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."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 Cantor'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.