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## THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI (2002)

### Citations

1169 | Word and Object - Quine - 1960 |

547 | Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38 - Gödel - 1931 |

128 |
Parts of Classes
- Lewis
- 1991
(Show Context)
Citation Context ... of their more metaphysical preoccupations, the approach on the mathematical side was directed toward an increasingly extensional view of functions regarded as arbitrary correspondences. 25 See Lewis =-=[1991]-=-.sTHE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR 289 Frege [1891] had two fundamental categories, function and object, with a function being “unsaturated” and supplemented by objects as arguments.... |

110 |
Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. d
- Frege
- 1879
(Show Context)
Citation Context ...in the passages immediately following the oft-quoted definition of set [Menge]. The first to draw the inclusion vs. membership distinction generally in logic was Frege. Indeed, in his Begriffsschrift =-=[1879]-=- the distinction is manifest for concepts, and in his Grundlagen [1884] and elsewhere he emphasized the distinction in terms of “subordination” and “falling under a concept.” The a vs. {a} distinction... |

57 |
1889) Arithmetices Principia—Nova Methodo Exposita
- Peano
(Show Context)
Citation Context ...d already lodged his criticism in the Grundgesetze [1893:2–3] and written that in Husserl’s review “the problems are not solved.”sTHE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR 275 Giuseppe Peano =-=[1889]-=-, the first to axiomatize mathematics in a symbolic language, used “Λ” to denote both the falsity of propositions (Part II of Logical Notations) and the null class (Part IV). He later [1897] provided ... |

56 |
Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die Reine und Angewandte Mathematik 77:258–262
- Cantor
(Show Context)
Citation Context ... the basic distinction between inclusion, ⊆, and membership, ∈, a distinction without which abstract set theory could not develop. Set theory as a field of mathematics began, of course, with Cantor’s =-=[1874]-=- result that the reals are uncountable. But it was only much later in [1891] that Cantor gave his now famous diagonal proof, showing in effect that for any set X the collection of functions from X int... |

56 |
Was sind und was sollen die Zahlen
- Dedekind
(Show Context)
Citation Context ...aking place between the two great pioneers of set theory, years after their foundational work on arithmetic and on the transfinite! The following is the preface to the third, 1911 edition of Dedekind =-=[1888]-=-, with some words italicized for emphasis: 11 When I was asked roughly eight years ago to replace the second edition of this work (which was already out of print) by a third, I had misgivings about do... |

53 |
Vorlesungen uber die Algebra der Logik (exakte Logik), volume 3, part 1: Algebra und Logik der Relative
- Schröder
- 1895
(Show Context)
Citation Context ...ing’ and ‘universe,’ which as ‘classes’ should be understood to comprise respectively ‘no beings,’ ‘all beings.’ ” Note how Boole had already entertained the singleton. 2 Edmund Husserl in his review =-=[1891]-=- of Schröder’s [1890] also criticized Schröder along similar lines. Frege had already lodged his criticism in the Grundgesetze [1893:2–3] and written that in Husserl’s review “the problems are not sol... |

50 | The Mathematical Analysis of Logic, Being an Essay Toward a Calculus of Deductive Reasoning. London and Cambridge - Boole |

37 |
Axiomatic Set Theory
- Bernays
- 1958
(Show Context)
Citation Context ...ne, and is arguably a significant reflection of his philosophy. Already in [1936:50] Quine had provided a reinterpretation of a type theory that associated individuals with their unit classes. In his =-=[1937]-=- Quine formulated a set theory now known as New Foundations (NF), and in his book Mathematical Logic [1940] he extended the theory to include classes. 23 In §22 of [1940] Quine extended the “x ∈ y” no... |

37 | Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics - Ferreir’os - 1999 |

34 |
Über Grenzzahlen und Mengenbereiche
- Zermelo
- 1930
(Show Context)
Citation Context ... theory began. The generative view of sets based on the iteration of the “set of” {} operation would be further codified by the assumption of the Axiom of Foundation in Zermelo’s final axiomatization =-=[1930]-=-, the source of the nowstandard theory ZFC. Foundation disallows a ∈ a, and hence precludes {a} = a, for any set a. In any case, having any sets a such that {a} = a is immediately antithetical to the ... |

