remark later added in Hilbert's hand in Hellinger's lecture notes.
30Hilbert even argues that the natural numbers can be defined from finite sets using the addition principle.
Volker Peckhaus and Reinhard Kahle Lemma 4 F ⊆ U. From this lemma it follows that there exists a function of U in F whose rangeis the whole set F. Therefore, we can apply Cantor's diagonalization methodto define a function from U to U which is distinct from each element of F.
Proposition 5 There exists a total function g from U to U such that g 6∈ F. But by definition of F, this set contains all total function from U to U. Thus,we get as a Corollary 6 The system of sets defined by 1 is contradictory. Analysis of the Paradox The reconstruction given above reveals the source of the paradox. Obviouslythe addition principle is too vague. Hilbert allows "to unite several sets andeven infinitely many into a union," he even allows to "unite them all," i. e., allsets defined by addition and self-mapping. He does not determine, however,the domain of the universal quantifier. The definition of the set U is, thus,based on an impredicative construction, because U itself has to belong tothis domain. In short: The definition of U depends on a totality containingU itself.
These problems can be overcome by restricting the addition principle. It has to be demanded that the sets united have to be elements of another setalready established. And this is, in fact, the way in which Zermelo proceededin his axiomatization of set theory. This axiomatic system, refined by Fraenkeland Skolem and called ZFC, is still today generally accepted as the basis ofmathematics. In ZFC we have a union axiom corresponding to the additionprinciple. But in contrast to the addition principle, a family of sets T beingitself a set is demanded which can be regarded as an index set giving somecontrol over the sets gathered in the union [Zermelo 1908b, 265]. Nowadays,the union axiom is stated as: ∀T ∃S∀x(x ∈ S ↔ ∃U(x ∈ U ∧ U ∈ T )) Fraenkel correctly saw that an unrestricted union axiom within axiomatizedset theory led to the same problems as the ones connected with Russell'sparadox. He saw the reason for these problems in an unconcerned use of thenotion "arbitrarily many." Fraenkel referred directly to the union axiom, sohis analysis reads like a diagnosis of the cause of Hilbert's paradox.31 31[Fraenkel 1927, p. 71]: "Will man [ . . ] zu etwas allgemeineren Prozessen [of set for- mation] fortschreiten, so muß man [ . . ] auch die Zusammenfassung der Elemente ver- Hilbert's Paradox Although Hilbert worked only in a restricted domain of sets, containing only those sets formed by addition and self-mapping, his addition principlewas itself too vague, so that it resulted in effects similar to those of Cantor'scomprehension.32 From another perspective the lack of a proper quantifica-tion theory is conspicuous. Hilbert's formulation is therefore affected by thegeneral problems of impredicativity.
Zermelo's axiomatization of set theory can thus be read as an answer to two different paradoxes. His strategy was to avoid unrestricted comprehen-sion, leading to Cantor's paradox (and also to the Zermelo-Russell paradox),and unrestricted union, leading to Hilbert's paradox. He easily prevented theformulation of Hilbert's paradox by introducing the family set T in the unionaxiom (axiom V). The paradoxes resulting from unrestricted comprehensionwere avoided by introducing the separation axiom (axiom III) which ensuresthat each set M has at least one subset M0 not being element of M [Zermelo1908b, 264].
In contrast to the addition principle, the mapping principle is "innocent" of the emergence of Hilbert's paradox. If we replace the total functions fromM to M by total functions from M to the set {0, 1} we get the set of chara-teristic functions of all subsets of M. Thus, the mapping principle is closelyrelated to the power set axiom as it is used in modern set theory. Hilbertdemanded for the mapping principle that the set of all self-mappings is ob-tained over sets already established, a restriction also valid for the modernpower set axiom.
schiedener Mengen anstreben. Einen Fingerzeig, wie dies zu erfolgen hat, liefert uns dieBildung der Vereinigungsmenge in der Cantorschen Mengenlehre, wo die s¨amtlichen Ele-mente beliebig vieler Mengen zu einer neuen Menge, der Vereinigungsmenge, vereinigtwerden k¨onnen [ . . ]. Hinsichtlich der gefahrdrohenden Folgen eines unbek¨ brauchs des Begriffs ‘beliebig viele' sind wir freilich, z. B. durch das Russellsche Para-doxon, hinl¨anglich gewitzigt; wir gehen daher nicht wie fr¨ uher von beliebig vielen Mengen aus, sondern setzen voraus, daß diese Mengen als die Elemente einer bereits als legitimerkannten Menge s¨auberlich gegeben sind." 32This is also the conclusion of Paul Bernays who reported in 1971, obviously referring to Hilbert's paradox: "Der Gedanke der Beschr¨ankung auf solche Mengen, die man, beginnendmit einer Ausgangsmenge (etwa der Menge der nat¨ urlichen Zahlen) durch Potenzmengen- bildungen, Vereinigungsprozesse und Aussonderungen bilden kann, wurde—wie ich ausErz¨ahlungen von Hilbert weiß—seinerzeit auch erwogen; er f¨ uhrte aber zun¨achst gerade zu einer Versch¨arfung der Paradoxien, da man die Vereinigungsprozesse nicht gen¨ deutlich normierte, vielmehr die Zusammenfassung der durch die angegebenen Prozessegewinnbaren Mengen zu einer Menge ihrerseits als einen zul¨assigen Vereinigungsprozeßansah" [Bernays 1971/1976, p. 199]. We would like to thank Jos´e F. Ruiz, Madrid, forbringing this quote to our attention.
