Mathematical Logic. On Numbers, Sets, Structures, and Symmetry

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Kossak R. Mathematical Logic: On Numbers, Sets, Structures, and Symmetry Series Title Copyright Preface About the Content Contents Part I Logic, Sets, and Numbers 1 First-Order Logic 1.1 What We Talk About When We Talk About Numbers 1.1.1 How to Choose a Vocabulary? 1.2 Symbolic Logic 1.2.1 Trivial Structures Exercises 2 Logical Seeing 2.1 Finite Graphs 2.2 Symmetry 2.3 Types and Fixed Points 2.4 Seeing Numbers Exercises 3 What Is a Number? 3.1 How Natural Are the Natural Numbers? 3.1.1 Arithmetic Operations and the Decimal System 3.1.2 How Many Numbers Are There? 3.1.3 Zero 3.1.4 The Set of Natural Numbers Exercises 4 Seeing the Number Structures 4.1 What Is the Structure of the Natural Numbers? 4.1.1 Sets and Set Notation 4.1.2 Language of Formal Arithmetic 4.1.3 Linearly Ordered Sets 4.1.4 The Ordering of the Natural Numbers 4.2 The Arithmetic Structure of the Natural Numbers 4.3 The Arithmetic Structure of the Integers 4.3.1 Natural Numbers Are Integers, Naturally 4.4 Fractions! 4.5 The Arithmetic Structure of the Rationals 4.5.1 Equivalence Relations and the Rationals 4.5.2 Defining Addition and Multiplication of the Rational Numbers Formally 4.5.3 Dense Ordering of the Rationals Exercises 5 Points, Lines, and the Structure of R 5.1 Density of Rational Numbers 5.2 What Are Real Numbers, Really? 5.3 Dedekind Cuts 5.3.1 Dedekind Complete Orderings 5.3.2 Summary 5.4 Dangerous Consequences 5.5 Infinite Decimals Exercises 6 Set Theory 6.1 What to Assume About Infinite Sets? Exercises Part II Relations, Structures, Geometry 7 Relations 7.1 Ordered Pairs 7.2 Cartesian Products 7.3 What Is a Relation on a Set? 7.4 Definability: New Relations from Old 7.5 How Many Different Structures Are There? 7.5.1 A Very Small Complex Structure Exercises 8 Definable Elements and Constants 8.1 Definable Elements 8.2 Databases, Oracles, and the Theory of a Structure 8.3 Defining Real Numbers 8.4 Definability With and Without Parameters Exercises 9 Minimal and Order-Minimal Structures 9.1 Types, Symmetries, and Minimal Structures 9.2 Trivial Structures 9.3 The Ordering of the Natural Numbers 9.4 The Ordering of the Integers 9.5 The Additive Structure of the Integers 9.6 The Ordering of the Rational Numbers Exercises 10 Geometry of Definable Sets 10.1 Boolean Combinations 10.2 Higher Dimensions 10.2.1 Euclidean Spaces 10.3 Shadows and Complexity 10.3.1 Diophantine Equations and Hilbert's 10th Problem 10.3.2 The Reals vs. The Rationals Exercises 11 Where Do Structures Come From? 11.1 The Compactness Theorem 11.2 New Structures from Old 11.2.1 Twin Primes Exercises 12 Elementary Extensions and Symmetries 12.1 Minimality of (N,