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,<)
12.2 Building Symmetries
Exercises
13 Tame vs. Wild
13.1 Complex Numbers
13.1.1 Real Numbers and Order-Minimality
13.2 On the Wild Side
Exercises
14 First-Order Properties
14.1 Beyond First-Order
14.1.1 Finiteness
14.1.2 Minimality and Order-Minimality
14.2 Well-Ordered Sets and Second-Order Logic
Exercises
15 Symmetries and Logical Visibility One More Time
15.1 Structures
15.2 The Natural Numbers
15.3 The Integers
15.4 The Rationals
15.5 The Reals
15.6 The Complex Numbers
Exercises
16 Suggestions for Further Reading
Appendix A Proofs
A.1 Irrationality of 2
A.2 Cantor's Theorem
A.3 Theories of Structures with Finite Domains
A.4 Existence of Elementary Extensions
A.5 Ramsey's Theorem
A.5.1 A Stronger Version of Ramsey's Theorem
Appendix B Hilbert's Program
Bibliography
Index