33 | Zermelo’s Axiom of Choice: Its origins, development and influence - Moore - 1982 |

32 | Heijenoort, From Frege to Gödel, A Source Book - van - 1967 |

32 |
daß jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59:514–516
- Beweis
- 1904
(Show Context)
Citation Context ...aw Zermelo at Göttingen make his major advances in the development of set theory. 20 His first substantial result in set theory was his independent discovery of Russell’s Paradox. He then established =-=[1904]-=- the Well-Ordering Theorem, provoking an open controversy about this initial use of the Axiom of Choice. After providing a second proof [1908] of the Well-Ordering Theorem in response, Zermelo also pr... |

24 |
Eine axiomatisierung der mengenlehre. Journal für die reine und angewandte
- Neumann
- 1925
(Show Context)
Citation Context ...als” Dana Scott [1962] showed how to transform models of NF into models having a satisfying {a} = a. Moreover, his technique has led to a spate of recent results. 24 23 As in von Neumann’s set theory =-=[1925]-=- as recast by Bernays [1937], in Quine’s system only some classes are membership-eligible, i.e., capable of appearing to the left of the membership relation, and these “elements” correspond to the set... |

21 |
Eléments de Mathématique, I. Théorie des Ensembles, Fascicule de Résultats, Actualités Scientifiques et Industrielles
- Bourbaki
- 1939
(Show Context)
Citation Context ...ORI recasting of von Neumann’s system, Bernays [1937:68] also acknowledged Kuratowski [1921] and began with its definition for the ordered pair. It is remarkable that Nicolas Bourbaki in his treatise =-=[1954]-=- on set theory still took the ordered pair as primitive, only later providing the Kuratowski reduction in the [1970] edition. 35 As with the singleton, we end here with a return to philosophy through ... |

19 |
Sur la notion de l’ordre dans la théorie des ensembles, Fund
- Kuratowski
- 1921
(Show Context)
Citation Context ...null set yet having no members and capable of belonging to sets. Zermelo [1908a][1930] had conceived of set theory as a theory of collections built on a base of individuals. However, Abraham Fraenkel =-=[1921]-=-[1922:234ff] from the beginning of his articles on set theory emphasized that there is no need for individuals and generally advocated a minimalist approach as articulated by his Axiom of Restriction ... |

15 |
Set theory with a universal set
- Forster
- 1992
(Show Context)
Citation Context ...recast Cantor’s diagonal argument in terms of classes, 18 a formulation synoptic to the usual set-theoretic one about the power set. Cantor’s 15 See Grattan-Guinness [1978], Garciadiego [1992], Moore =-=[1995]-=- and Kanamori [1997] for more on the evolution of Russell’s Paradox. 16 Russell [1903:129] wrote: “The distinction of philosophy and mathematics is broadly one of point of view: mathematics is constru... |

13 |
1912-14]: 'A Simplification of the Logic of Relations
- WIENER
(Show Context)
Citation Context ... of ordered pairs, and a function extensionally as a kind of relation, referring to the final version of his Formulario Mathematico [1905–8:73ff] as the source. Capping this to and fro Norbert Wiener =-=[1914]-=- provided a definition of the ordered pair in terms of unordered pairs of classes only, thereby reducing relations to classes. Working in Russell’s theory of types, Wiener defined the ordered pair 〈x,... |

12 |
zur Begründung der transfiniten Mengenlehre
- Beiträge
(Show Context)
Citation Context ...he extravagant definition that the ordered pair of x and y is that class to which all and only the extensions of relations to which x stands to y belong. 26 On the other hand, Peirce [1883], Schröder =-=[1895]-=-, and Peano [1897] essentially regarded a relation from the outset as just a collection of ordered pairs. 27 Whereas Frege was attempting an analysis of thought, Peano was mainly concerned about recas... |

12 |
Zur Axiomatik der Mengenlehre (Fundierungs– und Auswahlaxiome), Zeitschrift für Math .Log
- Specker
- 1957
(Show Context)
Citation Context ...ed the independence of the Axiom of Foundation from the other axioms of his system and provided [1954:83] a proof based on having a’s such that {a} = a. Ernst Specker in his Habilitationsschrift (cf. =-=[1957]-=-) also provided such a proof. Moreover, he coordinated such a’s playing the role of individuals in his refinement of the Fraenkel-Mostowski method for deriving independence results related to the Axio... |

