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

Editor: New York : Wiley, c1965Edición: 2nd edDescripción: xvi, 327 p. ; 24 cmOtro título: Foundations of mathematicsOtra clasificación: 03B30
Contenidos:
```PART ONE • FUNDAMENTAL CONCEPTS AND METHODS OF MATHEMATICS
I • 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 [16]
5 • Source of the axioms [18]
Problems [21]
II • 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 [28]
4 • Completeness of an axiom system [31]
5 • Independence and completeness of undefined technical terms [39]
6 • Miscellaneous comment [43]
7 • Axioms for simple order [45]
8 • Axioms defining equivalence [48]
Problems [50]
III • Theory of Sets [54]
1 • Background of the theory [55]
2 • The Russell contradiction [56]
3 • Basic relations and operations [58]
4 • Finite and infinite sets [63]
5 • Relation between the ordinary infinite and the Dedekind infinite [66]
6 • The Choice Axiom [72]
Problems [76]
IV • Infinite Sets [80]
1 • Countable sets; the number ϰ0 [80]
2 • Uncountable sets [87]
3 • Diagonal procedures and their applications [91]
4 • Cardinal numbers and their ordering [101]
Problems [111]
V • Well-Ordered Sets; Ordinal Numbers [114]
1 • Order types [114]
2 • The order type ω [117]
3 • The general well-ordered set [120]
4 • The second class of ordinals [133]
5 • Equivalence of Choice Axiom, Well-Ordering Theorem, and Comparability [135]
Problems [139]
VI • The Linear Continuum and the Real Number System [142]
1 • Analysis of the structure of the real numbers as an ordered system [144]
2 • Operations in R [154]
3 • The real number system as based on the Peano axioms [158]
4 • The complex number system [162]
Problems [163]
VII • Groups and Their Significance for the Foundations [165]
1 • Groups [165]
2 • Applications in algebra and to number systems [174]
3 • The group notion in geometry [182]
4 • Topology [188]
5 • Concluding remarks [192]
Problems [193]
PART TWO • DEVELOPMENT OF VARIOUS
VIEWPOINTS ON FOUNDATIONS
VIII • The Early Developments [199]
1 • The eighteenth-century beginnings of analysis [199]
2 • The nineteenth-century foundation of analysis [200]
3 • The symbolizing of logic [205]
4 • The reduction of mathematics to logical form [208]
5 • Introduction of antinomies and paradoxes [210]
6 • Zermelo’s Well-Ordering Theorem [210]
7 • Poincare’s views [211]
8 • Zermelo’s set theory [212]
9 • Amendments to the Zermelo system [217]
Problems [218]
IX • The Frege-Russell Thesis: Mathematics an Extension of Logic [219]
1 • The Frege-Russell thesis [219]
2 • Basic symbols; propositions and propositional functions [221]
3 • Calculus of propositions [225]
4 • Forms of general propositions; the predicate calculus [234]
5 • Classes and relations as treated in P.M. [238]
6 • Concluding remarks [243]
Problems [244]
X • Intuitionism [246]
1 • Basic philosophy of Intuitionism [247]
2 • The natural numbers and the definition of set; spreads [248]
3 Species [251]
4 • Relations between species [252]
5 • Theory of cardinal numbers [252]
6 • Order and ordinal numbers [256]
7 • The intuitionist logic [256]
8 • General remarks [260]
Problems [262]
XI *. Formal Systems; Mathematical Logic [264]
1 • Hilbert’s “proof theory” [264]
2 • Actual development of the proof theory [266]
3 • Gödel’s incompleteness theorem [270]
4 • Consistency of a formal system [274]
5 • Formal systems in general [274]
6 • General significance of formal systems [277]
Problems [279]
XII • The Cultural Setting of Mathematics [281]
1 • The cultural background [282]
2 • The position of mathematics in the culture [283]
3 • The historical position of mathematics [284]
4 • The present-day position of mathematics [285]
5 • What is mathematics from the cultural point of view? [286]
6 • What we call “mathematics” today [290]
7 • The process of mathematical change and growth [292]
8 • Differences in the kind and quality of mathematics [295]
9 • Mathematical existence [298]
Bibliography [301]
Index of Symbols [315]
Index of Topics and Technical Terms [317]
Index of Names [325]```
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Libros ordenados por tema 03 W673-2 (Browse shelf) Available A-2799

Bibliografía: p. 301-313.

