## The Mathematical Development Of Set Theory - From Cantor To Cohen (1996)

Venue: | The Bulletin of Symbolic Logic |

Citations: | 8 - 2 self |

### BibTeX

@ARTICLE{Kanamori96themathematical,

author = {Akihiro Kanamori},

title = {The Mathematical Development Of Set Theory - From Cantor To Cohen},

journal = {The Bulletin of Symbolic Logic},

year = {1996},

volume = {2},

pages = {1--71}

}

### Years of Citing Articles

### OpenURL

### Abstract

This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

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Citation Context ...ctive not-both, he was not impressed with Wiener's reduction, which could not have been considered a genuine one. 50 Unlike Russell, Willard V.O. Quine in his major philosophical work Word and Object =-=[1960, 53]-=- regarded the reduction of the ordered pair as a paradigm for philosophical analysis. Making no intensional distinctions Hausdor# [1914, 32#., 70#.] defined an ordered pair in terms of unordered pairs... |

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Citation Context ...forth on its independent course as a distinctive field of mathematics was its full extensionalization in first-order logic. 66 However influential Zermelo's [1930] and despite his subsequent advocacy =-=[1931, 1935]-=- of infinitary logic, his e#orts to forestall Skolem were not to succeed, as stronger currents were at work in the direction of first-order formalization. Hilbert e#ected a basic shift in the developm... |

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Citation Context ...so turned out to require an inaccessible cardinal (Levy [1970, 131#.])---a remarkable turn of events for an apparently technical artifact at the beginning of forcing. Many years later, Saharon Shelah =-=[1980, 1984]-=- was able to establish the necessity of Solovay's inaccessible for the proposition that every set of reals is Lebesgue measurable; on the other hand, Shelah also showed that the inaccessible is not ne... |

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Citation Context ...1. The history may be continued to the present in a subsequent article. Van Dalen in van-Dalen-Monna [1972] also gives a history of set theory from Cantor to Cohen. See the texts Jech [1978] or Kunen =-=[1980]-=- for basic set-theoretic terminology or unelaborated results. 2. Dauben [1979], Meschkowski [1983], and Purkert-Ilgauds [1987] are mathematical biographies of Cantor. 3. See Kechris-Louveau [1987] for... |

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Citation Context ...at {# # S | f(#) = #} is stationary in #. 81. Genesis 2:20: "And Adam gave names to all cattle, and to the fowl of the air, and to every beast of the field . . . " See Chang-Keisler [1990] a=-=nd Hodges [1993]-=- for model theory and its development. 82. To define L(A), first recall def defined at the beginning of 3.4 and let tc(x) denote the smallest transitive set # x. Then define: L 0 (A) = tc({A}) (to ens... |

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Citation Context ...existential." See also Moore [1982, 39]. One exception to the misleading trend is Fraenkel [1930, 237][1953, 75], who from the beginning emphasized the constructive aspect of diagonalization. 9. =-=Gray [1994]-=- shows that Cantor's original [1874] argument can be implemented by an algorithm that generates n digits of a transcendental number with time complexity O(2 n 1/3 ), and his later diagonal argument, w... |

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Citation Context ...ility of intrinsic mathematical interest as well as a sense of steady mathematical progress. Notes 1. The history may be continued to the present in a subsequent article. Van Dalen in van-Dalen-Monna =-=[1972]-=- also gives a history of set theory from Cantor to Cohen. See the texts Jech [1978] or Kunen [1980] for basic set-theoretic terminology or unelaborated results. 2. Dauben [1979], Meschkowski [1983], a... |

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Citation Context ... incursions of a looming tradition can no longer be ignored. Gottlob Frege is regarded as the greatest philosopher of logic since Aristotle for developing quantificational logic in his Begri#sschrift =-=[1879]-=-, establishing a logical foundation for arithmetic in his Grundlagen [1884], and generally stimulating the analytic tradition in philosophy. The architect of that tradition was Bertrand Russell who in... |

