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

Venue: | Bull. Symbolic Logic |

Citations: | 7 - 2 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 ...definitions by recursion on the natural numbers. 35 In his axiomatization paper [1908a] Zermelo provided a new proof of the Schrsoder-Bernstein Theorem, and it is noteworthy that themes of Kuratowski =-=[1922]-=-, to be discussed in Section 3, were foreshadowed by the proof. Zermelo focused on the following formulation: If M # # M 1 # M and there is a bijection g : M # M # , then there is a bijection h : M # ... |

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Citation Context ...ely wrote: "Suppose A does not have a maximal element. Then for each x # A there exists an element yx # A such that xsyx ." It is as if the mighty struggles of the past never took place! 47 =-=See Aczel [1977] for a sur-=-vey of the theory of inductive definitions in extensions of recursion theory and Moschovakis [1974] for a detailed development (for the "positive" case) in abstract structures. 48 Kuratowski... |

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Citation Context ...er Russell extended Peano's symbolic approach to the logic of relations. Armed with the new insights Russell in the rest of the year completed most of the final draft of The Principles of Mathematics =-=[1903]-=-, a book he had been working on in various forms from 1898. However, the sudden light would also cast an abiding shadow, for by May 1901 quaint as this may now seem, in conversation with Felix Bernste... |

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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 ... 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 ...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 ...be solved through his theory of types. 39 The Schrsoder-Bernstein Theorem in Principia is #73--88. In #94 the Bernstein and Zermelo proofs are compared. 40 See for example the account Dugundji-Granas =-=[1982]-=-. 41 The Brouwer fixed point theorem asserts that any continuous function from a closed simplex into itself must have a fixed point; to illustrate without bothering to define these terms, any continuo... |

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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 ...tay at the Mathematischen Forschungsinstitut Oberwolfach under the Research in Pairs program. This paper was partly inspired by a conversation with the late 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... |

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Citation Context ...1888]. 7 Having a such that a = {a} can serve certain technical purposes. For W. V. Quine [1940, 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 serve as the atoms of his Fraenkel-Mostowski permutation models for independence results relate... |

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Citation Context ...egarded as consisting of everything conceivable is inconsistent, and Husserl in a review [1891] had criticized Schrsoder'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-... |

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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 ...al 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: &q-=-uot;The introduction of the concept of `covering' is the most striking advance in the principles of the theory of transfinite numbers from 1885 to 1895 . . . ." THE MATHEMATICAL IMPORT OF ZERMELO... |

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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 ...xiom of Foundation on such a. For Ernst Specker [1957] such a serve as the atoms of his Fraenkel-Mostowski permutation models for independence results related to the Axiom of Choice. Since Dana Scott =-=[1962] "Qui-=-nean atoms" a = {a} have figured in the model-theoretic investigation of Quine's best known set theory, New Foundations; see Forster [1995, Chapter 3]. THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-O... |

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Citation Context ...ristotle. Indeed, the inclusion vs. membership distinction is fundamental to the development of logic in Frege's Begri#- sschrift [1879], and the a vs. {a} distinction is explicit in his Grundgesetze =-=[1893]-=-. These distinctions for sets are also basic 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 pio... |

2 |
and the origins of the set-theoretic "paradoxes", Birkh auser
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Citation Context ...ut Russell's predisposition 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 Ga=-=rciadiego [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 ... |

2 |
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Citation Context ... [1888, IV]", and that Peano [1906] published a proof that was "quite similar". 37 Russell on first reading Zermelo [1908a] expressed delight with his proof of Schrsoder-Bernstein but 3=-=6 See Goldfarb [1988]-=- for more about Poincar e against the logicists. 37 Zermelo's footnote is Footnote 11 of van Heijenoort [1967, p. 209]. The reference to Dedekind [1888] raises two points: First, Dedekind [1888] had a... |

2 |
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Citation Context ... 40 Of the pioneering work, fixed points figured crucially in Poincar e's classic analysis [1884] of the three-body problem. The best-known fixed point theorem in mathematics, due to L. E. J. Brouwer =-=[1909]-=-, is fundamental to algebraic topology. 41 And a few pages after Kuratowski's [1922] in the same journal, Stefan Banach [1922] published work from his thesis including what has become another basic fi... |

