Abstract
Bernstein’s Division Theorem (BDT) states that if an infinite cardinal number is divisible by a finite number then the quotient is unique, namely, if \( {\text{km}} = {\text{kn}} \) then \( {\text{m = n}} \), where k is a natural number, m, n cardinal numbers. The theorem is included (p 122) in Bernstein’s doctorate dissertation of 1901 (published in 1905). It was reproduced in Hobson 1907 pp 159–162. BDT is sometimes called Bernstein’s theorem but since there are other results that bear Bernstein’s name we use ‘Bernstein’s Division Theorem’.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It should be observed that despite the use of the notion of cardinal number here and below, BDT (and the inequality-BDT) are theorems about sets and mappings, just as CBT. Thus the proofs below do not seem to require that the sets involved be consistent.
- 2.
In this paragraph we give an example of proof-processing performed with no intention for its application at sight. The quest is for a certain symmetry (gestalt) obtained by reversal of roles (metaphor). Here the gestalt and metaphor are at a higher level above the context than in our other examples. Cf. 9.3 for a similar remark concerning Theorem 68 and CBT.
- 3.
But Bernstein wanted, no doubt, to avoid the well-ordering doctrine.
- 4.
Carrying out the here described scheme without introducing some notation seems almost impossible, leading to the well-known conclusion that even for a sketch of mathematical reasoning, some notation is necessary.
- 5.
Theorem 1 (of the first chapter of the paper, p 121) is CBT which is given in its two-set formulation taken from 1895 Beiträge.
- 6.
Bernstein does not use the notion ‘power’, except once with regard to ℵ0.
- 7.
Our comments to the proof, rather numerous due mostly to Bernstein’s rather loose style, will be given in footnotes. Cantor must have been aware of this shortcoming of Bernstein when he remarked, after hearing J. Kőnig’s 1904 Heidelberg lecture, which relied on a theorem from Bernstein’s dissertation, that he suspects the king less than the king’s ministers (Dauben 1979 p 249).
- 8.
Bernstein must be talking here about 1–1 mappings from S onto S. Lemma 1 is not used in the proof.
- 9.
Apparently the ′, here and below, belongs to μ not to the χ.
- 10.
Clearly Bernstein is talking about mappings from S onto S. Bernstein does not define the composition of mapping which he denotes by the same sign he sometimes uses for the multiplication of cardinal numbers. Condition (5) is rather: ν≠μ, ν, μ = 1, 2, 3, ….
- 11.
Identifying 1 with χ0, as Bernstein does below, makes this premise redundant.
- 12.
There is a typo in the original and ‘≠’ is printed instead of ‘ = ’.
- 13.
The sentence should have ended: ‘a contradiction to the premise’. Note that contrary to current use (and Zermelo’s) Bernstein writes \( {\chi_{\nu }} \cdot {\chi_{\mu }}^\prime({\text{s}}^\prime) \) for \( {\chi_{\mu }}^\prime\left( {{\chi_{\nu }}({\text{s}}^\prime)} \right) \). We stay here with Bernstein’s convention.
- 14.
Bernstein is making use here of Cantor’s convenient convention of denoting by {t} (only they both use () instead of our {}) a set whose members are denoted by t with various subscripts or superscripts. Thus t′ also belongs to T.
- 15.
Bernstein tacitly adopts the convention that χ(T) is the set of all \( \chi ({{t}}),{{ t}} \in {\text{T}} \).
- 16.
In this equation, S, T are the powers of the corresponding sets, not the sets themselves. Interpreting them as sets trivializes (11). Lemma 5 requires rather the conclusion of lemma 4 as premise not its premises, namely: T, χ1(T), χ2(T), … form a disjoint system, then S = S + T.
- 17.
Bernstein’s use of (11) for the proof of (11) seems to be a mistake and it is not necessary for under the premises of lemma 4, or assuming the conclusion of lemma 4, there exists a disjoint system of equivalents of T in S.
- 18.
R is the residue obtained from S after removing the disjoint system. Note that the equation S = T·ℵ0 + R is an equation of powers and Bernstein uses the sign · for the multiplication of powers operation which he never introduced.
- 19.
Bernstein drops here and henceforth the use of ‘·’ to denote the multiplication operation between sets, powers or mappings, and instead uses juxtaposition.
- 20.
