In the Dictionary of Scientific Biography (Gillespie 1970–1980 p 58), further circumstances of Bernstein’s finding of his CBT proof are described: The date of the finding is given as “1895 or 1896 while [Bernstein was] a student in the Gymnasium” and it is said that:
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In the Dictionary of Scientific Biography (Gillespie 1970–1980 p 58), the circumstances of Bernstein’s finding of his CBT proof are described: The date of the finding is given as “1895 or 1896 while [Bernstein was] a student in the Gymnasium” and it is said that:
Cantor, who had been working on the equivalence problem, had left for a holiday and Bernstein had volunteered to correct proofs of his book on transcendental numbers. In that interval, the idea for a solution came to Bernstein one morning while shaving. Cantor then worked with this approach for several years before formulating it to his satisfaction.
The dictionary seems erroneous in this passage. First, there is no book of Cantor on transcendental numbers. Second, according to Cantor himself (letter to Dedekind of August 30, 1899), Bernstein first presented his proof of CBT, in a Halle seminar in Easter 1897 (which was in April, cf. Purkert-Ilgaud 1987, p 139). Third, Bernstein was born in February 1878 and so he was 19 years old at the time of his discovery, so most likely already a student at the university not in the gymnasium. Fourth, the suggestion that Cantor was working at the time on CBT conflicts the strong evidence that Cantor worked on CBT in 1882. Fifth, it is not conceivable that Cantor continued to work on the theorem for several years following Bernstein’s approach for the theorem is simple and the proof simple too.
In Bernstein’s internet biographyFootnote 1 the story is slightly changed and the book is Cantor’s 1895/7 Beiträge, though Bernstein is still depicted as a gymnasium student. In Ebbinghaus 2007 the “book” is the more reasonable 1897 Beiträge and the time is the winter semester of 1896/7,Footnote 2 which fits the letter to Dedekind. Then we can understand the reference to transcendental numbers to mean the ordinal numbers, which are the subject of Cantor’s 1897Beiträge.
It is, however, possible that Bernstein found the proof when he was reading the proofs of 1895 Beiträge, in early 1895, when he was still in the gymnasium. Then the cardinal numbers are the transcendental numbers mentioned in the dictionary. Under this explanation the presentation of the proof in the Halle seminar was 2 years after the discovery of the proof.
The anecdote of the Dictionary of Scientific Biography about the shaving is, nevertheless, intriguing because we can imagine Bernstein in front of a mirror and by placing another mirror in front of the first, experiments the conditions of CBT. Then must have occurred the inspiring moment of gestalt switch, when he concentrated on the frames of the images of the mirrors instead of concentrating on the images of the whole mirrors. We will get back to this thought experiment.
Strangely, Bernstein never published his original proof.Footnote 3 Instead, it was published by Borel, in an appendix on the notion of power to his book on the theory of functions (1898 pp 102–107). There Borel recounted (p 103 footnote 3) that the proof was communicated to him by Cantor, whom he approached during the first international congress of mathematicians held in Zurich in August 1897.
In the appendix, Borel thanked Cantor for the permission to publish the proof. Two things appear peculiar in Borel’s story: That Cantor made a decision on Bernstein’s behalf and that he ignored the benefit to Bernstein that could have resulted if Bernstein himself would have published the result. With regard to the first point, the answer could be that Cantor had a paternalistic attitude towards Bernstein, who was the son of his friend and colleague (Gillespie 1970–1980 p 58). With regard to the second point, perhaps Cantor did not regard the proof as very important because he himself already had a proof of CBT in 1882 (see Chap. 1). In addition, Cantor was probably aware, when he met with Borel, of Schröder’s attempted proof of the theorem. As Cantor told Dedekind in the August 30, 1899, letter, he considered Bernstein’s proof to be similar to that of Schröder. So, Cantor perhaps thought that Bernstein’s proof was a nice finding by a young man but no great discovery in set theory research. Still, Cantor had high regard for the proof, perhaps because the proof was not leaning on Cantor’s scale of number-classes but still used his idea of abstraction (see below). In the dictionary (p 58) it is said that Bernstein was influenced to forsake his studies in fine arts at the university of Pisa and return to mathematics, by two mathematics professors in Pisa who heard his praise from Cantor at a mathematical congress (perhaps the very same congress when Borel met Cantor).