11 | From Kant to Hilbert: A source book in the foundations of mathematics - Ewald - 1996 |

9 | unendliche, lineare Punktmannigfaltigkeiten V [On infinite, linear point-manifolds (sets - Über |

8 | The mathematical import of Zermelo’s well-ordering theorem - Kanamori - 1997 |

8 |
Ich habe mich wohl gehütet, alle Patronen auf einmal zu verschießen’. Ernst Zermelo
- Peckhaus
- 1990
(Show Context)
Citation Context ...an P(M) �⊆ M, but the connection between subsets and characteristic functions was hardly appreciated then, and Zermelo was just making the first moves toward his abstract view of sets. 22 20 Peckhaus =-=[1990]-=- provides a detailed account of Zermelo’s years 1897–1910 at Göttingen. 21 See Rang-Thomas [1981]. 22 In Zermelo’s axiomatization paper [1908a], the first result of his axiomatic theory was just the r... |

7 |
Über die Zermelosche Begründung der Mengenlehre, abstract
- Fraenkel
- 1921
(Show Context)
Citation Context ...null set yet having no members and capable of belonging to sets. Zermelo [1908a][1930] had conceived of set theory as a theory of collections built on a base of individuals. However, Abraham Fraenkel =-=[1921]-=-[1922:234ff] from the beginning of his articles on set theory emphasized that there is no need for individuals and generally advocated a minimalist approach as articulated by his Axiom of Restriction ... |

7 |
Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time
- Hamilton
(Show Context)
Citation Context ...in the study of analytic geometry. However, to focus the historical background it should be noted that ordered pairs are not explicit in Descartes and in the early work on analytic geometry. Hamilton =-=[1837]-=- may have been the first to objectify ordered pairs in his reconstrual of the complex numbers as ordered “couples” of real numbers. In logic, the ordered pair is fundamental to the logic of relations ... |

6 |
Eine elementare frage der mannigfaltigkeitslehre, Jahresberich der Deutschen
- Über
(Show Context)
Citation Context ...ing’ and ‘universe,’ which as ‘classes’ should be understood to comprise respectively ‘no beings,’ ‘all beings.’ ” Note how Boole had already entertained the singleton. 2 Edmund Husserl in his review =-=[1891]-=- of Schröder’s [1890] also criticized Schröder along similar lines. Frege had already lodged his criticism in the Grundgesetze [1893:2–3] and written that in Husserl’s review “the problems are not sol... |

5 |
die von drei Moduln erzeugte Dualgruppe
- Über
- 1900
(Show Context)
Citation Context ...f the ‘essence’ of the cardinal number to philosophy.” 32 Before Hausdorff and going beyond Cantor, Dedekind was first to consider non-linear orderings, e.g., in his remarkably early, axiomatic study =-=[1900]-=- of lattices. 33 As to historical precedence, Wiener’s note was communicated to the Cambridge Philosophical Society, presented on 23 February 1914, while the preface to Hausdorff’s book is dated 15 Ma... |

5 | scientiWc papers - Selected - 1991 |

5 |
Sur la logique des relations avec des applications à la théorie des séries, Revue de mathématiques/Rivista di Matematiche
- Russell
- 1901
(Show Context)
Citation Context ...nal Congress of Philosophy in Paris.” There in August he met Peano and embraced his symbolic logic, particularly his use of different signs for inclusion and membership. During September Russell (see =-=[1901]-=-) extended Peano’s symbolic approach to the logic of relations. Armed with the new insights, in the rest of the year Russell completed most of the final draft of The Principles of Mathematics [1903], ... |

5 | Philosophical Development - My - 1959 |

5 |
Quine’s individuals,” in Logic, methodology and philosophy of science
- Scott
- 1962
(Show Context)
Citation Context ...s Logic Quine [1964:§4] carried out his stipulatory approach and emphasized how it leads to the seamless incorporation of individuals. Stimulated by Quine’s discussion of his “individuals” Dana Scott =-=[1962]-=- showed how to transform models of NF into models having a satisfying {a} = a. Moreover, his technique has led to a spate of recent results. 24 23 As in von Neumann’s set theory [1925] as recast by Be... |

