Lecture 1:
Construction & Extension:
Story of Numbers
Addressed by Z.Liu
0 Outline
Complex
Numbers
Real
Numbers
Rational Numbers
Integers
Natural Numbers
1 Natural Numbers
Motivation
Peano
Axioms
Binary
Operations
Construction
1.0 Motivation
1
2
3
4
5
1.1 Peano Axioms
P1
P2
P3
P4
1.2 Construction
Preliminaries
• Set Theory
• Relations & Functions
Construction
• Axiom of Infinity
• Linear Ordering & Well Ordering
Induction
• Mathematical Induction
• Ordinals & Transfinite Induction
1.2.0 Preliminaries
Axiom of Extensionality
Axiom of Pairing
Axiom of Separation
Axiom of Union
Cartesian
Axiom of Power Set
Product
Axiom of Infinity
Axiom of Replacement
Axiom of Regularity
Axiom of Choice
Relation
Function
1.2.1 Construction
1.2.2 Induction
Natural
Mathematical
Numbers
Induction
Ordinal
Transfinite
Numbers
Induction
1.3 Binary Operation
Addition
Multiplication
Exponentiation
2 Integers
Motivation
Construction
Modular
Arithmetic
Algebraic
Structure
2.0 Motivation
2.1 Construction
2.2 Algebraic Structure
Group
Field
Division
Monoid
Ring
Or
Integral Domain
Ring
Semigroup
Or
Commutative Ring
2.3 Modular Arithmetic
3 Rational Numbers
Motivation
p-adic
Numbers
Construction
Extension
&
&
Quotient Field
Vector Space
3.0 Motivation
3.1 Construction
3.2 Extension
3.3 p-adic Numbers
4 Real Numbers
Motivation
Cardinal
Number
Measure
Theory
Metric Space
&
Topology
4.0 Motivation
Dedekind
Cauchy
Completion
Completion
4.1 Cardinal Number
4.2 Metric Space & Topological Space
Inner Product
Space
Normed Space
Metric Space
Topological Space
4.3 Measure Theory
5 Complex Numbers
Motivation
Construction
Complex
Analysis
Linear
Representation
5.0 Motivation
5.1 Construction
5.2 Linear Representation
5.3 Complex Analysis
Differentiation
Laurent
Series
Integral
Thank you
For your attention!