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Lecture 1:
Construction & Extension:Story of Numbers
Addressed by Z.Liu
0 OutlineComplex Numbers
Real Numbers
Rational Numbers
Integers
Natural Numbers
1 Natural Numbers
Motivation
PeanoAxioms
Construction
BinaryOperations
1.0 Motivation
54321
1.1 Peano Axioms
P 1
P 2
P 3
P 4
1.2 Construction
•Set Theory
•Relations & Functions
Preliminaries
•Axiom of Infinity
•Linear Ordering & Well Ordering
Construction
•Mathematical Induction
•Ordinals & Transfinite Induction
Induction
1.2.0 PreliminariesAxiom of Extensionality
Axiom of Pairing
Axiom of Separation
Axiom of Union
Axiom of Power Set
Axiom of Infinity
Axiom of Replacement
Axiom of Regularity
Axiom of Choice
CartesianProduct
Relation Function
1.2.1 Construction
1.2.2 Induction
NaturalNumbers
OrdinalNumbers
MathematicalInduction
TransfiniteInduction
1.3 Binary Operation
Addition Multiplication Exponentiation
2 Integers
Motivation
ConstructionAlgebraicStructure
ModularArithmetic
2.0 Motivation
2.1 Construction
-1=0-1 =1-2 =2-3
2.2 Algebraic Structure
Group
Monoid
Semigroup
Field
DivisionRingOr
Integral Domain
RingOr
Commutative Ring
2.3 Modular Arithmetic
3 Rational Numbers
Motivation
Construction&
Quotient Field
Extension&
Vector Space
p-adicNumbers
3.0 Motivation
3.1 Construction
0.5=1/2 =2/4 =3/6 =4/8
3.2 Extension
3.3 p-adic Numbers
4 Real Numbers
Motivation
CardinalNumber
Metric Space&
Topology
MeasureTheory
4.0 Motivation
Dedekind
Completion
Cauchy
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
ConstructionLinear
Representation
ComplexAnalysis
5.0 Motivation
5.1 Construction
5.2 Linear Representation
5.3 Complex Analysis
Differentiation
IntegralLaurentSeries
Thank youFor your attention!