In terms of the cardinality of the two sets, this classically implies that if |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|; that is, A and B are equipotent. This is a useful feature in the ordering of cardinal numbers.
Assume without loss of generality that A and B are disjoint. For any a in A or b in B we can form a unique two-sided sequence of elements that are alternately in A and B, by repeatedly applying and to go from A to B and and to go from B to A (where defined; the inverses and are understood as partial functions at this stage of the proof.)
For any particular a, this sequence may terminate to the left or not, at a point where or is not defined.
By the fact that and are injective functions, each a in A and b in B is in exactly one such sequence to within identity: if an element occurs in two sequences, all elements to the left and to the right must be the same in both, by the definition of the sequences. Therefore, the sequences form a partition of the (disjoint) union of A and B. Hence it suffices to produce a bijection between the elements of A and B in each of the sequences separately, as follows:
Call a sequence an A-stopper if it stops at an element of A, or a B-stopper if it stops at an element of B. Otherwise, call it doubly infinite if all the elements are distinct or cyclic if it repeats. See the picture for examples.
- For an A-stopper, the function is a bijection between its elements in A and its elements in B.
- For a B-stopper, the function is a bijection between its elements in B and its elements in A.
- For a doubly infinite sequence or a cyclic sequence, either or will do ( is used in the picture).
The traditional name "Schröder–Bernstein" is based on two proofs published independently in 1898. Cantor is often added because he first stated the theorem in 1887, while Schröder's name is often omitted because his proof turned out to be flawed while the name of Richard Dedekind, who first proved it, is not connected with the theorem. According to Bernstein, Cantor had suggested the name equivalence theorem (Äquivalenzsatz).
- 1887 Cantor publishes the theorem, however without proof.
- 1887 On July 11, Dedekind proves the theorem (not relying on the axiom of choice) but neither publishes his proof nor tells Cantor about it. Ernst Zermelo discovered Dedekind's proof and in 1908 he publishes his own proof based on the chain theory from Dedekind's paper Was sind und was sollen die Zahlen?
- 1895 Cantor states the theorem in his first paper on set theory and transfinite numbers. He obtains it as an easy consequence of the linear order of cardinal numbers. However, he could not prove the latter theorem, which is shown in 1915 to be equivalent to the axiom of choice by Friedrich Moritz Hartogs.
- 1896 Schröder announces a proof (as a corollary of a theorem by Jevons).
- 1897 Bernstein, a 19-year-old student in Cantor's Seminar, presents his proof.
- 1897 Almost simultaneously, but independently, Schröder finds a proof.
- 1897 After a visit by Bernstein, Dedekind independently proves the theorem a second time.
- 1898 Bernstein's proof (not relying on the axiom of choice) is published by Émile Borel in his book on functions. (Communicated by Cantor at the 1897 International Congress of Mathematicians in Zürich.) In the same year, the proof also appears in Bernstein's dissertation.
- 1898 Schröder publishes his proof which, however, is shown to be faulty by Alwin Reinhold Korselt in 1902 (just before Schröder's death), (confirmed by Schröder), but Korselt's paper is published only in 1911.
Both proofs of Dedekind are based on his famous 1888 memoir Was sind und was sollen die Zahlen? and derive it as a corollary of a proposition equivalent to statement C in Cantor's paper, which reads A ⊆ B ⊆ C and |A| = |C| implies |A| = |B| = |C|. Cantor observed this property as early as 1882/83 during his studies in set theory and transfinite numbers and was therefore (implicitly) relying on the Axiom of Choice.
The 1895 proof by Cantor relied, in effect, on the axiom of choice by inferring the result as a corollary of the well-ordering theorem. However, König's proof given above shows that the result can also be proved without using the axiom of choice.
On the other hand, König's proof uses the principle of excluded middle, to do the analysis into cases, so this proof does not work in constructive set theory. Even more, no proof at all can exist from constructive set theory alone (i.e. dispensing with the principle of excluded middle), since the Schröder–Bernstein theorem implies the principle of excluded middle. Therefore, intuitionists do not accept the theorem.