PART ONE • FUNDAMENTAL CONCEPTS AND METHODS OF MATHEMATICS --
I • 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 [16] --
5 • Source of the axioms [18] --
Problems [21] --
II • 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 [28] --
4 • Completeness of an axiom system [31] --
5 • Independence and completeness of undefined technical terms [39] --
6 • Miscellaneous comment [43] --
7 • Axioms for simple order [45] --
8 • Axioms defining equivalence [48] --
Problems [50] --
III • Theory of Sets [54] --
1 • Background of the theory [55] --
2 • The Russell contradiction [56] --
3 • Basic relations and operations [58] --
4 • Finite and infinite sets [63] --
5 • Relation between the ordinary infinite and the Dedekind infinite [66] --
6 • The Choice Axiom [72] --
Problems [76] --
IV • Infinite Sets [80] --
1 • Countable sets; the number ϰ0 [80] --
2 • Uncountable sets [87] --
3 • Diagonal procedures and their applications [91] --
4 • Cardinal numbers and their ordering [101] --
Problems [111] --
V • Well-Ordered Sets; Ordinal Numbers [114] --
1 • Order types [114] --
2 • The order type ω [117] --
3 • The general well-ordered set [120] --
4 • The second class of ordinals [133] --
5 • Equivalence of Choice Axiom, Well-Ordering Theorem, and Comparability [135] --
Problems [139] --
VI • The Linear Continuum and the Real Number System [142] --
1 • Analysis of the structure of the real numbers as an ordered system [144] --
2 • Operations in R [154] --
3 • The real number system as based on the Peano axioms [158] --
4 • The complex number system [162] --
Problems [163] --
VII • Groups and Their Significance for the Foundations [165] --
1 • Groups [165] --
2 • Applications in algebra and to number systems [174] --
3 • The group notion in geometry [182] --
4 • Topology [188] --
5 • Concluding remarks [192] --
Problems [193] --
PART TWO • DEVELOPMENT OF VARIOUS --
VIEWPOINTS ON FOUNDATIONS --
VIII • The Early Developments [199] --
1 • The eighteenth-century beginnings of analysis [199] --
2 • The nineteenth-century foundation of analysis [200] --
3 • The symbolizing of logic [205] --
4 • The reduction of mathematics to logical form [208] --
5 • Introduction of antinomies and paradoxes [210] --
6 • Zermelo’s Well-Ordering Theorem [210] --
7 • Poincare’s views [211] --
8 • Zermelo’s set theory [212] --
9 • Amendments to the Zermelo system [217] --
Problems [218] --
IX • The Frege-Russell Thesis: Mathematics an Extension of Logic [219] --
1 • The Frege-Russell thesis [219] --
2 • Basic symbols; propositions and propositional functions [221] --
3 • Calculus of propositions [225] --
4 • Forms of general propositions; the predicate calculus [234] --
5 • Classes and relations as treated in P.M. [238] --
6 • Concluding remarks [243] --
Problems [244] --
X • Intuitionism [246] --
1 • Basic philosophy of Intuitionism [247] --
2 • The natural numbers and the definition of set; spreads [248] --
3 Species [251] --
4 • Relations between species [252] --
5 • Theory of cardinal numbers [252] --
6 • Order and ordinal numbers [256] --
7 • The intuitionist logic [256] --
8 • General remarks [260] --
Problems [262] --
XI *. Formal Systems; Mathematical Logic [264] --
1 • Hilbert’s “proof theory” [264] --
2 • Actual development of the proof theory [266] --
3 • Gödel’s incompleteness theorem [270] --
4 • Consistency of a formal system [274] --
5 • Formal systems in general [274] --
6 • General significance of formal systems [277] --
Problems [279] --
XII • The Cultural Setting of Mathematics [281] --
1 • The cultural background [282] --
2 • The position of mathematics in the culture [283] --
3 • The historical position of mathematics [284] --
4 • The present-day position of mathematics [285] --
5 • What is mathematics from the cultural point of view? [286] --
6 • What we call “mathematics” today [290] --
7 • The process of mathematical change and growth [292] --
8 • Differences in the kind and quality of mathematics [295] --
9 • Mathematical existence [298] --
Bibliography [301] --
Index of Symbols [315] --
Index of Topics and Technical Terms [317] --
Index of Names [325] --

MR, 32 #35

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