Volker Peckhaus and Reinhard Kahle Hilbert's paradox is closely related to Cantor's own paradox. Both Cantorand Hilbert construct "sets" which lead to contradictions being proved withthe help of Cantor's diagonalization argument. However, the ways in whichthese "sets" are constructed differ essentially. According to Cantor ([Cantor1883, § 11], cf. [Cantor 1932, pp. 195–197]), there are three principles forthe generation of cardinals. The first principle ("erstes Erzeugungsprinzip")concerns the generation of real whole numbers [reale ganze Zahlen, i. e., or-dinal numbers] by adding a unit to a given, already generated number. Thesecond principle allows the formation of a new number, if a certain succes-sion of whole numbers with no greatest number is given. This new number isimagined as the limit of this succession. Cantor adds a third principle, the in-hibition or restriction principle ("Hemmungs- oder Beschr¨ankungsprinzip")which grants that the second number class has not only a higher cardinalitythan the first number class, but exactly the next higher cardinality. Con-sidering Cantor's general definition of a set as the comprehension of certainwell-distinguished objects of our intuition or our thinking as a whole ([Cantor1895/97], [Cantor 1932, p. 282]), one can justly ask whether the sets of allcardinals, of all ordinals or the universal set of all sets are sets according tothis definition, i. e., whether an unrestricted comprehension is possible. Can-tor denies this, justifying his opinion with the help of a reductio ad absurdumargument, but he doesn't exclude the possibility of forming the paradoxes byprovisions in his formalism.
Hilbert, on the other hand, introduces two alternative set formation prin- ciples, the addition principle and the mapping principle, but they lead toparadoxes as well. In avoiding concepts from transfinite arithmetic Hilbertbelieves that the purely mathematical nature of his paradox is guaranteed.
For him, this paradox appears to be much more serious for mathematics thanCantor's, because it concerns an operation that is part of everyday practiceof working mathematicians.
The significance of Hilbert's paradox for the history of mathematics should now be obvious. The paradox shows the importance of the end 19th centurydiscussion on universal sets and classes, e. g., Cantor's absolutely infinite to-talities, Dedekind's infinite totality of all things which might become objectsof our thinking, and Boole's universe of discourse. From the beginning thelimitation of size argument played a role (cf. [Hallett 1984]). This discussionmarked a latent foundational crisis in mathematics. The mathematicians in-volved were dealing with paradoxes, i. e., contradictions that are, they be-lieved, avoidable. The foundational crisis became manifest in 1903, whenBertrand Russell and Gottlob Frege published the insight that "Russell'sparadox" could be derived from Frege's system of the Grundgesetze. Now Hilbert's Paradox mathematicians were dealing with antinomies, i. e., intrinsic contradictionsthat could not easily be solved. Even this new move was closely connected tothe earlier discussion because Russell found his own paradox while investi-gating Cantor's set theory (cf. [Garciadiego Dantan 1992], [Grattan-Guinness1978], [Grattan-Guinness 2000, pp. 310–315], [Moore 1980, pp. 104–105]).
Hilbert himself had to change his axiomatic programme. Now logic and settheory moved into the focus of his foundational research (cf. [Peckhaus 1990,pp. 61–75]).