5 |
über die Grundlagen der Mengenlehre', Mathematische Annalen 65
- `Untersuchungen
- 1908
(Show Context)
Citation Context ... as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo =-=[1908a]-=- and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’ {} operation. So it is surprising that, while the... |

4 |
Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung uber den Begri# der Zahl, Wilhelm K obner, Breslau, translated with German text by John L. Austin, as The foundations of arithmetic, a logico-mathematical enquiry into the concept
- Die
- 1950
(Show Context)
Citation Context ...spired to the logical analysis of mathematics rather than the mathematical analysis of logic, and what in effect is the null class played a key role in his analysis of number. Frege in his Grundlagen =-=[1884]-=- eschewed the terms “set” [“Menge”] and “class” [“Klasse”], but in any case the extension of the concept “not identical with itself” was key to his definition of zero as a logical object. Schröder, in... |

4 |
on the Logics of Russell and Schröder: An Account of His Doctoral Thesis, and of His Discussion of It with Russell”, Annals of Science 32
- “Wiener
- 1975
(Show Context)
Citation Context ...ould itself have to be given sense, which would be a circular or an inadequate exercise, and “It seems therefore more correct to take an intensional view of relations . . . ”. 30 See Grattan-Guinness =-=[1975]-=- for more on Wiener’s work and his interaction with Russell. 31 Hausdorff’s mathematical attitude is reflected in a remark following his explanation of cardinal number in a revised edition [1937:§5] o... |

4 |
Grundzüge der Mengenlehre, W. de Gruyter
- Hausdorff
- 1914
(Show Context)
Citation Context ... of ordered pairs, and a function extensionally as a kind of relation, referring to the final version of his Formulario Mathematico [1905–8:73ff] as the source. Capping this to and fro Norbert Wiener =-=[1914]-=- provided a definition of the ordered pair in terms of unordered pairs of classes only, thereby reducing relations to classes. Working in Russell’s theory of types, Wiener defined the ordered pair 〈x,... |

4 |
Definition by induction in Frege’s Grundgesetze der Arithmetik
- Heck
- 1995
(Show Context)
Citation Context ...recast Cantor’s diagonal argument in terms of classes, 18 a formulation synoptic to the usual set-theoretic one about the power set. Cantor’s 15 See Grattan-Guinness [1978], Garciadiego [1992], Moore =-=[1995]-=- and Kanamori [1997] for more on the evolution of Russell’s Paradox. 16 Russell [1903:129] wrote: “The distinction of philosophy and mathematics is broadly one of point of view: mathematics is constru... |

4 | The Ignorance of Bourbaki
- Mathias
- 1992
(Show Context)
Citation Context ...informatively recast Cantor’s diagonal argument in terms of classes, 18 a formulation synoptic to the usual set-theoretic one about the power set. Cantor’s 15 See Grattan-Guinness [1978], Garciadiego =-=[1992]-=-, Moore [1995] and Kanamori [1997] for more on the evolution of Russell’s Paradox. 16 Russell [1903:129] wrote: “The distinction of philosophy and mathematics is broadly one of point of view: mathemat... |

4 |
Zermelo’s Discovery of the “Russell Paradox
- Rang, Thomas
- 1981
(Show Context)
Citation Context ...ted then, and Zermelo was just making the first moves toward his abstract view of sets. 22 20 Peckhaus [1990] provides a detailed account of Zermelo’s years 1897–1910 at Göttingen. 21 See Rang-Thomas =-=[1981]-=-. 22 In Zermelo’s axiomatization paper [1908a], the first result of his axiomatic theory was just the result in the Husserl note, that every set M has a subset { x ∈ M | x /∈ x } not a member of M, wi... |

4 |
logic as based on the theory of types
- Mathematical
- 1967
(Show Context)
Citation Context ...he singleton problem in conversation, briefly with Felix Bernstein on 13 June 1897 and then with Cantor himself on 4 September 1899 pointing out the contradiction. 10 It is hard to imagine such a his =-=[1908]-=- on he used �‘a. Frege [1893:§11] wrote a boldface backslash before a to denote its unit class. It was Zermelo [1908a:262] who introduced the now familiar use of {a}, having written just before: “The ... |