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Citation Context ...len in van-Dalen-Monna [1972] also gives a history of set theory from Cantor to Cohen. See the texts Jech [1978] or Kunen [1980] for basic set-theoretic terminology or unelaborated results. 2. Dauben =-=[1979]-=-, Meschkowski [1983], and Purkert-Ilgauds [1987] are mathematical biographies of Cantor. 3. See Kechris-Louveau [1987] for recent developments in the Cantorian spirit about uniqueness for trigonometri... |

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Citation Context ...onvert many of the Fraenkel-Mostowski consistencies in algebra established by L auchli [1962]. David Pincus [1972] significantly extended the Jech-Sochor transfer theorem. See Felgner [1971] and Jech =-=[1973]-=- for more on independence of weak versions of AC and transfers. 90. See Gitik-Magidor [1992] and Jech [1995] for the recent work on the Singular Cardinals Problem. 91. The original application of the ... |

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Citation Context .... Independence. 4.1. Forcing. Paul Cohen (1934--), born just before Gsodel established his relative consistency results, established the independence of AC from ZF and the independence of CH from ZFC =-=[1963, 1964]-=-. These were, of course, the inaugural examples of forcing, soon to become a remarkably general and flexible method with strong intuitive underpinnings for extending models of set theory. 86 If Gsodel... |

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Citation Context ...y set of reals has the Baire property. Jean Raisonnier [1984] provided a simpler proof of Shelah's result on Lebesgue measurability. 92. There is a telling antecedent in the result of Gerhard Gentzen =-=[1936, 1943]-=- that the consistency strength of arithmetic can be exactly gauged by an ordinal # 0 , i.e., transfinite induction up to that ordinal in a formal system of notations. Although Hilbert's program of est... |

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Citation Context ...el in which AC holds. The basic results about this inner model were to be rediscovered several times, and the formal definition turns on some form of the Reflection Principle for ZF (see Myhill-Scott =-=[1971, 278]-=-). In these several ways reflection phenomena both as heuristic and as principle became incorporated into set theory, bringing to the forefront what was to become a basic feature of the study of well-... |

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Citation Context ...aration Axiom those expressible in first-order logic. After Leopold Lsowenheim [1915] had broken the ground for model theory with his result about the satisfiability of a first-order sentence, Skolem =-=[1920, 1923]-=- had located the result solidly in first-order logic THE MATHEMATICAL DEVELOPMENT OF SET THEORY FROM CANTOR TO COHEN 27 and generalized it to the Lsowenheim-Skolem Theorem: If a countable collection o... |

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Citation Context ...[1943] (cf., 3.5) in combinatorial formulations that was later seen to imply that a strongly compact cardinal is measurable, and a measurable cardinal is weakly compact. Tarski's student William Hanf =-=[1964]-=- then established, using the satisfaction relation for infinitary languages, that there are many inaccessible cardinals (and Mahlo cardinals) below a weakly compact cardinal. A fortiori, (Tarski [1962... |

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Citation Context ... years Gsodel speculated about the possibility of deciding propositions like CH with large cardinal hypotheses based on the heuristics of reflection and generalization. In 1946 remarks he (see Gsodel =-=[1990, 151]) suggested the-=- consideration of "stronger and stronger axioms of infinity," and reflection as follows: "Any proof for a set-theoretic theorem in the next higher system above set theory (i.e., any pro... |

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Citation Context ...a property needed. The situation would be considerably clarified, but only two decades later. 77 The detailed investigation of partition properties began in the 1950's, with Erdsos and Richard Rado's =-=[1956]-=- being representative. 78 For a cardinal # set # 0 (#) = # and # n+1 (#) = 2 # n (#) . What became known as the Erdsos-Rado Theorem asserts: For any infinite cardinal # and n # #, # n (#) + -# (# + ) ... |

