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## Introduction to the foundations of mathematics / Raymond L. Wilder.

Editor: New York : Wiley, c1952Descripción: xiv, 305 p. ; 24 cmOtra clasificación: 03B30
Contenidos:
```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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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
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] --
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] --
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] --
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] --
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] --
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] --
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] --
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] --
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] --
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] --
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] --
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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