- Myhill isomorphism theorem
- Netto's theorem, according to which the bijections constructed by the Schröder–Bernstein theorem between spaces of different dimensions cannot be continuous
- Schröder–Bernstein theorem for measurable spaces
- Schröder–Bernstein theorems for operator algebras
- Schröder–Bernstein property
- J. König (1906). "Sur la théorie des ensembles". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. 143: 110–112.
- Felix Hausdorff (2002), Egbert Brieskorn; Srishti D. Chatterji; et al. (eds.), Grundzüge der Mengenlehre (1. ed.), Berlin/Heidelberg: Springer, p. 587, ISBN 978-3-540-42224-2 – Original edition (1914)
- Georg Cantor (1887), "Mitteilungen zur Lehre vom Transfiniten", Zeitschrift für Philosophie und philosophische Kritik, 91: 81–125
Reprinted in: Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo (eds.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Berlin: Springer, pp. 378–439 Here: p.413 bottom
- Richard Dedekind (1932), Robert Fricke; Emmy Noether; Øystein Ore (eds.), Gesammelte mathematische Werke, vol. 3, Braunschweig: Friedr. Vieweg & Sohn, pp. 447–449 (Ch.62)
- Ernst Zermelo (1908), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen, 65 (2): 261–281, here: p.271–272, doi:10.1007/bf01449999, ISSN 0025-5831, S2CID 120085563
- Richard Dedekind (1888), Was sind und was sollen die Zahlen? (2., unchanged (1893) ed.), Braunschweig: Friedr. Vieweg & Sohn
- Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo (eds.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Berlin: Springer, pp. 285 ("Satz B")
- Georg Cantor (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512 (Theorem see "Satz B", p.484). doi:10.1007/bf02124929. S2CID 177801164.
- (Georg Cantor (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994.)
- Friedrich M. Hartogs (1915), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Mathematische Annalen, 76 (4): 438–443, doi:10.1007/bf01458215, ISSN 0025-5831, S2CID 121598654
- Ernst Schröder (1896). "Über G. Cantorsche Sätze". Jahresbericht der Deutschen Mathematiker-Vereinigung. 5: 81–82.
- Oliver Deiser (2010), Einführung in die Mengenlehre – Die Mengenlehre Georg Cantors und ihre Axiomatisierung durch Ernst Zermelo, Springer-Lehrbuch (3rd, corrected ed.), Berlin/Heidelberg: Springer, pp. 71, 501, doi:10.1007/978-3-642-01445-1, ISBN 978-3-642-01444-4
- Patrick Suppes (1972), Axiomatic Set Theory (1. ed.), New York: Dover Publications, pp. 95 f, ISBN 978-0-486-61630-8
- Émile Borel (1898), Leçons sur la théorie des fonctions, Paris: Gauthier-Villars et fils, pp. 103 ff
- Felix Bernstein (1901), Untersuchungen aus der Mengenlehre, Halle a. S.: Buchdruckerei des Waisenhauses
Reprinted in: Felix Bernstein (1905), Felix Klein; Walther von Dyck; David Hilbert (eds.), "Untersuchungen aus der Mengenlehre", Mathematische Annalen, 61 (1): 117–155, (Theorem see "Satz 1" on p.121), doi:10.1007/bf01457734, ISSN 0025-5831, S2CID 119658724
- Ernst Schröder (1898), Kaiserliche Leopoldino-Carolinische Deutsche Akademie der Naturforscher (ed.), "Ueber zwei Definitionen der Endlichkeit und G. Cantor'sche Sätze", Nova Acta, 71 (6): 303–376 (proof: p.336–344)
- Alwin R. Korselt (1911), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über einen Beweis des Äquivalenzsatzes", Mathematische Annalen, 70 (2): 294–296, doi:10.1007/bf01461161, ISSN 0025-5831, S2CID 119757900
- Korselt (1911), p.295
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- This article incorporates material from the Citizendium article "Schröder-Bernstein_theorem", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.