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Citation Context ...heoretic methods. Model theory began in earnest with the method of diagrams 81 of Abraham Robinson's thesis [1951] and the related method of constants from Leon Henkin's thesis which gave a new proof =-=[1949]-=- of the Gsodel Completeness Theorem. Tarski had set the stage with his definition of truth and more generally his casting of formal languages and structures in set-theoretic terms, and with him establ... |

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Citation Context ...ball. Raphael Robinson [1947] later showed that there is such a decomposition into just five pieces with one of them containing a single point, and moreover that five is the minimal number. See Wagon =-=[1985]-=- for more on these and similar results; they stimulated interesting developments in measure theory that, rather than casting doubt on AC, embedded it further into mathematical practice (cf., 2.6). 52 ... |

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Citation Context ...nerated L set by set with a transfinite recursion in terms of eight elementary set generators, a sort of Gsodel numbering into the transfinite. These generators were based on Bernays's axiomatization =-=[1937]-=- which itself was based on von Neumann's axiomatization [1925, 1928]. Providing a rigorous formalization of his metamathematical construction Gsodel now emphasized how it yields a class model and a fi... |

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Citation Context ...ndation, though this was not his main purpose. In its formality and purpose this work was a precursor of Gsodel's construction of L (see 3.4). 63. Shoenfield [1967, 238#.][1977], Wang [1974a], Boolos =-=[1971]-=-, and Scott [1974] motivate the axioms of set theory in terms of an iterative concept of set based on stages of construction. Parsons [1977] raises issues about this approach. Potter [1990] is a histo... |

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Citation Context ...a reworking [1964] of the independence of CH. The concept was generalized and simplified in a series of papers on the so-called #-models from the active Prague seminar founded by Vop enka (see H ajek =-=[1971, 78]-=-), culminating in the exposition Vop enka [1967]. However, the earlier papers did not have much impact, partly because of an involved formalism in which formulas were valued in a complete lattice rath... |

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Citation Context ... the stress brought on by the lack of firm ground led Brouwer [1911] to definitively establish the invariance of dimension in a seminal paper for algebraic topology. 12. This is emphasized by Hallett =-=[1984] as Cantor-=-'s "finitism." 48 AKIHIRO KANAMORI 13. After describing the similarity between # and # 2 as limits of sequences, Cantor [1887, 99] interestingly correlated the creation of the transfinite nu... |

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Citation Context ...existential." See also Moore [1982, 39]. One exception to the misleading trend is Fraenkel [1930, 237][1953, 75], who from the beginning emphasized the constructive aspect of diagonalization. 9. =-=Gray [1994]-=- shows that Cantor's original [1874] argument can be implemented by an algorithm that generates n digits of a transcendental number with time complexity O(2 n 1/3 ), and his later diagonal argument, w... |

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Citation Context ...s of later developments, Zermelo's axioms had the advantages of schematic simplicity and open-endedness. The generative set formation axioms, especially Power Set and Union, were to lead to Zermelo's =-=[1930]-=- cumulative hierarchy picture of sets, and the vagueness of the definit property in the Separation Axiom was to invite Thoralf Skolem's [1923] proposal to base it on first-order logic, enforcing exten... |

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Citation Context ...ball. Raphael Robinson [1947] later showed that there is such a decomposition into just five pieces with one of them containing a single point, and moreover that five is the minimal number. See Wagon =-=[1985]-=- for more on these and similar results; they stimulated interesting developments in measure theory that, rather than casting doubt on AC, embedded it further into mathematical practice (cf., 2.6). 52 ... |

25 |
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Citation Context ...y being specified by positing a large cardinal. 92 Forcing quickly led to the conclusion that there could be no direct implication for CH: Levy and Solovay (Levy [1964], Solovay [1965a], Levy-Solovay =-=[1967]-=-) established that measurable cardinals neither imply nor refute CH, with an argument generalizable to most inaccessible large cardinals. Rather, the subsumption for many other propositions would be i... |