4 | eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre, Journal für die reine und angewandte - Über - 1929 |

3 |
den Grundlagen der Cantor-Zermeloschen Mengenlehre
- Zu
- 1922
(Show Context)
Citation Context ...ative hierarchy view based on the empty set provided a uniform, set-theoretic universe. Nevertheless, individuals have continued to be used for various adaptations of the set-theoretic view. Fraenkel =-=[1922]-=- himself, in the Hilbertian axiomatic tradition and notwithstanding his own Axiom of Restriction, concocted a domain of individuals to argue for the independence of the Axiom of Choice, in the inaugur... |

3 |
Beleuchtung einiger Punkte in E. Schroders Vorlesungen uber die Algebra der Logik, Archiv f ur systematische Philosophie
- Kritische
(Show Context)
Citation Context ...ject. Schröder, in the first volume [1890] of his major work on the algebra of logic, held a traditional view that a class is merely a collection of objects, without the {} so to speak. In his review =-=[1895]-=- of Schröder’s [1890], Frege argued that Schröder cannot both maintain this view of classes and assert that there is a null class, since the null class contains no objects. 2 For Frege, logic enters i... |

3 |
Russell discovered his paradox
- Bertrand
(Show Context)
Citation Context ...n hand, Russell had informatively recast Cantor’s diagonal argument in terms of classes, 18 a formulation synoptic to the usual set-theoretic one about the power set. Cantor’s 15 See Grattan-Guinness =-=[1978]-=-, Garciadiego [1992], Moore [1995] and Kanamori [1997] for more on the evolution of Russell’s Paradox. 16 Russell [1903:129] wrote: “The distinction of philosophy and mathematics is broadly one of poi... |

3 |
Nine Letters from Giuseppe Peano to Bertrand Russell”, History and Philosophy of Logic 13
- KENNEDY
- 1975
(Show Context)
Citation Context ...ould itself have to be given sense, which would be a circular or an inadequate exercise, and “It seems therefore more correct to take an intensional view of relations . . . ”. 30 See Grattan-Guinness =-=[1975]-=- for more on Wiener’s work and his interaction with Russell. 31 Hausdorff’s mathematical attitude is reflected in a remark following his explanation of cardinal number in a revised edition [1937:§5] o... |

3 |
A theory of probable inference. Note B. The logic of relatives, Studies in Logic by Members of the
- Peirce
(Show Context)
Citation Context ...:§144] provided the extravagant definition that the ordered pair of x and y is that class to which all and only the extensions of relations to which x stands to y belong. 26 On the other hand, Peirce =-=[1883]-=-, Schröder [1895], and Peano [1897] essentially regarded a relation from the outset as just a collection of ordered pairs. 27 Whereas Frege was attempting an analysis of thought, Peano was mainly conc... |

3 | A System of Logistic - Quine - 1934 |

3 | les ensembles définissables de nombres réels, Fund - Tarski, Sur - 1931 |

3 | die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre - Uber |

3 |
Beweis fur die Moglichkeit einer Wohlordnung
- Neuer
(Show Context)
Citation Context ...he singleton problem in conversation, briefly with Felix Bernstein on 13 June 1897 and then with Cantor himself on 4 September 1899 pointing out the contradiction. 10 It is hard to imagine such a his =-=[1908]-=- on he used �‘a. Frege [1893:§11] wrote a boldface backslash before a to denote its unit class. It was Zermelo [1908a:262] who introduced the now familiar use of {a}, having written just before: “The ... |

2 |
definizione di funzione, Atti della Accademia nazionale dei Lincei, Rendiconti, Classe di scienze fisiche, matematiche e naturali
- Sulla
(Show Context)
Citation Context ...t of view. They [1910:∗56] used their ordered pair initially to define the ordinal number 2.s290 AKIHIRO KANAMORI he overlooked this shortcoming in Peano. 29 Commenting obliviously on Principia Peano =-=[1911,1913]-=- simply reaffirmed an ordered pair as basic, defined a relation as a class of ordered pairs, and a function extensionally as a kind of relation, referring to the final version of his Formulario Mathem... |