Appendix I: Hilbert's Paradox (English Translation) [Marginal note: 18th lecture, 10 July] [ . . ] 204 In addition, I now come to twoexamples of contradictions which are much more convincing, the first, beingof purely mathematical nature, appears to be especially important; when Ifound it, I thought in the beginning that it causes invincible problems for settheory that would lead to the latter's eventual failure; now I firmly believe,however, that everything essential can be kept after a revision of the founda-tions, as always in science up to now. I have not published this contradiction,but it is known to set theorists, especially to G. Cantor. Anyhow, we regardfinite sets, represented by finitely many numbers, as the operational basispermitted, and also the countable infinite set 1, 2, 3 . . of all natural num-bers. Furthermore, it seems to be allowed to unite two such sets (1, 2, 3 . .)and (a1, a2, a3 . .) into a new conceptual unit (1, 2, 3 . . , a1, a2, a3 . .), i. e., anew set that contains each element of either sets. In the same way, we areable to unite several sets and even infinitely many into a union. We designatethis as the addition principle, and write 205 in short for the set obtained fromM1, M2 . ., M1 + M2 + · · · These unions are operations, generally applied in logic in even much morecomplicated cases without any hesitation. Therefore, it seems to be possibleto apply them here without further ado. Besides this addition principle, weuse a further consideration for forming new sets. Let y = f (x) be a numbertheoretic function which maps to every integer value x an integer y; in asense immediately to be understood, we can designate such a function asa mapping [Belegung] of the number sequence to itself, by imagining forinstance a scheme: x = 1, 2, 3, 4 . . y = 2, 3, 6, 9 . . The system of all these number theoretic functions f (x), or of all possiblemappings of the number sequence to its own elements, forms a new set "re- Volker Peckhaus and Reinhard Kahle sulting from the number sequence M by self-mapping," we write it MM. 206As a principle following from the laws of uniting in ordinary logic and, ac-cording to it, completely unobjectionable, we can now regard the opinionthat in every case well-defined sets arise from well-defined sets by the self-mapping operation (mapping principle). For instance, by using this principle,from the continuum of all real numbers results the set of all real functions.
We want to use only these two principles unobjectionable according to allprevious mathematics and logic.
We start with all finite sets of numbers and the infinite series 1, 2, 3 . . of natural numbers already derived therefrom by addition, and take all setsresulting from them by applying the operations of addition and self-mappingan arbitrary number of times; these sets form again a well-defined unit, foraccording to the addition principle I unite them all into a sum set U whichis well-defined. If I form now the set F = UU of self-mappings of U, this setarises from the original number sequence via the two operations of additionand 207 self-mapping only; it, therefore, also is one of the sets from whoseaddition U just resulted and, therefore, must be a subset of U: F is contained in U.
[Marginal note: 19th lecture, 11 July] I will now show that this leads to acontradiction. Let u1, u2, u3 . . be the elements of U; then, each element f ofF = UU represents a mapping of U to itself, i. e., in a way a function, thatassigns to each element ui of U another uf(i), where it is not at all necessary that the uf(i) have to be distinct from one another; we, therefore, represent this element f most conveniently in schematic form: f (u1) = uf(1), f(u2) = uf(2), f(u3) = uf(3) . . Our result (1), that F is contained in U, can now be expressed more exactlyin the following way: we can definitely assign to each single element ui ofU a fi of F so that all fi will thereby be used, maybe even repeatedly, butthat, in any case, to each ui only corresponds exactly one fi; this means,obviously, nothing else than that there are "not more" elements fi than ui.
We now consider such an assignment: 208 u1 f1, u2 f2, u3 f3 . . , and from this I will form a new mapping g of U to itself that differs from allfi, i. e., it is not an element of F because, in our assignment, all elements ofF had to be used up; but since F includes all possible mappings, we have,thus, derived the contradiction. We again apply the principle of Cantor'sdiagonalization method. In the mapping f1, let the element u1 correspond tothe u f1(u1) = u Hilbert's Paradox if ug(1) is an element different from u , then we construct the new mapping g which assigns u1 to it: g(u1) = ug(1) 6= u We proceed further according to this principle; by the way, the designationof elements of U and F by number indices is not essential, and it should byno means insinuate that these sets are countable which is not at all the case.
If u2 is some element of U, a mapping [Belegung] f2 209 belongs to it in themapping [Abbildung] of f to u; we look for the element f2(u2) = u it [the mapping f2] assigns to u2, choose ug(2) 6= u and define a mapping g which assigns it to u2: g(u2) = ug(2) 6= uf(2) The mapping g which we obtain in this way has the scheme g(u1) = ug(1) 6= u , g(u , g(u 2) = ug(2) 6= uf(2) 3) = ug(3) 6= uf(3) It is distinct from any mapping fk of F in at least one assignment; namely,if uk is the element (or one of these) corresponding to fk in the mapping[Abbildung] of F to U, then it follows from the definition of g that: fk(uk) = u f (k) k) = ug(k) 6= uf(k) By this, we indeed have the contradiction that the well-defined mapping gcannot be a member of the set of all mappings. We could also formulate thiscontradiction so that, according to the last consideration, the set UU is alwaysbigger [note in Hilbert's hand: "of greater cardinality"] than U but, accordingto the former, is an element of U. This contradiction is not at all yet solved;anyway, one can see that it must depend upon the fact that the operations ofuniting arbitrary sets or objects into 210 new sets or totalities, respectively,is, nevertheless, not allowed, although it is always used in traditional logic,and although we have carefully applied it only to natural numbers and setsarising from them, i. e., to purely mathematical objects.