25 |
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Citation Context ...ide excellent illustrations of two vastly di#erent approaches toward proving the existence of mathematical objects. Liouville's is purely constructive; Cantor's is purely existential." See also M=-=oore [1982, 39]-=-. One exception to the misleading trend is Fraenkel [1930, 237][1953, 75], who from the beginning emphasized the constructive aspect of diagonalization. 9. Gray [1994] shows that Cantor's original [18... |

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Citation Context ...L DEVELOPMENT OF SET THEORY FROM CANTOR TO COHEN 17 ordinal numbers and hence his function hierarchy as merely une facon de parler, and continued to view infinite concepts only in potentiality. Borel =-=[1898]-=- took a pragmatic approach and seemed to accept the countable ordinal numbers. Lebesgue was more equivocal but still accepting; recalling Cantor's early attitude Lebesgue regarded the ordinal numbers ... |

22 |
dass jede Menge wohlgeordnet werden kann (Aus einem an Herrn Hilbert gerichteten Briefe
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Citation Context ...monplace in mathematics, but THE MATHEMATICAL DEVELOPMENT OF SET THEORY FROM CANTOR TO COHEN 49 several authors at the time referred specifically to Cantor's concept of covering, most notably Zermelo =-=[1904]. Jou-=-rdain in his introduction to his English translation [1915, 82] of the Beitr age wrote: "The introduction of the concept of `covering' is the most striking advance in the principles of the theory... |

21 |
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Citation Context ... used AC to provide what is now known as Hausdor#'s Paradox, an implausible decomposition of the sphere and the source of the better known Banach-Tarski Paradox from Stefan Banach and Alfred Tarski's =-=[1924]-=-. 46 Hausdor#'s Paradox was the first, and a dramatic, synthesis of classical mathematics and the Zermelian abstract view. Hausdor#'s reduction of functions through a defined ordered pair highlights t... |

21 |
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Citation Context ... not to a restriction or mutilation but rather to a presently unsurveyable unfolding and enrichment, of mathematical science." 65. See Kanamori [1994, Chapter 5]. 54 AKIHIRO KANAMORI 66. See Gold=-=farb [1979]-=- and Moore [1988b] for more on the emergence of first-order logic. 67. The historical development is clarified by the fact that while this book was published in light of the developments of the 1920's... |

21 |
Mathematische Probleme, Vortrag, gehalten auf dem internationalen Mathematiker-Kongreß zu Paris
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Citation Context ...uncovered by his diagonal argument between well-ordering and arbitrary functions (and hence power set), remains central to set theory as the main source of its vitality and fascination. David Hilbert =-=[1900]-=- when he presented his famous list of 23 problems at the 1900 International Congress of Mathematicians in Paris made establishment of CH the very first problem and pointed out Cantor's main di#culty b... |

20 |
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Citation Context ...forth on its independent course as a distinctive field of mathematics was its full extensionalization in first-order logic. 66 However influential Zermelo's [1930] and despite his subsequent advocacy =-=[1931, 1935]-=- of infinitary logic, his e#orts to forestall Skolem were not to succeed, as stronger currents were at work in the direction of first-order formalization. Hilbert e#ected a basic shift in the developm... |

18 |
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Citation Context ... years Gsodel speculated about the possibility of deciding propositions like CH with large cardinal hypotheses based on the heuristics of reflection and generalization. In 1946 remarks he (see Gsodel =-=[1990, 151]) suggested the-=- consideration of "stronger and stronger axioms of infinity," and reflection as follows: "Any proof for a set-theoretic theorem in the next higher system above set theory (i.e., any pro... |

17 |
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Citation Context ... a clear account of Zermelo's first proof [1904] of the Well-Ordering Theorem, Hausdor# (p. 140#.) emphasized its maximality aspect by giving synoptic versions of Zorn's Lemma two decades before Zorn =-=[1935]-=-, one of them now known as Hausdor#'s Maximality Principle. 45 Also, Hausdor# (p. 304) provided the now standard account of the Borel hierarchy of sets, with the still persistent F # and G # notation.... |