As it happened, Bernstein was not negatively affected by Borel’s publication. He still got his name attached to an important theorem though he never publicly produced his original proof for it. What is nevertheless strange, is that even in his doctorate dissertation of 1901, published in his 1905 paper, he only mentioned his proof in a by-the-way manner, referencing its presentation to Borel’s book (Bernstein 1905 p 117 footnote). His reason could have been that by then, with the publication of the proofs of Schröder (1898) and Zermelo (1901) which he mentions in his paper, the importance of his own proof was diluted. We believe, however, that such an attitude is unnatural and that Bernstein had a deeper reason for not presenting his proof: it seems to us that Bernstein’s original proof was cruder than the one presented by Borel. So, Bernstein stood nothing to gain by repeating his original argument. We will first review the proof of CBT given by Borel in 1898. Then we will interpolate the original argument of Bernstein, never published. We will then compare both proofs to the earlier proofs of Cantor, Dedekind and Schröder.
1 Borel’s Proof
We quote Borel’s proof from his 1898 book (p 104ff) with our comments in the footnotes.
There exists a proper subsetFootnote 4 A1, of A, that has same powerFootnote 5 as B and there exists a proper subset B1, of B, that has same power as A. It is required to prove that A has same power as B.Footnote 6
As B and A1 have same power, there exists a projectionFootnote 7 from B on A1,Footnote 8 that is to say a law after which the elements of B and of A1 correspond in a unique and reciprocal fashion.Footnote 9 There exists even an infinity of such projections;Footnote 10 but we choose from them one well determined.Footnote 11 It is clear that, by such a projection, to every proper subset of B correspond a proper subset of A1; let A2 be the proper subset of A1 that thus corresponds to B1; A2 has, by the very definition of the power,Footnote 12 same power as B1 and, as a result, same power as A. Moreover, A2 is a proper subset of A1, which is itself a proper subset of A. It all comes down to saying that, A2 having same power as A, A1 also has same power as A (because B has same power as A1).Footnote 13
By the hypothesis A2 has same power as A; we choose a determined projection from A on A2; A1 which is a proper subset of A becomes a proper subset A3 of A2, and A2 becomes a proper subset A4 of A3. This is what we indicate on the schematic figure below, where the projection from A on A2 is an homothetic transformation.Footnote 14 Therefore A3 has same power as A1 and A4 same power as A2 and as A. If we project A on A4, A1 and A2 will project upon A5 and A6; A5 will be a proper subset of A4 and will have same power as A1 and A3; A6 will be a proper subset of A5 and will have same power as A,Footnote 15 A2, A4. Thus continuing, we form a sequence of sets A, A1, A2, A3, A4, … such that each of them is a proper subset of its preceding set and in addition such that all the sets of even index have same power as A and all the sets of odd indices have same power as A1 (Fig. 11.1).
This sequence can be continued indefinitely, that is to say that the set An is defined, whatever the integer n; it contains, incidentally, elements, because it has same power as A or A1, accordingly if n is even or odd.
Consider nowFootnote 16 the set D formed of the elements common to all the sets A, A1, A2, …, An, …. This set D can contain, incidentally, no element. It is clear that this set D can be obtained by successively removing from A the sets: A-A1, A1-A2, A2-A3, …, An-An+1, ….Footnote 17 We can therefore write
A = D + (A-A1) + (A1-A2) + (A2-A3) + (A3-A4) + (A4-A5)… and each of the symbols between parenthesis designates a determined set, since An+1 is a proper subset of An. We can similarly write
A1 = D + (A1-A2) + (A2-A3) + (A3-A4)….
It is now easy to demonstrate that the sets A and A1 have same power; it is enough to remark that we can regard them as formed of a denumerable infinitude of sets having pairwise same power. It results, in effect, from the nature of the projectionFootnote 18 by which we obtained the sets A3, A4,… that A-A1 has same power as A2-A3, as A4-A5, as A6-A7, …. Similarly, the sets A1-A2, A3-A4, A5-A6, … all have same power. It suffices, hence, to write the expression of A1 in the form
A1 = D + (A2-A3) + (A1-A2) + (A4-A5) + (A3-A4) + (A5-A7) + …, to recognize that each of the terms of this series has same power as the term of the same place in the series which defines A.Footnote 19 The theorem is thus demonstrated.Footnote 20
2 Bernstein’s Original Proof
Borel’s footnote to his definition of the set D suggests that the proof given by him is not exactly the proof given by Bernstein (as presented to Borel by Cantor). The original proof, according to Borel, contained reference to transfinite numbers. In Cantor’s set theory, transfinite numbers, cardinal numbers for our context here, are introduced by abstraction. So, Bernstein’s original argument probably was that when passing from the frames to their cardinal numbers, A and A1 are partitioned to partitions with the same cardinal numbers. Thus, they too have the same cardinal number. Whatever differences there are in the frames they disappear after abstraction. This metaphor Bernstein could have picked from Cantor who used it to prove the commutativity and associativity of the addition operation of powers (cardinal numbers) in his 1895 Beiträge §3.Footnote 21 We seem to have here a case of proof-processing. Cantor may have had his high regard for Bernstein’s proof because of this use of abstraction as an operator, just as envisaged by Cantor.
