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Set Theory

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Set Theory

The foundations of modern mathematics

144Modules

8Units

1

Naive Set Theory

Intuitive foundations and basic operations

20 modules

What is Set Theory?INTRO

Sets Are EverywhereINTRO

A Glimpse of ParadoxesINTRO

Set Notation and MembershipLESSON

Set Operations ExamplesEXAMPLE

Set Operations PracticeACTIVITY

Set Notation PracticePRACTICE

Set Operations PracticePRACTICE

Set-Builder NotationLESSON

Set Builder ToolACTIVITY

Subsets and EqualityLESSON

Power SetsLESSON

Union and IntersectionLESSON

Complement and DifferenceLESSON

Cartesian ProductsLESSON

Indexed Families of SetsLESSON

Venn Diagram ExamplesEXAMPLE

Naive Set Theory SummarySUMMARY

Naive Set Theory ReferenceFORMULA SHEET

Naive Set Theory QuizQUIZ

2

Axiomatic Set Theory

The Zermelo-Fraenkel axioms

20 modules

Why Do We Need Axioms?INTRO

Russell's ParadoxINTRO

Axiom of ExtensionalityLESSON

Axiom of Empty SetLESSON

Axiom of PairingLESSON

Axiom of UnionLESSON

Applying the AxiomsEXAMPLE

ZFC Axiom ExplorerACTIVITY

Axiom of Power SetLESSON

Axiom Schema of SeparationLESSON

Axiom Schema of ReplacementLESSON

Axiom of InfinityLESSON

Axiom of RegularityLESSON

Cumulative HierarchyLESSON

Constructing Sets from AxiomsEXAMPLE

ZFC Axioms PracticePRACTICE

Using Axioms PracticePRACTICE

Axiomatic Set Theory SummarySUMMARY

ZFC Axioms ReferenceFORMULA SHEET

Axiomatic Set Theory QuizQUIZ

3

Relations and Functions

Ordered pairs, relations, and function theory

21 modules

What are Relations?INTRO

Functions as SetsINTRO

Ordered PairsLESSON

Relations DefinitionLESSON

Relations PracticePRACTICE

Properties of RelationsLESSON

Relation VisualizerACTIVITY

Equivalence RelationsLESSON

Equivalence Relation ExamplesEXAMPLE

Order Relation ExamplesEXAMPLE

Equivalence Classes and PartitionsLESSON

Partial OrdersLESSON

Total OrdersLESSON

Well-OrdersLESSON

Functions as RelationsLESSON

Functions PracticePRACTICE

Injective and Surjective FunctionsLESSON

BijectionsLESSON

Relations and Functions SummarySUMMARY

Relations ReferenceFORMULA SHEET

Relations and Functions QuizQUIZ

4

Ordinal Numbers

Transfinite ordinals and ordinal arithmetic

20 modules

Beyond Finite NumbersINTRO

Ordering the InfiniteINTRO

Well-Ordered SetsLESSON

Ordinal Number DefinitionLESSON

Comparing OrdinalsLESSON

Successor OrdinalsLESSON

Limit OrdinalsLESSON

Ordinal ArithmeticLESSON

Ordinal Arithmetic ExamplesEXAMPLE

Ordinal Basics PracticePRACTICE

Ordinal Arithmetic PracticePRACTICE

Ordinal NotationsLESSON

Ordinal CalculatorACTIVITY

Epsilon NumbersLESSON

Transfinite InductionLESSON

Transfinite Induction ExamplesEXAMPLE

Transfinite RecursionLESSON

Ordinals SummarySUMMARY

Ordinals ReferenceFORMULA SHEET

Ordinals QuizQUIZ

5

Cardinal Numbers

Cardinality and transfinite cardinals

20 modules

Counting the InfiniteINTRO

Different Sizes of InfinityINTRO

Cardinality DefinitionLESSON

Cardinality ExplorerACTIVITY

Finite CardinalityLESSON

Countable SetsLESSON

Uncountable SetsLESSON

Cantor's TheoremLESSON

Schröder-Bernstein TheoremLESSON

Comparing CardinalsLESSON

Cardinal ArithmeticLESSON

Cardinal Arithmetic PracticePRACTICE

Aleph NumbersLESSON

The Continuum HypothesisLESSON

Countability Proof ExamplesEXAMPLE

Diagonal Argument ExamplesEXAMPLE

Countability PracticePRACTICE

Cardinals SummarySUMMARY

Cardinals ReferenceFORMULA SHEET

Cardinals QuizQUIZ

6

The Axiom of Choice

AC and its equivalents

20 modules

Choosing ElementsINTRO

The Controversial AxiomINTRO

Statement of the Axiom of ChoiceLESSON

Choice FunctionsLESSON

Choice Function ExplorerACTIVITY

Zorn's LemmaLESSON

Using Zorn's LemmaEXAMPLE

Well-Ordering TheoremLESSON

Equivalents of ACLESSON

AC Equivalents PracticePRACTICE

Applications of ACLESSON

Consequences of ACLESSON

AC Basics PracticePRACTICE

Vitali SetsLESSON

Vitali Set ConstructionEXAMPLE

The Banach-Tarski ParadoxLESSON

Weak Forms of ChoiceLESSON

Axiom of Choice SummarySUMMARY

Axiom of Choice ReferenceFORMULA SHEET

Axiom of Choice QuizQUIZ

7

Models and Consistency

Models of set theory and independence results

18 modules

What is a Model?INTRO

Can We Prove Consistency?INTRO

Models of Set TheoryLESSON

Models PracticePRACTICE

The Constructible Universe LLESSON

Building L ExamplesEXAMPLE

Gödel's Consistency ProofLESSON

Relative Consistency ExamplesEXAMPLE

Consistency PracticePRACTICE

Introduction to ForcingLESSON

Independence of CHLESSON

Independence of ACLESSON

Large CardinalsLESSON

Foundations DebateLESSON

Model ExplorerACTIVITY

Models and Consistency SummarySUMMARY

Models ReferenceFORMULA SHEET

Models and Consistency QuizQUIZ

8

Course Resources

Comprehensive references and practice materials

5 modules

Problem Solving GuideSUMMARY

Set Theory Master ReferenceFORMULA SHEET

Complete Formula SheetFORMULA SHEET

Axiom Summary SheetFORMULA SHEET

Comprehensive Set Theory TestQUIZ