Introduction to the foundations of mathematics / Raymond L. Wilder.
Editor: New York : Wiley, c1952Descripción: xiv, 305 p. ; 24 cmOtra clasificación: 03B30PART I • FUNDAMENTAL CONCEPTS AND METHODS OF MATHEMATICS T • The Axiomatic Method [3] 1 • Evolution of the method [3] 2 • Description of the method; the undefined terms and axioms [9] 3 • Description of the method; the proving of theorems [13] 4 • Comment on the above theorems and proofs [17] 5 • Source of the axioms [19] Suggested reading [21] Problems [21] Tl • Analysis of the Axiomatic Method [23] 1 • Consistency of an axiom system [23] 2 • The proof of consistency of an axiom system [26] 3 • Independence of axioms [29] 4 • Completeness of an axiom system [32] 5 • The logical basis [40] 6 • Miscellaneous comment [41] 7 • Axioms for simple order [44] 8 • Axioms defining equivalence [46] Suggested reading [49] Problems [49] III * Theory of Sets [52] 1 • Background of the theory [53] 2 • The Russell contradiction [55] 3 • Basic relations and operations [57] 4 • Finite and infinite sets [62] 5 • Relation between the ordinary infinite and the Dedekind infinite [66] 6 • The Choice Axiom [71] Suggested reading [74] Problems [75] IV • Infinite Sets [78] 1 • Countable sets; the number ϰ0 [79] 2 • Uncountable sets [85] 3 • Diagonal procedures and their applications [88] 4 • Cardinal numbers and their ordering [98] 5 • Further enumerations of the rational numbers [106] Suggested reading [107] Problems [107] V • Well-Ordered Sets; Ordinal Numbers [110] 1 • Order types [110] 2 • The order type ω [112] 3 • The general well-ordered set [115] 4 • The second class of ordinals [126] 5 • Equivalence of Choice Axiom, Well-ordering Theorem, and Comparability [129] Suggested reading [132] Problems [132] VI • The Linear Continuum and the Real Number System [134] 1 • Analysis of the structure of R as an ordered system [134] 2 • Operations in R [145] 3 • The real number system as based on the Peano axioms [148] 4 • The complex number system [156] Suggested reading [157] Problems [157] VII • Groups and Their Significance for the Foundations [158] 1 • Groups [158] 2 • Applications in algebra and to number systems [166] 3 • The group notion in geometry [176] 4 • Concluding remarks [183] Suggested reading [183] Problems [184] PART II • DEVELOPMENT OF VARIOUS VIEWPOINTS ON FOUNDATIONS VIII • The Early Developments [189] 1 • The eighteenth-century beginnings of analysis [189] 2 • The nineteenth-century foundation of analysis [190] 3 • The symbolizing of logic [195] 4 • The reduction of mathematics to logical form [198] 5 • Introduction of antinomies and paradoxes [200] 6 • Zermelo’s Well-ordering Theorem [201] 7 • Poincare’s views [201] 8 • Zermelo’s set theory [203] 9 • Amendments to the Zermelo system [207] Additional bibliography [208] IX • The Frege-Russell Thesis: Mathematics an Extension of Logic [209] 1 • The Frege-Russell thesis [209] 2 • Basic symbols; propositions and propositional functions [211] 3 • Calculus of propositions [215] 4 • Forms of general propositions [222] 5 • Classes and relations [224] 6 • Concluding remarks [229] Additional bibliography [229] X • Intuitionism [230] 1 • Basic philosophy of Intuitionism [231] 2 • The natural numbers and the definition of set [233] 3 • Species [235] 4 • Relations between species [235] 5 • Theory of cardinal numbers [237] 6 • Order and ordinal numbers [243] 7 • The intuitionist logic [243] 8 • General remarks [247] Additional bibliography [249] XI • Formalism [250] 1 • Hilbert’s “proof theory” [251] 2 • Actual development of the proof theory [252] 3 • Gödel’s incompleteness theorem ffijg 4 . Consistency of a formal system [260] 5 • Formal systems in general [261] 6 • Many-valued logics [263] Additional bibliography [263] XII • The Cultural Setting of Mathematics [264] 1 • The cultural background [265] 2 • The position of mathematics in the culture [266] 3 • The historical position of mathematics [267] 4 • The present-day position of mathematics [269] 5 • What is mathematics from the cultural point of view? [270] 6 • What we call “mathematics” today [274] 7 • The process of mathematical change and growth [276] 8 • Differences in the kind and quality of mathematics [280] 9 • Mathematical existence [283] Bibliography [285] Index of Symbols [296] Index of Topics and Technical Terms [297] Index of Names [303]
Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
---|---|---|---|---|---|---|---|---|
Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 03 W673 (Browse shelf) | Available | A-316 |
Incluye referencias bibliográficas (p. 285-295) e índices.