Bernstein’s argument comes out well from the figure that Borel gave. If we float the image of A1 and its subsets above that of A, it is easy to gain with one coup the metaphor that the difference between the two sets in their sequence of even frames of A is irrelevant to their powers. Borel, however, uses the figure in an inessential way, when he points out the nesting character of the sequence of sets and that this sequence is in fact composed of two interchanging sequences of equivalent sets. These points are easy enough to understand from Borel’s prose and Borel does not refer to the figure to switch the gestalt from the sequence of nesting sets to the sequence of frames. Thus, it seems that the figure was perhaps suggested to Borel by Cantor and it may have originated with Bernstein (being an art student).Footnote 22 Borel may have brought the figure to keep a trace of the proof presented to him by Cantor, from which he departed when he partitioned the sets into frames and residue and proved the equivalence of the sets as a result of the equivalence of the partitions.
This observation brings us back to the story of the discovery of Bernstein’s proof while shaving. Let us note the following thought experiment suggested by a drawing provided in Abian 1965 p 254. Take a round mirror and a square mirror and place them facing each other. In the square mirror the image of the round mirror will be shown and in the round mirror the image of the square mirror (Fig. 11.2).
Moreover, in the image of the round mirror in the square mirror, the image of the image of the square mirror that is in the round mirror will be shown and likewise in the image of the square mirror in the round mirror the image of the image of the round mirror that is in the square mirror. This sequence of images continues for (countable) ever. Now this is nice but disturbing because even though the two mirrors seem to correlate they do not perfectly match. Or do they? To realize that they do, and establish the proof of CBT, one has only to shift focus and observe that in both mirrors we have two infinite sequences of frames: one sequence of frames is square on the outside and round on the inside and the other round on the outside and square on the inside. Separating the second type of frames from both A and A1, it is immediate to realize that the remaining part and the separated parts have the same power after abstraction. Of course, in the general case, the nesting sets can be diffused and are not limited to the plane, but the clue derived from the thought experiment holds. Our thought experiment and the shaving story lead us to believe that Bernstein’s proof was for the two-set formulation, with no shift to the single-set formulation as in Borel’s rendering of it.
Incidentally, the mirrors drawings leads us to the following ethical version of CBT: In the pupil of my eye there is an image of the whole of you; in the pupil of your eye there is an image of the whole of me; by CBT we are equivalent!
3 Comparison with Earlier Proofs
With Bernstein, a gestalt switch occurred, from the gestalt of the nesting sets identified by Schröder, to the gestalt of corresponding frames. Abstraction, in Cantor’s sense, is the metaphor of Bernstein’s proof,Footnote 23 and it differs from Schröder’s metaphor of passage to the residue. Bernstein’s descriptors differ also from Cantor’s descriptors: the gestalt of number-classes and the metaphor of enumerating the subset by the whole set, which underlie his proof. Note that Cantor perhaps interpreted Bernstein’s proof on his own terms whereby abstraction can be performed only on consistent sets. Thus, for Cantor, Bernstein’s proof had no wider scope than his own Grundlagen proof.
With regard to Dedekind, there is some common ground between Bernstein’s gestalt of the frames and Dedekind’s chain gestalt but the similarity is not spelled out. If we grant Bernstein with separating one type of frames from the sets then Bernstein had obtained Dedekind’s partitioning metaphor; however, Bernstein did not use Dedekind’s pushdown the chain metaphor.
Bernstein’s frames gestalt was adopted by Borel, and many others who later provided proofs of CBT, but the abstraction metaphor, that probably excited Cantor, did not. The idea of defining cardinal numbers by abstraction was rejected by the generation that followed Cantor. This is then an especially nice and important feature of Borel’s proof, that it stays in the language of sets and mappings used to formulate the theorem. No appeal is made to extraneous notions such as enumeration-by, or abstraction, invoked in the proofs of Cantor and Bernstein. In this Borel is similar to Dedekind. For this reason the proofs provided by Borel and others were not limited to consistent sets.