Bernstein is using here the equality ℵ0 + 1 = ℵ0. Bernstein applies here the reemergence argument we encountered in Zermelo (1901) (see Sect. 13.2). Bernstein uses the reemergence argument also in other parts of his paper, e.g., p 129f. While Bernstein deduced from S = T·ℵ0 + R that \( {\text{S}} = {\text{S}} + {\text{T}} \), Zermelo (1901 theorem II) obtained S = T·ℵ0 + R from \( {\text{S}} = {\text{S}} + {\text{T}} \); thus we can say that Bernstein and Zermelo walked the same road in opposite directions.
- 21.
Bernstein takes \( {\text{2M}} = {\text{2N}} \) to imply that there is a set S of power 2M (2N) which has two partitionings: the partitions of the first partitioning are x 1, x 2, and those of the second are x 3, x 4. See the previous section.
- 22.
In b), c), ‘=’ should be read as equivalence, ‘~’. While it is possible to read the x i here as powers, later they surely mean sets.
- 23.
The introduction of ϕ a is pointless; it can be taken as the identity. Indeed, ϕ a is not used in the proof.
- 24.
Since x 1 is disjoint from x 2, given ϕ b from x 1 onto x 2, we can extend it over x 2 by defining ϕ b (s) for every \( {{s}} \in {{{x}}_{{2}}} \) to be equal to \( {\varphi_{\rm{b}}}^{{{ - 1}}}({\text{s}}) \). A similar argument holds for x 3, x 4 and ϕ c . Thus property (13) for (the extended) ϕ b and ϕ c is established.
- 25.
The given proof can be conducted for each of the quarters, in the role of x 13, and its diagonally opposing quarter, instead of x 24.
- 26.
Bernstein’s way of expression in this passage is confusing: he suddenly uses the equivalence sign ‘ ~ ’ which he avoided before; then he designates by (14*) something that is not related to (14); he continues as if theorem 2 provides a conclusion with regard to x 1 and x 2 only and an analogous case needs to be considered for x 3 and x 4. Obviously interchanging T1 and T2 affects all four halves. Note that once ϕ b and ϕ c are regarded as transformations of S they are not affected by an interchange. Bernstein is correct, of course, in maintaining that if theorem 2 is proved for the transformed halves it is also proved for the original halves, and this is the main point for it is through interchange transformation that Bernstein intends to change the original partitionings of S if they do not come under the conditions of his proof-plan.
- 27.
Here the equality sign should be replaced by the equivalence sign, which Bernstein briefly used in the preceding equations.
- 28.
Bernstein did not define χ1 and it caused some blunder below.
- 29.
From the sequence of χ ν ; the subsequences of mappings with odd or even indices are not closed under multiplication.
- 30.
We don’t know why Bernstein thought it necessary to stress the finitude of the index when no other possibility was discussed or even seems possible in the context.
- 31.
- 32.
In each of x 23, x 14 a disjoint system is formed. The attribute ‘simply infinite’ (it must mean ‘denumerable’) is redundant.
- 33.
The equality signs should be read as equivalence’ ~ ’ or the x ij as powers. Lemma 5, however, cannot be applied as given to obtain the desired results \( {{{x}}_{{{13}}}} + {{{x}}_{{{14}}}}\sim {{{x}}_{{{14}}}} \), \( {{{x}}_{{{13}}}} + {{{x}}_{{{23}}}}\sim {{{x}}_{{{23}}}} \). Instead the following lemma 5* is necessary: if S′, \( {\text{T}} \subseteq {\text{S}} \) and S′ has a disjoint system of subsets all equivalent to T, then S′ + T ~ S′. The proof of lemma 5* is the same as the proof of lemma 5. Figuratively speaking, lemma 5* (5) is the convex (concave) version of lemma 5 (5*).
- 34.
This and the next two paragraphs, justify the claim with regard to case 2).
- 35.
There is a typo here in the original: x 4 is printed instead of χ4 and actually it should be χ3 as explained in the next footnote.
- 36.
There is a typo here in the original: x 24 is printed instead of x 14. Besides, Bernstein confuses the cases and does not point out that for the odd cases the analogy is not straight but crossed. The correct assignment is: \( {\chi_{{4n}}},{\text{ \it n}} > 0 \), maps x 13 into x 14 and likewise \( {\chi_{{{{4n}} + {3}}}};{\chi_{{{{4n}} + {2}}}} \) maps x 13 into x 23 and likewise \( {\chi_{{{{4n}} + {1}}}},{\text{ \it n}} > 0 \). So the alternation occurs every third (not every second) step. Why all the images of x 13 are disjoint is explicated below.
- 37.