Appendix II: Hilbert's Paradox (German Original) [Marginalie: 18. Vorles. 10. VII.] [ . . ] 204 Ich komme nun noch zu 2 Beispie-len f¨ur Widerspr¨uche, die viel ¨uberzeugender sind, der erste, der rein ma-thematischer Natur ist, scheint mir besonders bedeutsam; als ich ihn fand,glaubte ich zuerst, daß er der Mengentheorie un¨uberwindliche Schwierig-keiten in den Weg legte, an denen sie scheitern m¨ußte; ich glaube jedoch Volker Peckhaus and Reinhard Kahle jetzt sicher, daß wie stets bisher in der Wissenschaft, nach der Revision derGrundlagen alles Wesentliche erhalten bleiben wird. Ich habe diesen Wi-derspruch nicht publiciert; er ist aber den Mengentheoretikern, insbeson-dere G. Cantor, bekannt. Wir sehen die endlichen Mengen, durch endlichviele Zahlen repr¨asentiert, jedenfalls als erlaubte Operationsbasis an, undebenso die abz¨ahlbar unendliche Menge 1, 2, 3 . . aller nat¨urlichen Zahlen.
Ferner erscheint es erlaubt, 2 solche Mengen (1, 2, 3 . .) und (a1, a2, a3 . .)zu einer neuen Begriffseinheit (1, 2, 3 . . , a1, a2, a3 . .), einer neuen Menge,zusammenzufassen, die jedes Element der beiden Mengen enth¨alt. Ebensok¨onnen wir auch mehrere Mengen und sogar unendlich viele zu einer Vereini-gungsmenge zusammenfassen. Wir bezeichnen das als Additionsprincip, undschreiben 205 die so aus M1, M2 . . hervorgehende Menge kurz M1 + M2 + · · · Diese Zusammenfassungen sind Processe, die man in der Logik stets ohnejedes Bedenken in noch weit komplicierteren F¨allen anwendet; es scheint also,daß man auch hier ohne weiteres davon Gebrauch machen k¨onnte. Außerdiesem Additionsprincip verwenden wir noch eine weitere Betrachtung zurBildung neuer Mengen. Es sei y = f (x) eine zahlentheoretische Funktion,die zu jedem ganzzahligen Wert x ein ganzzahliges y zuordnet; in sofort zuverstehendem Sinne k¨onnen wir eine solche Funktion auch als eine Belegungder Zahlenreihe mit sich selbst bezeichnen, indem wir etwa an ein Schemadenken: x = 1, 2, 3, 4 . . y = 2, 3, 6, 9 . . Das System aller solcher zahlentheoretischen Funktionen f (x) oder allerm¨oglichen Belegungen der Zahlenreihe mit Elementen ihrer selbst bildeteine neue Menge, die "durch Selbstbelegung aus der Zahlenreihe M entste-hende," wir schreiben sie MM. Als aus den 206 Zusammenfassungsgesetzender ¨ublichen Logik folgendes und nach ihr g¨anzlich unbedenkliches Principk¨onnen wir nun das ansehen, daß aus wohldefinierten Mengen durch Selbstbe-legung immer wieder wohldefinierte Mengen entstehen. (Belegungsprincip).
Durch dies Princip entsteht aus dem Continuum aller reellen Zahlen bei-spielsweise die Menge aller reellen Funktionen. Allein mit diesen beiden nachaller bisherigen Mathematik und Logik unbedenklichen Principen wollen wirarbeiten.
Wir gehen von allen endlichen Mengen von Zahlen und der aus ihnen be- reits durch Addition entstehenden unendlichen Reihe 1, 2, 3 . . der nat¨urlichenZahlen aus, und fassen alle Mengen auf, die aus ihnen durch die beiden be-liebig oft anzuwendenden Processe der Addition und Selbstbelegung entste-hen; diese Mengen bilden wieder eine wohldefinierte Gesammtheit, nach dem Hilbert's Paradox Additionsprincip vereinige ich sie alle zu einer Summenmenge U, die wohl-definiert ist. Bilde ich nun die Menge F = UU der Selbstbelegungen von U,so entsteht diese auch aus der urspr¨unglichen Zahlenreihe lediglich durch diebeiden Processe der Addition und 207 Selbstbelegung; sie geh¨ort also auchzu den Mengen, aus deren Addition erst U entstand, und muß daher ein Teilvon U sein: F ist in U enthalten.