2 |
Grundbegri#e der mengenlehre, Vandenhoeck & Ruprecht, G ottingen, reprinted from Abhandlungen der Fries'schen Schule
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Citation Context ...eijenoort [1967, p. 117]. 30 Moore-Garciadiego [1978] and Garciadiego [1992] describe the evolution of the BuraliForti paradox. 31 See also the discussion in Hallett [1984, pp. 180--1]. While Russell =-=[1906]-=- discussed Zermelo [1904], it is unlikely that Russell made a conscious adaptation along the lines of Theorem 2.1. 296 AKIHIRO KANAMORI {x | x / # x} contains a series similar to all the ordinal numbe... |

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Citation Context ... "sa Nancago". THE MATHEMATICAL IMPORT OF ZERMELO'S WELL-ORDERING THEOREM 303 Referring to Bourbaki's [1939] "fundamental lemma" and Zermelo's two proofs for the Well-Ordering Theo=-=rem, Helmuth Kneser [1950]-=-, a student of David Hilbert, provided a proof of Theorem 3.2 much as given above, i.e., in the style of Zermelo's first [1904] proof. Kneser also pointed out that the partially ordered set need not b... |

2 |
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Citation Context ...definitions by recursion on the natural numbers. 35 In his axiomatization paper [1908a] Zermelo provided a new proof of the Schrsoder-Bernstein Theorem, and it is noteworthy that themes of Kuratowski =-=[1922]-=-, to be discussed in Section 3, were foreshadowed by the proof. Zermelo focused on the following formulation: If M # # M 1 # M and there is a bijection g : M # M # , then there is a bijection h : M # ... |

1 |
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Citation Context ... of X , the P(X ) in Corollary 2.3 can be replaced by the following set: PWO (X ) = {Z # X | Z is well-orderable} . That this set, like P(X ), is not bijective with X was first shown by Alfred Tarski =-=[1939]-=- through a less direct proof. Tarski [1939] (Theorem 3) did have a version of Theorem 2.1; substantially the same version appeared in the expository work of Nicolas Bourbaki [1956, p. 43] (Chapter 3, ... |

1 |
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Citation Context ... 40 Of the pioneering work, fixed points figured crucially in Poincar e's classic analysis [1884] of the three-body problem. The best-known fixed point theorem in mathematics, due to L. E. J. Brouwer =-=[1909]-=-, is fundamental to algebraic topology. 41 And a few pages after Kuratowski's [1922] in the same journal, Stefan Banach [1922] published work from his thesis including what has become another basic fi... |

1 |
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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 ... |

1 |
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Citation Context ... Couturat's contention [1899, pp. 628--9] that Peano's introduction of # was an unnecessary complication beyond Schrsoder's system of logic. Russell's [1901] was meant to be a counterpart to Couturat =-=[1899], tho-=-ugh it was never published. See Moore's comments in Russell [1993, pp. 350--1]. Years later, Russell [1959, pp. 66--7] wrote: "The enlightenment that I derived from Peano came mainly from two pur... |

1 |
A note on Cantor's theorem and Russell's paradox
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Citation Context ... `a man.' The fact that a word can be framed with a wider meaning than term raises grave logical problems." 21 The formal transition from {x | x / # f(x)} to {x | x / # x} was pointed out by Cros=-=sley [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 IMP... |

1 |
and Akihiro Kanamori, Hilbert and set theory
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Citation Context ...egarded as consisting of everything conceivable is inconsistent, and Husserl in a review [1891] had criticized Schrsoder'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-... |

1 |
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Citation Context ...e embodiment of the inclusion vs. membership distinction became important in set theory after the work of John von Neumann [1923] on ordinals, and central to the subject since the work of Kurt Gsodel =-=[1938]-=- on the constructible universe L. The claim that Theorem 2.1 anticipates later developments is bolstered by its proof being essentially the argument for the Transfinite Recursion Theorem, the theorem ... |

1 | und Wohlordnung, Journal f ur die reine und angewandte - Kettentheorie |