Efnisyfirlit

• The Foundations of Mathematics - Second Edition
• Copyright
• Dedication
• Preface to the Second Edition
• Preface to the First Edition
• Contents
• Part I The Intuitive Background
  • 1 Mathematical Thinking
    • Concept Formation
    • Schemas
    • An Example
    • Natural and Formal Mathematics
    • Building Formal Ideas on Human Experience
    • Formal Systems and Structure Theorems
    • Using Formal Mathematics More Flexi
    • Exercises
  • 2 Number Systems
    • Natural Numbers
    • Fractions
    • Integers
    • Rational Numbers
    • Real Numbers
    • Inaccurate Arithmetic in Practical Drawing
    • A Theoretical Model of the Real Line
    • Different Decimal Expansions for Different Numbers
    • Rationals and Irrationals
    • The Need for Real Numbers
    • Arithmetic of Decimals
    • Sequences
    • Order Properties and the Modulus
    • Convergence
    • Completeness
    • Decreasing Sequences
    • Different Decimal Expansions for the Same Real Number
    • Bounded Sets
    • Exercises
• Part II The Beginnings of Formalisation
  • 3 Sets
    • Members
    • Subsets
    • Is There a Universe?
    • Union and Intersection
    • Complements
    • Sets of Sets
    • Exercises
  • 4 Relations
    • Ordered Pairs
    • Mathematical Precision and Human Insight
    • AlternativeWays to Conceptualise Ordered Pairs
    • Relations
    • Equivalence Relations
    • Example: Arithmetic Modulo n
    • Subtle Aspects of Equivalence Relations
    • Order Relations
    • Exercises
  • 5 Functions
    • Some Traditional Functions
    • The General Function Concept
    • General Properties of Functions
    • The Graph of a Function
    • Composition of Functions
    • Inverse Functions
    • Restriction
    • Sequences and n-tuples
    • Functions of Several Variables
    • Binary Operations
    • Indexed Families of Sets
    • Exercises
  • 6 Mathematical Logic
    • Statements
    • Predicates
    • All and Some
    • More Than One Quantifier
    • Negation
    • Logical Grammar: Connectives
    • The Link with Set Theory
    • Formulas for Compound Statements
    • Logical Deductions
    • Proof
    • Exercises
  • 7 Mathematical Proof
    • Axiomatic Systems
    • Proof Comprehension and Self-Explanation
    • Examination Questions
    • Exercises
• Part III The Development of Axiomatic Systems
  • 8 Natural Numbers and Proof by Induction
    • Natural Numbers
    • Definition by Induction
    • Laws of Arithmetic
    • Ordering the Natural Numbers
    • Uniqueness of N0
    • Counting
    • Von Neumann’s Brainwave
    • Other Forms of Induction
    • Division
    • Factorisation
    • The Euclidean Algorithm
    • Reflections
    • Exercises
  • 9 Real Numbers
    • Preliminary Arithmetical Deductions
    • Preliminary Deductions about Order
    • Construction of the Integers
    • Construction of Rational Numbers
    • Construction of Real Numbers
    • Sequences of Rationals
    • The Ordering on R
    • Completeness of R
    • Exercises
  • 10 Real Numbers as a Complete Ordered Field
    • Examples of Rings and Fields
    • Examples of Ordered Rings and Fields
    • Isomorphisms Again
    • Some Characterisations
    • The Connection with Intuition
    • Exercises
  • 11 Complex Numbers and Beyond
    • Historical Background
    • Construction of the Complex Numbers
    • Complex Conjugation
    • The Modulus
    • Euler’s Approach to the Exponential Function
    • Addition Formulas for Cosine and Sine
    • The Complex Exponential Function
    • Quaternions
    • The Change in Approach to Formal Mathematics
    • Exercises
• Part IV Using Axiomatic Systems
  • 12 Axiomatic Systems, Structure Theorems, and Flexible Thinking
    • Structure Theorems
    • Psychological Aspects of Different Approaches to Mathematical Thinking
    • Building Formal Theories
      • Semigroups and Groups
      • Rings and Fields
      • Vector Spaces
    • The Way Ahead
    • Exercises
  • 13 Permutations and Groups
    • Permutations
    • Permutations as Cycles
    • Group Properties for Permutations
    • Axioms for a Group
    • Subgroups
    • Isomorphisms and Homomorphisms
    • Partitioning a Group to Obtain a Quotient Group
    • The Number of Elements in a Group and a Subgroup
    • Partitions that Define a Group Structure
    • The Structure of Group Homomorphisms
    • The Structure of Groups
    • Major Contributions of Group Theory throughout Mathematics
    • The Way Ahead
  • 14 Cardinal Numbers
    • Cantor’s Cardinal Numbers
    • The Schröder–Bernstein Theorem
    • Cardinal Arithmetic
    • Order Relations on Cardinals
    • Exercises
  • 15 Infinitesimals
    • Ordered Fields Larger than the Real Numbers
    • Super Ordered Fields
    • The Structure Theorem for Super Ordered Fields
    • Visualising Infinitesimals on a Geometric Number Line
    • Magnification in Higher Dimensions
    • Calculus with Infinitesimals
    • Non-standard Analysis
    • Amazing Possibilities in Non-standard Analysis
    • Exercises
• Part V Strengthening the Foundations
  • 16 Axioms for Set Theory
    • Some Difficulties
    • Sets and Classes
    • The Axioms Themselves
    • The Axiom of Choice
    • Consistency
    • Exercises
• Appendix—How to Read Proofs: The 'Self-Explanation' Strategy
  • How to Self-Explain
  • Example Self-Explanations
  • Self-Explanation Compared with Other Comments
    • Paraphrasing
    • Monitoring
  • Practice Proof 1
  • Practice Proof 2
  • Remember . . .
• References and Further Reading
  • References
  • Further Reading
  • Online Reading
• Index

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