“ausserdem bilden sie den Teil einer Gruppe von solchen Abbildungen, wo es zu jedem Element ein und nur ein inverse Element der Gruppe gibt.” Bernstein seems to say that the series of χ′s forms a sub-group of the group of all 1–1 mappings from S onto itself, denoted in lemma 1 by Φ S. Why he mentions only the property of having an inverse and not the property of closure – we don’t know. Anyway, the group nature of the χ′s, the existence of an inverse, is apparently applied to establish that the χ′s are 1–1, though this seems to follow from the fact that the χ′s are compositions of 1–1 mappings.
- 38.
Lemma 1 is not mentioned in the proof. Lemmas 2–4 are not necessary for lemma 5 or 5*, which only require that the images of x 13 under the χ′s be different. This point can be established directly: By ϕ b , namely χ2, x 13 is mapped into x 23 and its copy is thus disjoint from x 13. ϕ c then maps the copy of x 13 in x 23 into x 14 and this copy is thus disjoint from x 13. This copy then is a result of applying χ4 to x 13. ϕ b then maps the copy from x 14 into x 23 and this new copy is disjoint from the previous one because they are both obtained from disjoint subsets of x 1 by the same 1–1 mapping. This new copy is the result of applying χ6 to x 13. Continuing in this way we see that each χ4n+2 produces a new copy of x 13 in x 23, disjoint from the previous ones, and χ4n likewise produces a new copy of x 13 in x 14 which is again disjoint from the previous ones. In Bernstein’s proof this procedure is unnecessarily doubled by the oddly indexed sequence of χ’s. In fact it is enough for the proof to take the sequence of even (or odd) χ′s. ϕ b ϕ c (ϕ c ϕ b ) map each copy of x 13 in x 23 (x 14) to the next one.
- 39.
We should read: the set of those elements.
- 40.
Bernstein writes χ1, which he did not define. Therefore, here and below, we increase the index of the χ′s mentioned by 1. This is the blunder we mentioned earlier.
- 41.
Namely: the set of those elements [of x 13] different from the elements in x′13.
- 42.
Bernstein is not worried about elements of x 13 transferred by χ3 to elements of x 24 already hit by χ2 because after the interchange consummated below these elements of x 24 will be moved and so χ3 will no longer take members of x 13 to x 24.
- 43.
Thus, for \( \nu > {1} \), \( {{{x}}^{{(\nu )}}}_{{{13}}} = {\chi_{{\nu + {1}}}}^{{ - {1}}}\left( {{\chi_{{\nu + {1}}}}({{{x}}_{{{13}}}}) \cap {{{x}}_{{{24}}}} - {\Sigma_{{\mu = {1}...(\nu - {1})}}}{\chi_{{\mu + {1}}}}\left( {{{{x}}^{{(\mu )}}}_{{{13}}}} \right)} \right) - {\Sigma_{{\mu = {1}...(\nu - {1})}}}{{{x}}^{{(\mu )}}}_{{{13}}} \).
- 44.
Scheme 17 is graphically similar to the scheme in Zermelo 1901 proof of theorem I (see 0). Apparently, both mathematicians were influenced by a similar scheme used in Schoenflies 1900 proof of CBT. We wonder if in circumstances around BDT begins the bitter feelings that Zermelo developed towards Bernstein (Peckhaus 1990 p 48; Ebbinghaus 2007 §2.8.4).
- 45.
In (18) Bernstein defines \( {\overline{\overline {{x}}}_{{{13}}}} \) and \( {\overline{\overline {{x}}}_{{{24}}}} \).
- 46.
There is a typo here in the original and \( \overline { x} \) 14 is written instead of \( {\overline{{\bar{x}}}_{{{24}}}} \).
- 47.
Symmetrically, \( {{\overline{{x}}}_{{{13}}}} \) could have been transferred to x 14 and \( \overline { x} \) 24 to x 23.
- 48.
By combining the \( {{x}}{^*_{{ij}}} \) according to the equations in (14).
- 49.
In the original ℵ is written instead of μ, which is surely a typo.
- 50.
“Sind die betrachteten Mengen endlich, so muss der nach der angegebenen Vorschriftvollzogene Austausch zur Folge haben, dass x 13 und dann aber auch x 24 völlig verschwenden sind. Dies kann man sich leicht veranschaulichen.” It seems that Bernstein wanted to emphasize here that if the exchange empties the sets, which can happen even if the sets are infinite, the result still holds.
- 51.
Surely Bernstein means x 1 ~ x 3 ~ x 2 ~ x 4; likewise it should be x 1* ~ x 3*, as implied by lemma 5*.
- 52.
For the generalized theorem Bernstein changed his notation calling by y the partitions of the second partitioning. This change suggests that the generalized theorem was proved after the case k = 2.
- 53.
