Cunningham D.W. Set Theory: A First Course

Cambridge: Cambridge University Press, 2016. — xiv, 250 p. — (Cambridge Mathematical Textbooks). — ISBN: 978-1-107-12032-7.Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.Preface page
The Greek AlphabetElementary Set Theory
Logical Notation
Predicates and Quantifiers
A Formal Language for Set Theory
The Zermelo–Fraenkel AxiomsBasic Set-Building Axioms and Operations
The First Six Axioms
The Extensionality Axiom
The Empty Set Axiom
The Subset Axiom
The Pairing Axiom
The Union Axiom
The Power Set Axiom
Operations on Sets
De Morgan’s Laws for Sets
Distributive Laws for SetsRelations and Functions
Ordered Pairs in Set Theory
Relations
Operations on Relations
Reflexive, Symmetric, and Transitive Relations
Equivalence Relations and Partitions
Functions
Operations on Functions
One-to-One Functions
Indexed Sets
The Axiom of Choice
Order Relations
Congruence and PreorderThe Natural Numbers
Inductive Sets
The Recursion Theorem on ω
The Peano Postulates
Arithmetic on ω
Order on ωOn the Size of Sets
Finite Sets
Countable Sets
Uncountable Sets
CardinalityTransfinite Recursion
Well-Ordering
Transfinite Recursion Theorem
Using a Set Function
Using a Class FunctionThe Axiom of Choice (Revisited)
Zorn’s Lemma
Two Applications of Zorn’s Lemma
Filters and Ultrafilters
Ideals
Well-Ordering TheoremOrdinals
Ordinal Numbers
Ordinal Recursion and Class Functions
Ordinal Arithmetic
The Cumulative HierarchyCardinals
Cardinal Numbers
Cardinal Arithmetic
Closed Unbounded and StationaryNotesIndex of Special SymbolsTrue PDF