Borel’s metaphor lies in the rearrangement of the frames in equivalent pairs. It has nothing to do with Cantor’s enumeration or Dedekind’s pushdown, but it is clearly a translation of the abstraction metaphor to the language of sets and mappings. Borel begins with an observation similar to Schröder’s Scheere but switches to the frames gestalt and then to the single-set formulation. Since Borel does not mention Schröder it seems that Borel had obtained the Scheere construction of the nesting sets independently of Schröder, as well as his appreciation of the place of the residue in the decomposition of the sets involved. Perhaps because of their similar point of departure in the Scheere construction, their similar appreciation of the residue and their use of drawings, Schoenflies (1900 p 16 footnote 2) remarked that Bernstein’s proof (he meant Borel’s proof of course) is “essentially identical” with the proof of Schröder.
Let us recall again that in his letter to Dedekind of August 30, 1899, Cantor too said that Bernstein’s proof is similar to the proof of Schröder and to that of Dedekind. We leave it to the reader to decide how much inaccurate this judgment is in terms of gestalt and metaphor. Cantor’s view teaches us that mathematicians are quick to translate from one set of descriptors to another. Yet we maintain that not every set of alternative descriptors can provoke in a mathematician a solution to a problem he is occupied with. More study is necessary to establish this point of view.
Another metaphor of Borel is the use of complete induction. Schröder used complete induction too in defining his Scheere. Dedekind, however, consciously avoided complete induction, using instead an impredicative definition to define the chain of a set as the minimal chain that contains the first frame. Unlike Dedekind, Borel remains silent about the obvious fact that the sequence of odd frames and D (which compose Dedekind’s complement to the chain pushed down) belong to the two sets and that therefore no argument on their equivalence is necessary.
An important peculiarity of Borel’s proof is that he chooses at each step a new projection; thus he proves that the sets generated at each step are equivalent and then from the many mappings that can establish this equivalence he chooses one. In this way Borel unintentionally invoked the axiom of choice in his proof. Dedekind uses only the projection from A to A2, which even if not assumed as given, it could be chosen from among the mappings that establish this equivalence, using only a single choice. (See in this regard Harward’s criticism of Jourdain in Sect. 17.5.) Borel may have been aware that he is making infinitely many arbitrary choices and he may have been aware of Peano’s 1890 paper where infinitely many choices were rejected. Of course, the axiom of choice was not yet stated in 1898 so Borel was not aware that he is employing it. After the axiom of choice was explicitly stated by Zermelo in 1904, Borel became one of its main antagonists.
In 1906 Bernstein did publish a proof of CBT, in the context of the debate between Poincarè and the logicists, which resembled a 1906 proof of Peano (see Sect. 21.3).
4.
We use ‘subset’ for Borel’s ‘partie’.
5.
Borel uses ‘same power’ (méme puissance) for Cantor’s ‘equal power’ (gleiche Mächtigkeit). Borel avoids using ‘the’ before ‘same power’ surely to avoid any interpretation that a power is an entity. Borel rejected this possibility, despite the respect which he expresses for Cantor’s results, because he thought that Cantor had not resolved the question of comparability. Borel, however, thought that the notions of one set having a smaller power (Borel 1898 p 104) than another, or of two sets having the same power, notions introduced in Cantor’s 1878Beitrag (see Sect. 3.1), are clear. Hence, he was ready to discuss CBT. Still it is strange that he is using the term “same power” at all, for Cantor introduced the terminology “same power” in his 1878 Beitrag together with its synonym “equivalence” and Cantor himself, after 1878 Beitrag, preferred the latter (1887Mitteilungen p 413, 1895Beiträge §2).
6.
Cantor and Schröder also used for the subsets the same letter used for the whole-set, but Bernstein 1905 (p 121) used for the subsets the letter used for the sets to which they are equivalent.
7.
This is Borel’s term for ‘1–1 mapping’.
8.
Borel is using here Cantor’s definition of ‘same power’ (or ‘equal power’) from the first paragraph of Cantor’s 1878 Beitrag.
9.
Like his predecessors, Cantor (1932 p 283), Dedekind (1963 p 50) and Schröder (1898 p 337), and some later mathematicians too (J. Kőnig 1906), Borel characterizes ‘projection’ as a law after which the elements of the two sets correspond. For some notes on the historical development of the notion of mapping from the analytic notion of function, see Lakatos 1976 pp 151–152.
10.
This observation is also taken from the first paragraph of Cantor 1878Beitrag.
11.