PART I • FUNDAMENTAL CONCEPTS AND METHODS OF MATHEMATICS --
T • The Axiomatic Method [3] --
1 • Evolution of the method [3] --
2 • Description of the method; the undefined terms and axioms [9] --
3 • Description of the method; the proving of theorems [13] --
4 • Comment on the above theorems and proofs [17] --
5 • Source of the axioms [19] --
Suggested reading [21] --
Problems [21] --
Tl • Analysis of the Axiomatic Method [23] --
1 • Consistency of an axiom system [23] --
2 • The proof of consistency of an axiom system [26] --
3 • Independence of axioms [29] --
4 • Completeness of an axiom system [32] --
5 • The logical basis [40] --
6 • Miscellaneous comment [41] --
7 • Axioms for simple order [44] --
8 • Axioms defining equivalence [46] --
Suggested reading [49] --
Problems [49] --
III * Theory of Sets [52] --
1 • Background of the theory [53] --
2 • The Russell contradiction [55] --
3 • Basic relations and operations [57] --
4 • Finite and infinite sets [62] --
5 • Relation between the ordinary infinite and the --
Dedekind infinite [66] --
6 • The Choice Axiom [71] --
Suggested reading [74] --
Problems [75] --
IV • Infinite Sets [78] --
1 • Countable sets; the number ϰ0 [79] --
2 • Uncountable sets [85] --
3 • Diagonal procedures and their applications [88] --
4 • Cardinal numbers and their ordering [98] --
5 • Further enumerations of the rational numbers [106] --
Suggested reading [107] --
Problems [107] --
V • Well-Ordered Sets; Ordinal Numbers [110] --
1 • Order types [110] --
2 • The order type ω [112] --
3 • The general well-ordered set [115] --
4 • The second class of ordinals [126] --
5 • Equivalence of Choice Axiom, Well-ordering Theorem, and Comparability [129] --
Suggested reading [132] --
Problems [132] --
VI • The Linear Continuum and the Real Number System [134] --
1 • Analysis of the structure of R as an ordered system [134] --
2 • Operations in R [145] --
3 • The real number system as based on the Peano axioms [148] --
4 • The complex number system [156] --
Suggested reading [157] --
Problems [157] --
VII • Groups and Their Significance for the Foundations [158] --
1 • Groups [158] --
2 • Applications in algebra and to number systems [166] --
3 • The group notion in geometry [176] --
4 • Concluding remarks [183] --
Suggested reading [183] --
Problems [184] --
PART II • DEVELOPMENT OF VARIOUS VIEWPOINTS ON FOUNDATIONS --
VIII • The Early Developments [189] --
1 • The eighteenth-century beginnings of analysis [189] --
2 • The nineteenth-century foundation of analysis [190] --
3 • The symbolizing of logic [195] --
4 • The reduction of mathematics to logical form [198] --
5 • Introduction of antinomies and paradoxes [200] --
6 • Zermelo’s Well-ordering Theorem [201] --
7 • Poincare’s views [201] --
8 • Zermelo’s set theory [203] --
9 • Amendments to the Zermelo system [207] --
Additional bibliography [208] --
IX • The Frege-Russell Thesis: Mathematics an Extension of Logic [209] --
1 • The Frege-Russell thesis [209] --
2 • Basic symbols; propositions and propositional functions [211] --
3 • Calculus of propositions [215] --
4 • Forms of general propositions [222] --
5 • Classes and relations [224] --
6 • Concluding remarks [229] --
Additional bibliography [229] --
X • Intuitionism [230] --
1 • Basic philosophy of Intuitionism [231] --
2 • The natural numbers and the definition of set [233] --
3 • Species [235] --
4 • Relations between species [235] --
5 • Theory of cardinal numbers [237] --
6 • Order and ordinal numbers [243] --
7 • The intuitionist logic [243] --
8 • General remarks [247] --
Additional bibliography [249] --
XI • Formalism [250] --
1 • Hilbert’s “proof theory” [251] --
2 • Actual development of the proof theory [252] --
3 • Gödel’s incompleteness theorem ffijg --
4 . Consistency of a formal system [260] --
5 • Formal systems in general [261] --
6 • Many-valued logics [263] --
Additional bibliography [263] --
XII • The Cultural Setting of Mathematics [264] --
1 • The cultural background [265] --
2 • The position of mathematics in the culture [266] --
3 • The historical position of mathematics [267] --
4 • The present-day position of mathematics [269] --
5 • What is mathematics from the cultural point of view? [270] --
6 • What we call “mathematics” today [274] --
7 • The process of mathematical change and growth [276] --
8 • Differences in the kind and quality of mathematics [280] --
9 • Mathematical existence [283] --
Bibliography [285] --
Index of Symbols [296] --
Index of Topics and Technical Terms [297] --
Index of Names [303] --
MR, 14,441d
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