[Marginalie: 19. Vorles. 11. VII.] Ich zeige nun, dass dies zu einem Widers-pruch f¨uhrt. Es seien u1, u2, u3 . . die Elemente von U; jedes Element fvon F = UU repr¨asentiert dann eine Belegung von U mit sich selbst, d. h.
eine Funktion gewissermaßen, die jedem Elemente ui von U ein anderes uf(i) zuordnet, wobei die uf(i) keineswegs untereinander verschieden zu sein brau- chen; wir stellen dies Element f am besten also durch ein Schema dar: f (u1) = uf(1), f(u2) = uf(2), f(u3) = uf(3) . . Unser Resultat (1), daß F in U enthalten ist, kann man nun n¨aher so ausspre-chen: Man kann jedem Elemente ui von U eines fi von F eindeutig zuordnen,so daß alle fi dabei verwendet werden, eventuell sogar mehrfach, aber immerjedem ui nur genau ein fi entspricht; das heißt ja offenbar nichts anderes,als daß es "nicht mehr" Elemente fi gibt, als ui. Eine solche Zuordnungbetrachten wir nun: 208 u1 f1, u2 f2, u3 f3 . . , und daraus werde ich eine neue Belegung g von U mit sich selbst bilden, dievon allen fi verschieden ist, also gar nicht in F enthalten w¨are, da ja beiunserer Zuordnung alle Elemente von F zur Verwendung kommen sollten;da aber F alle m¨oglichen Belegungen enth¨alt, so haben wir hier den Wider-spruch. Wir wenden wieder das Princip des Cantorschen Diagonalverfahrensan. In der Belegung f1 entspreche dem Element u1 dasjenige u f1(u1) = u ist ug(1) ein von u verschiedenes Element, so ordnen wir in der neu zu konstruierenden Belegung g dies dem u1 zu: g(u1) = ug(1) 6= u Nach diesem Princip verfahren wir weiter; die Bezeichnung der Elemente vonU und F durch Zahlenindices ist ¨ubrigens unwesentlich und soll nicht etwaandeuten, daß diese Mengen abz¨ahlbar sind, was keineswegs der Fall ist. Istu2 irgend ein Element von U, so geh¨ort ihm in der Abbildung von f auf u Volker Peckhaus and Reinhard Kahle eine Belegung f2 209 zu; wir suchen das Element f2(u2) = u u2 zuordnet, w¨ahlen ug(2) 6= u und definieren eine Belegung g, die dies dem u2 zuordnet: g(u2) = ug(2) 6= uf(2) Die Belegung g, die wir so erhalten, hat das Schema g(u1) = ug(1) 6= u , g(u , g(u 2) = ug(2) 6= uf(2) 3) = ug(3) 6= uf(3) Sie unterscheidet sich von jeder Belegung fk aus F in mindestens einer Zuord-nung; ist n¨amlich uk das in der Abbildung von F auf U dem fk entsprechendeElement (oder eines derselben), so ist nach der Definition von g: fk(uk) = u f (k) k) = ug(k) 6= uf(k) Wir haben damit in der Tat den Widerspruch, daß die wohldefinierte Be-legung g nicht in der Menge aller Belegungen enthalten sein k¨onnte. Wirk¨onnten ihn auch dahin formulieren, daß gem¨aß der letzten Betrachtung dieMenge UU stets gr¨oßer [von Hilberts Hand: von gr¨osserer M¨achtigkeit] als Uist, nach der ersten aber in U enthalten. Dieser Widerspruch ist noch kei-neswegs gekl¨art; es ist wohl zu sehen, daß er jedenfalls darauf beruhen muß,daß die Operationen des Zusammenfassens irgend welcher Mengen, Dinge zu 210 neuen Mengen, Allheiten doch unerlaubt ist, obwohl es die traditionelleLogik doch stets gebraucht, und wir es in vorsichtiger Weise stets nur aufganze Zahlen und daraus entstehende Mengen, also auf rein mathematischesanwandten.
The first author is responsible for the historical part, the second author forthe systematic part. During the preparation of the final version the secondauthor visited the Mittag-Leffler Institute (Djursholm, Sweden). We wouldlike to thank several commentators for helpful hints and corrections, amongthem Roy Dyckhoff (St. Andrews), Heinz-Dieter Ebbinghaus (Freiburg i. Br.),Ivor Grattan-Guinness (Bengeo, UK), Akihiro Kanamori (Boston), Jos´e F.
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