## The Mathematical Import Of Zermelo's Well-Ordering Theorem (1997)

Venue: | Bull. Symbolic Logic |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Kanamori97themathematical,

author = {Akihiro Kanamori},

title = {The Mathematical Import Of Zermelo's Well-Ordering Theorem},

journal = {Bull. Symbolic Logic},

year = {1997},

volume = {3},

pages = {281--311}

}

### OpenURL

### Abstract

this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :

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Citation Context ...rsoder's argument for not distinguishing between inclusion and 22 See Dreben-Kanamori [1997] for the line of development from Russell's theory of types to Gsodel's constructible universe. 23 Peckhaus =-=[1990]-=- provides a detailed account of Zermelo's years 1897--1910 at Gsottingen. 24 See Rang-Thomas [1981]. THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-ORDERING THEOREM 291 membership. Zermelo was pointing out... |

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Citation Context ...g essentially the argument for the Transfinite Recursion Theorem, the theorem that justifies definitions by recursion along well-orderings. This theorem was articulated and established by von Neumann =-=[1923, 1928]-=- in his system of set theory. However, the argument as such first appeared in Zermelo's [1904]: Proof of Theorem 2.1. Call Y # X an F -set i# there is a well-ordering R of Y such that for each x # Y ,... |

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Citation Context ...disputable. Frege in his review [1895] of Schrsoder's [1890] soundly took him to task for these shortcomings. 4 Richard Dedekind in his classic essay on arithmetic Was sind und was sollen die Zahlen? =-=[1888, 3]-=- used the same symbol for inclusion and membership and subsequently identified an individual a with {a}. 5 In a revealing note found in his Nachlass Dedekind was to draw attention to the attendant dan... |

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Citation Context ...ziehung]. Arbitrary functions on arbitrary domains are now of course commonplace in mathematics, but several authors at the time referred specifically to the concept of covering, most notably Zermelo =-=[1904] (see-=- Section 2). Jourdain in the introduction to his English translation [1915, p. 82] of Cantor's [1895, 1897] wrote: "The introduction of the concept of `covering' is the most striking advance in t... |

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Citation Context ...isposition in 1900 to embrace Peano's "ideography" and Cantor's theory, both in terms of Russell's rejection a few years earlier of a neo-Hegelian idealism in favor 8 See Garciadiego [1992] =-=and Moore [1995]-=- for the evolution of Russell's Paradox. 9 Russell had first mentioned Peano in a letter dated 9 October 1899 to the philosopher Louis Couturat. There Russell had expressed agreement with Couturat's r... |

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Citation Context ...y [1939, p. 37] (6, Item 10) for his series El ements de Math ematique. 42 The idea of rendering well-orderings in set theory in terms of # occurred in Hessenberg [1906] and was pursued by Kuratowski =-=[1921]-=-. See Hallett [1984, p. 256#.] for an analysis of Zermelo's [1908] proof in this light. 43 The first to provide a proposition similarly related to Zorn's Lemma was Felix Hausdor# [1909, p. 300]. See C... |

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Citation Context ...shing the Cantorian extensional and operational view proceeded to resolve the problem of well-ordering sets mathematically. As noted in Footnote 2, in describing abstract functions Cantor had written =-=[1895, 4]: " . . . by a -=-`covering [Belegung] of N with M ,' we understand a law . . . ", and thus had continued his frequent use of the term "law" to refer to functions. Zermelo [1904, p. 514] specifically use... |

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Citation Context ...p. 136] such a are the individuals for a theory of sets, now known as Mathematical Logic. Paul Bernays [1954] based a proof of the independence of the Axiom of Foundation on such a. For Ernst Specker =-=[1957] such a se-=-rve as the atoms of his Fraenkel-Mostowski permutation models for independence results related to the Axiom of Choice. Since Dana Scott [1962] "Quinean atoms" a = {a} have figured in the mod... |

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Citation Context ... gave his now famous diagonal proof, showing in e#ect that for any set X the collection of functions from X into a two-element set is of a strictly higher cardinality than that of X . Much earlier in =-=[1874]-=-, the paper that began set theory, Cantor had established the uncountability of the real numbers by using their completeness under limits. In retrospect the diagonal proof can be drawn out from the [1... |