This choice, which Borel repeats at every step of his construction, involves the axiom of choice, quite unnecessarily, in Borel’s proof.
12.
In the original “d’après la definition meme de la puissance”; it seems that the definite article was inserted here by mistake.
13.
At this point Borel shifts the proof from proof of the two-set formulation of CBT to proof of its single-set formulation.
14.
Namely, ‘shrinkage’. Obviously, for the general case, the geometric aspects of the figure ought to be disregarded. It is interesting that in many of the early proofs of CBT a figure is used.
15.
There is here a typo in the original where it is written ‘A1’ instead of ‘A’.
16.
Here Borel inserted the following footnote (p 105): “By introducing the set D, I modify lightly the demonstration that Mr. G. Cantor had communicated me (see note for page 104), in order to avoid the introduction of transfinite numbers. This modification is, for that matter, without importance”. We explained above Borel’s reason for avoiding the cardinal numbers. Medvedev (1966 p 235) thought that Borel made here the first attempt to remove CBT from the context of the Comparability Theorem for cardinal numbers. Potter (2004 pp 165–6) says that Borel produced a simplified version of Bernstein’s proof; perhaps because of this footnote (why ‘simplified’ rather than ‘modified’ we do not know). Incidentally, Potter’s reference to the note in Borel 1898 pp 103–4 is mistakingly to p 105. We will get back to Borel’s footnote in the next section.
17.
Borel is using the difference operation between sets and + for the union operation. Cantor used (A, B) for the union of A and B (1895 Beiträge §3) and used A-B rarely. The notation A + B is attributed by Dedekind to Schröder (see Sect. 9.2.3).
18.
The original reads “du mode de projection”. This strange expression is explained by a footnote below.
19.
A similar conclusion from the equivalence of the partitioning to the equivalence of the sets appears in Cantor’s 1878 Beitrag (p 129), which is another evidence that Borel was influenced by this paper. Poincaré would later point out (see Sect. 26.7) that this step makes a special appeal to intuition.
20.
Here (p 106) Borel puts the following footnote: “It is important to point out a grave error that could be suggested by the preceding demonstration. We have seen that A-A1 and A2-A3 have same power, because of the projection by which we have obtained A3 [which carries the partitioning of A by A1 and A-A1 to the partitioning of A2 by A3 and A2-A3]; but this is not at all a consequence of the fact that A2 has same power as A and A3 same power as A1. It is easy to be assured that, two sets A and B having same power, and their proper subsets A1 and B1 having also the same power, nothing can hence be concluded for A-A1 and B-B1. It is enough to take, for example, for A the set of points included between 0 and 2, for B the set of points included between 0 and 1, and for A1, as well as for B1, the set of points with incommensurable abscissas [the irrationals] included between 0 and 1. The set A-A1 has now the power of the continuum, while the set B-B1, is denumerable. Likewise, it could have been supposed, for that matter, that both A1 and B1 are identical with B.”
21.
This linkage of Bernstein’s proof to the 1895 Beiträge, supports the possibility that Bernstein found his proof in 1895 when reading the proofs of 1895 Beiträge.
22.
Bernstein used figures also in his 1905 paper, when he discussed situations similar to the proof of CBT (see Sect. 14.2), while Borel used only one other figure in the whole of his 1898 book.
23.
With regard to the metaphor behind this gestalt switch, it seems to be “passage to the sequence of difference”, stemming from the theory of sequences and series of numbers.
References
Bernstein F. Untersuchungen aus der Mengenlehre. Mathematische Annalen. 1905;61:117–55.
Cantor G. Gesammelte Abhandlungen Mathematischen und philosophischen Inhalts, edited by Zermelo E. Springer, Berlin 1932. http://infini.philosophons.com/.
Dedekind R. Essays on the theory of numbers, English translation of Dedekind 1888, by Berman WW, New York: Dover; 1963. Also in Ewald 1996 vol 2 p 787–833.
Medvedev FA. 1966. Ранняя история теоремы эквивалентности (Early history of the equivalence theorem), Ист.-мат. исслед. (Research in the history of mathematics) 1966;17:229–46.
Schröder E. Über Zwei Defitionen der Endlichkeit und G. Cantorsche Sätze, Nova Acta. Abhandlungen der Kaiserlichen Leopold-Carolinschen deutchen Akademie der Naturfoscher. 1898;71:301–62.
Zermelo E. Über die Addition transfiniter Cardinalzahlen, Nachrichten von der Königlich Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse aus dem Jahre 1901;34–8.