Bernstein uses ‘n’ for our ‘k’ and he now uses ‘ ~ ’ where in theorem 2 he used ‘ = ’. Here, then, we can interpret n·M as {0, 1, …, n-1}·M, etc.
- 54.
Actually the problem is with the interchange not the choice of the χ′s.
- 55.
In the second and third rows, ‘ = ’ should be read as equivalence, ‘ ~ ’.
- 56.
The ϕ′s (ψ′s) are between the first row (column) and each of the others. Like in theorem 2 we assume that ϕ 1i (ψ 1i ) are defined on x i (y i ) by ϕ -1 1i (ψ -1 1i ). Expanding the mappings to S (the union of the rows (columns)) is not necessary.
- 57.
Bernstein does not say explicitly that x ij = x i ∩y j = y ji . All the ‘ = ’ signs in this paragraph are to be read as ‘ ~ ’. Obviously the proof is symmetrical with respect to which cell is chosen to be established as equivalent to the row. Bernstein takes x 11.
- 58.
In this paragraph too ‘ = ’ should be read as ‘ ~ ’.
- 59.
The inequality sign requires regarding x 1, y 1 as powers
- 60.
The rightmost ϕ in the definition of χ3 in the original is ϕ 13, which seems a typo for there is no reason to use ϕ 13 at this stage of the proof. The definition of the χ i here is simpler than in the proof of theorem 2 and corresponds to the even χ i there. We have noted previously that the same simplification could be applied to theorem 2. Note that the definition of χ0 is redundant as the χ i do not form a group (ϕ 12 ψ 12 does not have an inverse). The 1–1 character of the χ i is guaranteed directly by the 1–1 character of the ϕ′s and ψ′s. The simplified definition of the χ′s indicates, again, that the generalized theorem was found and proved after the case k = 2. We learn here something about Bernstein’s character: he did not bother to go back to his proof of theorem 2 to streamline it with the insight gained in the proof of theorem 3. Bernstein appears indifferent to perfecting his public results.
- 61.
To avoid cumbersome notation, we maintain the notations of the rows, columns, cells and their subsets after the interchanges.
- 62.
We have in fact to define by induction a sequence of nesting subsets of x 12, denoted by x 12(n), which contains what is left in x 12 after the n first interchanges. The intersection of x 12(n), which we denote by x 12 again, has the desired property.
- 63.
It will be interesting to find an algorithm with only one inductive process.
- 64.
x, a are powers not sets.
- 65.
x = x 1 means that x 1 is of power x, and naturally so is x 2. Similarly for x 3.
References
Bernstein F. Untersuchungen aus der Mengenlehre. Mathematische Annalen. 1905;61:117–55.
Cantor G. Contributions to the founding of the theory of Transfinite Numbers, English version of Cantor 1895 and Cantor 1897, translated by Jourdain PEB. Dover Publications Inc.
Dauben JW. Georg Cantor. His Mathematics and the Philosophy of the Infinite, Cambridge MA: Harvard University Press; 1979. Reprinted by Princeton University Press, 1990.
Ebbinghaus HD. Ernst Zermelo. An approach to his life and work. New York: Springer; 2007.
Hausdorff F. Grundzuge der Mengenlehre, Berlin; 1914a, reprinted by Chelsea, New York; 1949.
Hobson EW. The theory of functions of a real variable and the theory of Fourier’s series, Cambridge University Press; 1907. Digital version: http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACM2112.
Kőnig J. Sur le théorie des ensemble. Comptes Rendus Hebdomedaire des Séances de l’Academie des Science, Paris. 1906;143:110–2.
Lindenbaum A, Tarski A. Communication sur les recherches de la thèorie des ensembles. Comptes rendu des séances de la société Polonaise de Mathematique section Varsovie Annales de la Societe Polonaise Mathematique. 1926;19:299–330.
Peckhaus V. ‘Ich habe mich wohl gehütet alle patronen auf einmal zu verschiessen’; Ernst Zermelo in Göttingen. History and Philosophy of Logic. 1990a;11:19–58.
Schoenflies A. Die Entwicklung der Lehre von den Punktmannigfeltigkeiten, I, Jahresbericht der Deutschen Mathematiker-Vereinigung 1900;8.
Tarski A. Über Äquivalenz der Mengen in Bezug auf eine beliebige Klasse von Abbildungen. International Congress of Mathematicians. 1928;2:243–52.
Tarski A. Cardinal algebras. New York: Oxford University Press; 1949a.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Hinkis, A. (2013). Bernstein’s Division Theorem. In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0224-6_14
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0223-9
Online ISBN: 978-3-0348-0224-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)