Schröder–Bernstein theorem

Schröder–Bernstein theorem   (Redirected from Cantor–Bernstein–Schroeder theorem) Jump to navigation Jump to search In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.

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.

The theorem is named after Felix Bernstein and Ernst Schröder. It is also known as Cantor–Bernstein theorem, or Cantor–Schröder–Bernstein, after Georg Cantor who first published it without proof.

Contents 1 Proof 2 History 3 Prerequisites 4 See also 5 Notes 6 References 7 External links Proof König's definition of a bijection h:A → B from given example injections f:A → B and g:B → A. An element in A and B is denoted by a number and a letter, respectively. The sequence 3 → e → 6 → ... is an A-stopper, leading to the definitions h(3) = f(3) = e, h(6) = f(6), .... The sequence d → 5 → f → ... is a B-stopper, leading to h(5) = g−1(5) = d, .... The sequence ... → a → 1 → c → 4 → ... is doubly infinite, leading to h(1) = g−1(1) = a, h(4) = g−1(4) = c, .... The sequence b → 2 → b is cyclic, leading to h(2) = g−1(2) = b.

The following proof is attributed to Julius König.[1] 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 {displaystyle f} and {displaystyle g^{-1}} to go from A to B and {displaystyle g} and {displaystyle f^{-1}} to go from B to A (where defined; the inverses {displaystyle f^{-1}} and {displaystyle g^{-1}} are understood as partial functions.) {displaystyle cdots rightarrow f^{-1}(g^{-1}(a))rightarrow g^{-1}(a)rightarrow arightarrow f(a)rightarrow g(f(a))rightarrow cdots } For any particular a, this sequence may terminate to the left or not, at a point where {displaystyle f^{-1}} or {displaystyle g^{-1}} is not defined.

By the fact that {displaystyle f} and {displaystyle g} 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 {displaystyle f} is a bijection between its elements in A and its elements in B. For a B-stopper, the function {displaystyle g} is a bijection between its elements in B and its elements in A. For a doubly infinite sequence or a cyclic sequence, either {displaystyle f} or {displaystyle g} will do ( {displaystyle g} is used in the picture). History 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).[2] Cantor's first statement of the theorem (1887)[3] 1887 Cantor publishes the theorem, however without proof.[3][2] 1887 On July 11, Dedekind proves the theorem (not relying on the axiom of choice)[4] but neither publishes his proof nor tells Cantor about it. Ernst Zermelo discovered Dedekind's proof and in 1908[5] he publishes his own proof based on the chain theory from Dedekind's paper Was sind und was sollen die Zahlen?[2][6] 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.[7][8][9] 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.[2][10] 1896 Schröder announces a proof (as a corollary of a theorem by Jevons).[11] 1897 Bernstein, a 19-year-old student in Cantor's Seminar, presents his proof.[12][13] 1897 Almost simultaneously, but independently, Schröder finds a proof.[12][13] 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.[14] (Communicated by Cantor at the 1897 International Congress of Mathematicians in Zürich.) In the same year, the proof also appears in Bernstein's dissertation.[15][2] 1898 Schröder publishes his proof[16] which, however, is shown to be faulty by Alwin Reinhold Korselt in 1902 (just before Schröder's death),[17] (confirmed by Schröder),[2][18] 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,[7] 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.

Prerequisites 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.[8][9] 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.[19] Therefore, intuitionists do not accept the theorem.[20] There is also a proof which uses Tarski's fixed point theorem.[21] See also 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 Notes ^ J. König (1906). "Sur la théorie des ensembles". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. 143: 110–112. ^ Jump up to: a b c d e f 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) ^ Jump up to: a b 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 ^ Jump up to: a b Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo (eds.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Berlin: Springer, pp. 285 ("Satz B") ^ Jump up to: a 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. ^ Jump up to: a b (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. ^ Jump up to: a b 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 ^ Jump up to: a b 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 ^ Pradic, Pierre; Brown, Chad E. (2019). "Cantor-Bernstein implies Excluded Middle". arXiv:1904.09193 [math.LO]. ^ Ettore Carruccio (2006). Mathematics and Logic in History and in Contemporary Thought. Transaction Publishers. p. 354. ISBN 978-0-202-30850-0. ^ R. Uhl, "Tarski's Fixed Point Theorem", from MathWorld–a Wolfram Web Resource, created by Eric W. Weisstein. (Example 3) References Martin Aigner & Gunter M. Ziegler (1998) Proofs from THE BOOK, § 3 Analysis: Sets and functions, Springer books MR1723092, fifth edition 2014 MR3288091, sixth edition 2018 MR3823190 Hinkis, Arie (2013), Proofs of the Cantor-Bernstein theorem. A mathematical excursion, Science Networks. Historical Studies, vol. 45, Heidelberg: Birkhäuser/Springer, doi:10.1007/978-3-0348-0224-6, ISBN 978-3-0348-0223-9, MR 3026479 Searcóid, Míchaél Ó (2013). "On the history and mathematics of the equivalence theorem". Mathematical Proceedings of the Royal Irish Academy. 113A (2): 151–68. doi:10.1353/mpr.2013.0006. JSTOR 42912521. External links Weisstein, Eric W. "Schröder-Bernstein Theorem". MathWorld. Cantor-Schroeder-Bernstein theorem in nLab Cantor-Bernstein’s Theorem in a Semiring by Marcel Crabbé. 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. show vte Set theory show vte Mathematical logic show vte Logic Categories: Theorems in the foundations of mathematicsCardinal numbers

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