Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture

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""Cover ""; ""copyright page ""; ""Title page ""; ""Contents""; ""Preface""; ""Acknowledgments""; ""Notation""; ""0 Mathematics""; ""0.1 The Tree of Mathematics""; ""0.2 What Is Mathematics?""; ""0.3 Pure vs. Applied""; ""and supreme value of mathematical science.â€?90.4 What Kind of People Are Mathematicians?""; ""0.5 Mathematics Subject Classification""; ""1 Logic""; ""1.1 Mathematical Systems""; ""1.2 Warm-up Exercises""; ""1.3 Essentials of Logic""; ""1.4 Further Exercises""; ""1.5 The Axiomatic Method""; ""1.6 The Results of Gödel""; ""2 Set Theory""; ""2.1 Proofs"" ""Cover "" ""copyright page "" ""Title page "" ""Contents"" ""Preface"" ""Acknowledgments"" ""Notation"" ""0 Mathematics"" ""0.1 The Tree of Mathematics"" ""0.2 What Is Mathematics?"" ""0.3 Pure vs. Applied"" ""and supreme value of mathematical science.â€?90.4 What Kind of People Are Mathematicians?"" ""0.5 Mathematics Subject Classification"" ""1 Logic"" ""1.1 Mathematical Systems"" ""1.2 Warm-up Exercises"" ""1.3 Essentials of Logic"" ""1.4 Further Exercises"" ""1.5 The Axiomatic Method"" ""1.6 The Results of Gödel"" ""2 Set Theory"" ""2.1 Proofs"" ""2.2 Warm-up Exercises""""2.3 Essentials of Set Theory"" ""2.4 Further Exercises"" ""2.5 Paradoxes"" ""2.6 Axiomatic Set Theory"" ""3 The Natural Numbers"" ""3.1 Mathematical Symbols and Notation"" ""3.2 Warm-up Exercises"" ""3.3 Essentials of the Natural Number System"" ""3.4 Further Exercises"" ""3.5 Cantorâ€?s Infinite Arithmetic"" ""4 The Positive Rational Numbers"" ""4.1 Philosophy of Mathematics"" ""4.2 Warm-up Exercises"" ""4.3 Essentials of the Positive Rational Numbers"" ""4.4 Further Exercises"" ""4.5 Ethics"" ""5 The Real Numbers"" ""5.1 Famous Mathematical Objects""""5.2 Warm-up Exercises"" ""5.3 Essentials of the Positive Real Numbers"" ""5.4 Essentials of the Real Number System"" ""5.5 Further Exercises"" ""5.6 Important Properties of the Real Number Line"" ""6 The Complex Numbers"" ""6.1 Famous Mathematicians"" ""6.2 Warm-up Exercises"" ""6.3 Essentials of the Complex Number System"" ""6.4 Further Exercises"" ""6.5 Important Properties of the Complex Numbers"" ""7 And Beyond . . ."" ""7.1 What Is Mathematical Research?"" ""7.2 Famous Theorems"" ""7.3 Famous Unsolved Problems"" ""7.4 Professional Organizations""""7.5 Resources"" ""7.6 Extracurricular Activities"" ""References"" ""Index""