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!
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