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

✍ Scribed by Roman Kossak

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 258
• Series: Springer Graduate Texts in Philosophy, 4
• Edition: 2
• Category: Library

✦ Table of Contents

Preface
About the Content
Preface to the Second Edition
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 Inductive Definitions
1.2.2 Syntax
1.2.3 Semantics
1.2.4 First-Order Properties
1.2.5 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?
3.2 How Many Numbers are There?
3.2.1 Zero
3.2.2 The Set of Natural Numbers
3.3 Arithmetic Operations and the Decimal System
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 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
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 Structures as Sets of Sets
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
9.1.1 Trivial Structures are Minimal
9.2 The Ordering of the Natural Numbers
9.3 The Ordering of the Integers
9.4 The Additive Structure of the Integers
9.5 The Ordering of the Rational Numbers
Exercises
10 Geometry of Definable Sets
10.1 Boolean Combinations
10.1.1 Venn Diagrams
10.1.2 Higher Dimensions
10.2 Geometry of Real Numbers
10.2.1 Euclidean Spaces
10.3 Shadows and Complexity
10.4 Diophantine Equations and Hilbert's 10th Problem
10.4.1 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
15.5.1 The Complex Numbers
Exercises
Part III Inference, Models, Categoricity and Diversity
16 Logical Inference
16.1 Adding Function Symbols to Language
16.2 Deriving Consequences Formally: An Example
16.3 Proof Systems
16.4 Proving Unprovability
16.5 Completeness
16.6 Semantic Inference
Exercises
17 Categoricity
17.1 Countable Categoricity
17.1.1 Dense Linear Orderings
17.1.2 The Countable Random Graph
17.2 Uncountable Categoricity
17.2.1 Completeness of Categorical Theories
Exercises
18 Counting Countable Models
18.1 Counting Ordered Sets
18.2 Counting Successor Chains
18.3 Classifiability: Ord vs. Succ
18.4 Counting Types
18.5 True Arithmetic
18.5.1 Expansions and Reducts
Exercises
19 Infinitary Logics
19.1 The Meaning of ``…''
19.2 Syntax and Semantics of Lω1,ω
19.3 Logical Visibility Revisited
Exercises
20 Symmetry and Definability
20.1 The Number of Symmetric Images
20.1.1 Expanding the Successor Relation
20.2 Undefinability of Truth
20.2.1 Nonstandard Truth
20.3 Resplendence
20.4 Implicit Definability
Exercises
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
A.6 The Compactness Theorem
B Hilbert's Program
C Suggestions for Further Reading
References
Index