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

Editor: New York : Wiley, c1965Edición: 2nd edDescripción: xvi, 327 p. ; 24 cmOtro título: Foundations of mathematicsTema(s): Mathematics -- Philosophy | Logic, Symbolic and mathematicalOtra clasificación: 03B30PART 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] Suggested reading [21] 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] Suggested reading [50] 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] Suggested reading [76] 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] Suggested reading [111] 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] Suggested reading [139] 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] Suggested reading [163] 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] Suggested reading [193] 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] Additional bibliography [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] Additional bibliography [244] 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] Additional bibliography [262] 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] Additional bibliography [279] 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]

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 W673-2 (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] --

Suggested reading [21] --

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] --

Suggested reading [50] --

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] --

Suggested reading [76] --

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] --

Suggested reading [111] --

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] --

Suggested reading [139] --

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] --

Suggested reading [163] --

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] --

Suggested reading [193] --

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] --

Additional bibliography [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] --

Additional bibliography [244] --

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] --

Additional bibliography [262] --

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] --

Additional bibliography [279] --

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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