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Citation Context ...s 1 to rationals and 0 to irrationals is in class 2 and also observed with a non-constructive appeal to Cantor's cardinality argument that there are real functions that are not Baire. 41. See Hawkins =-=[1975]-=- for more on the development of Lebesgue measurability. See Oxtoby [1971] for an account of category and measure in juxtaposition. 42. See Moore [1982, 2.3]. 43. See Kanamori [1994, #] for more on lar... |

15 |
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Citation Context ... M is again L. Because of this Shepherdson [1953] noted that the relative consistency of hypotheses like the negation of CH cannot be established via inner models. 38 AKIHIRO KANAMORI However, Hajnal =-=[1956, 1961]-=- and Azriel Levy [1957, 1960] developed generalizations of L that were to become basic in a richer setting. For a set A, Hajnal formulated the constructible closure L(A) of A, i.e., the smallest inner... |

13 |
The Boolean prime ideal theorem does not imply the axiom of choice
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Citation Context ...el in which AC holds. The basic results about this inner model were to be rediscovered several times, and the formal definition turns on some form of the Reflection Principle for ZF (see Myhill-Scott =-=[1971, 278]-=-). In these several ways reflection phenomena both as heuristic and as principle became incorporated into set theory, bringing to the forefront what was to become a basic feature of the study of well-... |

13 | Sur la notion de l’ordre dans la théorie des ensembles. Fundamenta Mathematicae 2, 161–171 - Kuratowski - 1921 |

13 |
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Citation Context ... the stress brought on by the lack of firm ground led Brouwer [1911] to definitively establish the invariance of dimension in a seminal paper for algebraic topology. 12. This is emphasized by Hallett =-=[1984] as Cantor-=-'s "finitism." 48 AKIHIRO KANAMORI 13. After describing the similarity between # and # 2 as limits of sequences, Cantor [1887, 99] interestingly correlated the creation of the transfinite nu... |

12 |
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Citation Context ...[1943] (cf., 3.5) in combinatorial formulations that was later seen to imply that a strongly compact cardinal is measurable, and a measurable cardinal is weakly compact. Tarski's student William Hanf =-=[1964]-=- then established, using the satisfaction relation for infinitary languages, that there are many inaccessible cardinals (and Mahlo cardinals) below a weakly compact cardinal. A fortiori, (Tarski [1962... |

12 |
A remark on method in transfinite algebra
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Citation Context ... a clear account of Zermelo's first proof [1904] of the Well-Ordering Theorem, Hausdor# (p. 140#.) emphasized its maximality aspect by giving synoptic versions of Zorn's Lemma two decades before Zorn =-=[1935]-=-, one of them now known as Hausdor#'s Maximality Principle. 45 Also, Hausdor# (p. 304) provided the now standard account of the Borel hierarchy of sets, with the still persistent F # and G # notation.... |

11 |
Über die Unabhängigkeit des Wohlordnungssatzes vom
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Citation Context ...but it needed the articulation that would come with the formalization of the satisfaction relation. In the late 1930's Lindenbaum and Andrzej Mostowski so cast and extended Fraenkel's work. Mostowski =-=[1939]-=- forged a method according to post-Gsodelian sensibilities, bringing out the importance of groups of permutations leaving various urelements fixed, and the resulting models as well as later versions a... |

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10 |
zur Begründung der transfiniten Mengenlehre
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Citation Context ...erizing their cardinal number; unfortunately, he defined his ordered pair using his famously inconsistent Basic Law V. Peirce [1883] used i and j schematically to denote the components, and Schrsoder =-=[1895, 24]-=- adopted this and also introduced i : j. Peano [1897, 579] introduced (x, y) at the outset and regarded it as fundamental, switching to x; y in later writings. 48. Whitehead and Russell had first defi... |