2 | works of Giuseppe Peano - Selected |

2 |
logic
- Mathematical
(Show Context)
Citation Context ...einterpretation of a type theory that associated individuals with their unit classes. In his [1937] Quine formulated a set theory now known as New Foundations (NF), and in his book Mathematical Logic =-=[1940]-=- he extended the theory to include classes. 23 In §22 of [1940] Quine extended the “x ∈ y” notation formally to individuals (“non-classes”) y by stipulating for such y that “x ∈ y” should devolve to “... |

2 |
uber die Algebra der Logik (exakte Logik). Vol. 3: Algebra und Logik der Relative
- Vorlesungen
(Show Context)
Citation Context ...ject. Schröder, in the first volume [1890] of his major work on the algebra of logic, held a traditional view that a class is merely a collection of objects, without the {} so to speak. In his review =-=[1895]-=- of Schröder’s [1890], Frege argued that Schröder cannot both maintain this view of classes and assert that there is a null class, since the null class contains no objects. 2 For Frege, logic enters i... |

2 |
Appartenance et inclusion: un inedit de Richard Dedekind, Revue d'Histoire des Sciences et de leurs Applications
- Sinaceur
(Show Context)
Citation Context ...stems? Is a system a subjective figure in the individual soul? In that case is the constellation Orion a system? And what are its elements? The stars, or the molecules, or the atoms?” 10 See Sinaceur =-=[1971]-=-. In the note Dedekind proposed various emendations to his essay to clarify the situation. Undercutting the exclusion of the empty set [Nullsystem] in thesTHE EMPTY SET, THE SINGLETON, AND THE ORDERED... |

1 | to the founding of the theory of transfinite numbers, including translations of Cantor [1895] and Cantor [1897] above with introduction and notes by - Contributions - 1965 |

1 |
The null class nullified
- Carmichael
(Show Context)
Citation Context ...ts cannot be carried out without drawing the basic distinction between ⊆, inclusion, and ∈, membership. Surprisingly, neither this distinction nor the related distinction 4 See for example Carmichael =-=[1943]-=- [1943a]. 5 Something of the need of a place holder is illustrated by the use of {} in the computer typesetting program TEX. To render � 2, {} must be put in as a something to which � serves as a supe... |

1 |
on the theory of numbers (Wooster W
- Essays
- 1901
(Show Context)
Citation Context ...en in the 1890’s did introduce the empty set, denoting it by “0”, and used “[a]” for the singleton of a (see Dedekind [1932:450–60]). 11 cf. Ewald [1996:796]. This preface does not appear in Dedekind =-=[1963]-=-, which is an English translation of the second edition of Dedekind [1888]. 12 Actually, the sign appearing in Peano [1889] is only typographically similar to our “∈”, and only in Peano [1889a] was th... |

1 | der Arithmetik, Begriffsschriftlich abgeleitet - Grundgesetze - 1962 |

1 |
and the origins of the set-theoretic “paradoxes
- Garciadiego, Russell
(Show Context)
Citation Context ...informatively recast Cantor’s diagonal argument in terms of classes, 18 a formulation synoptic to the usual set-theoretic one about the power set. Cantor’s 15 See Grattan-Guinness [1978], Garciadiego =-=[1992]-=-, Moore [1995] and Kanamori [1997] for more on the evolution of Russell’s Paradox. 16 Russell [1903:129] wrote: “The distinction of philosophy and mathematics is broadly one of point of view: mathemat... |

1 |
third, revised edition of Hausdorff [1914]; translated by John R. Auman as Set theory
- Mengenlehre
- 1962
(Show Context)
Citation Context ...ne, and is arguably a significant reflection of his philosophy. Already in [1936:50] Quine had provided a reinterpretation of a type theory that associated individuals with their unit classes. In his =-=[1937]-=- Quine formulated a set theory now known as New Foundations (NF), and in his book Mathematical Logic [1940] he extended the theory to include classes. 23 In §22 of [1940] Quine extended the “x ∈ y” no... |

