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E-raamat: Set Theory And Foundations Of Mathematics: An Introduction To Mathematical Logic - Volume I: Set Theory

(Univ Of Florida, Usa), (Drake Univ, Usa), (Univ Of Florida, Usa), (Univ Of Florida, Usa)
  • Formaat: 224 pages
  • Ilmumisaeg: 04-Apr-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811201943
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  • Formaat: 224 pages
  • Ilmumisaeg: 04-Apr-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811201943
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This book provides an introduction to axiomatic set theory and descriptive set theory. It is written for the upper level undergraduate or beginning graduate students to help them prepare for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra. The book is designed as a flexible and accessible text for a one-semester introductory course in set theory, where the existing alternatives may be more demanding or specialized. Readers will learn the universally accepted basis of the field, with several popular topics added as an option. Pointers to more advanced study are scattered throughout the text.

Preface v
About the Authors vii
1 Introduction
1(6)
2 Review of Sets and Logic
7(36)
2.1 The Algebra of Sets
7(6)
2.2 Relations
13(9)
2.3 Functions
22(8)
2.4 Equivalence Relations
30(3)
2.5 Orderings
33(4)
2.6 Trees
37(6)
3 Zermelo-Fraenkel Set Theory
43(24)
3.1 Historical Context
43(3)
3.2 The Language of the Theory
46(1)
3.3 The Basic Axioms
47(4)
3.4 Axiom of Infinity
51(2)
3.5 Axiom Schema of Comprehension
53(5)
3.6 Axiom of Choice
58(4)
3.7 Axiom Schema of Replacement
62(2)
3.8 Axiom of Regularity
64(3)
4 Natural Numbers and Countable Sets
67(34)
4.1 Von Neumann's Natural Numbers
67(5)
4.2 Finite and Infinite Sets
72(5)
4.3 Inductive and Recursive Definability
77(13)
4.4 Cardinality
90(5)
4.5 Countable and Uncountable Sets
95(6)
5 Ordinal Numbers and the Transfinite
101(26)
5.1 Ordinals
101(5)
5.2 Transfinite Induction and Recursion
106(6)
5.3 Ordinal Arithmetic
112(10)
5.4 Ordinals and Well-Orderings
122(5)
6 Cardinality and the Axiom of Choice
127(22)
6.1 Equivalent Versions of the Axiom of Choice
127(4)
6.2 Applications of the Axiom of Choice
131(4)
6.3 Cardinal Numbers
135(14)
7 Real Numbers
149(26)
7.1 Integers and Rational Numbers
149(2)
7.2 Dense Linear Orders
151(2)
7.3 Complete Orders
153(5)
7.4 Countable and Uncountable Sets of Reals
158(7)
7.5 Topological Spaces
165(10)
8 Models of Set Theory
175(16)
8.1 The Hereditarily Finite Sets
176(7)
8.2 Transfinite Models
183(8)
9 Ramsey Theory
191(12)
9.1 Finite Patterns
191(5)
9.2 Countably Infinite Patterns
196(5)
9.3 Uncountable Patterns
201(2)
Bibliography 203(2)
Index 205