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Citation Context ...ate George Boolos and a reading of Moschovakis [1994]. The author would like to express his gratitude to Burton Dreben for his careful reading and numerous suggestions for improvement. 1 See Kanamori =-=[1996]-=- for the mathematical development of set theory from Cantor to Cohen. c # 1997, Association for Symbolic Logic 1079-8986/97/0303-0001/$4.10 281 282 AKIHIRO KANAMORI the motif as fully participating in... |

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Citation Context ...rsoder's argument for not distinguishing between inclusion and 22 See Dreben-Kanamori [1997] for the line of development from Russell's theory of types to Gsodel's constructible universe. 23 Peckhaus =-=[1990]-=- provides a detailed account of Zermelo's years 1897--1910 at Gsottingen. 24 See Rang-Thomas [1981]. THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-ORDERING THEOREM 291 membership. Zermelo was pointing out... |

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Citation Context ...with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. 1. Cantor's diagonal proof to Russell's paradox. Georg Cantor in =-=[1891]-=- gave his now famous diagonal proof, showing in e#ect that for any set X the collection of functions from X into a two-element set is of a strictly higher cardinality than that of X . Much earlier in ... |

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Citation Context ...order logic. But as with the Well-Ordering Theorem itself, the [1908a] axiomatization from early on served to ground the investigation of well-orderings, with the incisive result of Friedrich Hartogs =-=[1915]-=- on the comparability of cardinals being a prominent example. The early work also led to a new transformation of our motif. 3. Fixed point theorems. Kazimierz Kuratowski [1922] provided a fixed point ... |

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Citation Context ...e line of development from Russell's theory of types to Gsodel's constructible universe. 23 Peckhaus [1990] provides a detailed account of Zermelo's years 1897--1910 at Gsottingen. 24 See Rang-Thomas =-=[1981]-=-. THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-ORDERING THEOREM 291 membership. Zermelo was pointing out an inherent problem when inclusion implies membership as in the case of a universal class, but he ... |

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Citation Context ...nd range {0, 1}; regarded these as being enumerated by one super-function #(x, z) with enumerating variable z; and formulated the diagonalizing function g(x) = 1 - #(x, x). In his mature presentation =-=[1895]-=- of his theory of cardinality Cantor defined cardinal exponentiation in terms of the set of all functions from a set N into a set M , but such arbitrary functions were described in a convoluted way, r... |

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Citation Context ...ointed out by Crossley [1973]. Though not so explicit in The Principles, the analogy is clearly drawn in 1905 letters from Russell to G. H. Hardy and to Philip Jourdain (as quoted in Grattan-Guinness =-=[1978]-=-). THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-ORDERING THEOREM 289 was central to The Principles, the issues of whether the null-class exists and whether a term should be distinct from the class whose ... |

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Citation Context ...ziehung]. Arbitrary functions on arbitrary domains are now of course commonplace in mathematics, but several authors at the time referred specifically to the concept of covering, most notably Zermelo =-=[1904] (see-=- Section 2). Jourdain in the introduction to his English translation [1915, p. 82] of Cantor's [1895, 1897] wrote: "The introduction of the concept of `covering' is the most striking advance in t... |

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Citation Context ...e line of development from Russell's theory of types to Gsodel's constructible universe. 23 Peckhaus [1990] provides a detailed account of Zermelo's years 1897--1910 at Gsottingen. 24 See Rang-Thomas =-=[1981]-=-. THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-ORDERING THEOREM 291 membership. Zermelo was pointing out an inherent problem when inclusion implies membership as in the case of a universal class, but he ... |

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Citation Context ...c for Cantor's theory of cardinality and are evident from the beginning of his [1895], starting with its oft-quoted definition of set [Menge]. 3 Of other pioneers, Ernst Schrsoder in the first volume =-=[1890]-=- of his major work on the algebra of logic held to a traditional view that a class is merely a collection of objects (without the { }, so to speak), so that inclusion and membership could not be clear... |