1 | 85–97; translated in Peano [1973] below - pp |

1 |
principii di geometrica logicamente espositi
- unknown authors
(Show Context)
Citation Context ...edekind [1963], which is an English translation of the second edition of Dedekind [1888]. 12 Actually, the sign appearing in Peano [1889] is only typographically similar to our “∈”, and only in Peano =-=[1889a]-=- was the Greek noted. The epsilon “ε” began to be used from then on, and in Peano [1891:fn. 8] the ’ε��� connection was made explicit.s280 AKIHIRO KANAMORI 56, which in modern notation is: k ⊆ s → (k ... |

1 |
de l’intégrabilité des équations différentielles ordinaires
- Démonstration
(Show Context)
Citation Context ...s “set” [“Menge”] and “class” [“Klasse”], but in any case the extension of the concept “not identical with itself” was key to his definition of zero as a logical object. Schröder, in the first volume =-=[1890]-=- of his major work on the algebra of logic, held a traditional view that a class is merely a collection of objects, without the {} so to speak. In his review [1895] of Schröder’s [1890], Frege argued ... |

1 |
di logica matematica, Rivista di matematica
- Principii
(Show Context)
Citation Context ...ing’ and ‘universe,’ which as ‘classes’ should be understood to comprise respectively ‘no beings,’ ‘all beings.’ ” Note how Boole had already entertained the singleton. 2 Edmund Husserl in his review =-=[1891]-=- of Schröder’s [1890] also criticized Schröder along similar lines. Frege had already lodged his criticism in the Grundgesetze [1893:2–3] and written that in Husserl’s review “the problems are not sol... |

1 | de logique mathématique (introduction au formulaire de mathématiques - Notations |

1 |
di logica matematica, Atti della Accademia delle Scienze di Torino, Classe di Scienze Fisiche
- Studii
(Show Context)
Citation Context ...ppe Peano [1889], the first to axiomatize mathematics in a symbolic language, used “Λ” to denote both the falsity of propositions (Part II of Logical Notations) and the null class (Part IV). He later =-=[1897]-=- provided a definition of the null class as the intersection of all classes, making it more explicit that there is exactly one null class. More importantly, [1897] had the first occurrence of “∃”, use... |

1 |
Bollettino di bibliografia e storia delle scienze matematiche
- unknown authors
(Show Context)
Citation Context ...t of view. They [1910:∗56] used their ordered pair initially to define the ordinal number 2.s290 AKIHIRO KANAMORI he overlooked this shortcoming in Peano. 29 Commenting obliviously on Principia Peano =-=[1911,1913]-=- simply reaffirmed an ordered pair as basic, defined a relation as a class of ordered pairs, and a function extensionally as a kind of relation, referring to the final version of his Formulario Mathem... |

1 |
foundations for mathematical logic
- New
- 1937
(Show Context)
Citation Context ...ne, and is arguably a significant reflection of his philosophy. Already in [1936:50] Quine had provided a reinterpretation of a type theory that associated individuals with their unit classes. In his =-=[1937]-=- Quine formulated a set theory now known as New Foundations (NF), and in his book Mathematical Logic [1940] he extended the theory to include classes. 23 In §22 of [1940] Quine extended the “x ∈ y” no... |

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theory and its logic
- Set
- 1969
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Citation Context ...en in the 1890’s did introduce the empty set, denoting it by “0”, and used “[a]” for the singleton of a (see Dedekind [1932:450–60]). 11 cf. Ewald [1996:796]. This preface does not appear in Dedekind =-=[1963]-=-, which is an English translation of the second edition of Dedekind [1888]. 12 Actually, the sign appearing in Peano [1889] is only typographically similar to our “∈”, and only in Peano [1889a] was th... |

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mental development,The philosophy of
- My
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Citation Context ...was what spurred Bertrand Russell to his main achievements. §3. Russell. The turn of the century saw Russell make the major advances in the development of his mathematical logic. As he later wrote in =-=[1944]-=-: “The most important year in my intellectual life was the year 1900, and the most important event in this year was my visit to the International Congress of Philosophy in Paris.” There in August he m... |

1 | über die Algebra der Logik, three volumes - Vorlesungen |