Some admonitions.
IntroductionSome history. Role of set theory as foundation of mathematics. Russell's Paradox. Naive vs. axiomatic set theory.
PART ONE: NOT ENTIRELY NAIVE SET THEORY
1. Pairs, relations, and functionsAlso Cartesian products, equivalence relations, equivalence classes. Meant as a review of selected prerequisites.
2. Partial order relationsPartial order, linear order,wellorder, and wellfounded relations. Order types, operations on order types.
3. CardinalityEquipotency, intuitive notion of a cardinal, Cantor-Schröder-Bernstein Theorem (without proof), Cantor's diagonalization technique. Proof that the sets of integers, rationals, and algebraic numbers are countable. Multiplication and addition of cardinals (proofs only given for countable cardinals). Computation of cardinality for some uncountable sets.
4. InductionInduction vs. recursion. Recursive definition of natural numbers. Transitive closure of a set. Proof of the Cantor-Schröder-Bernstein Theorem.Characterization of the order types of rationals and reals. Dedekind finite sets (without using the name). Induction and recursion over wellfounded sets. Rank functions for wellfounded sets. Comparability for ordinals. Mostowski Collapse.
PART TWO: AN AXIOMATIC FOUNDATION OF SET THEORY
5. Formal languages and modelsFormal languages, terms, formulas, proofs. Models. Theories, consistency, independence. Gödel's Completeness Theorem. Compactness Theorem.
6. Power and limitations of the axiomatic methodComplete theories. Gödel's Incompleteness Theorems. What they mean, and what they don't mean. The interpretation of mathematics in set theory. Definable predicates. Impossibility of first-order definition of finiteness.
7. The axiomsAxioms of ZFC. Models for some of the axioms. Axiom schemata and nonconstructive axioms are discussed.
8. ClassesLegal and illegal ways of referring to proper classes in ZFC. Definitions of some classes that will be later used.
9. Versions of the axiom of choiceZorn's Lemma, Zermelo's Theorem, Hausdorff Maximum Principle, DC, Dedekind finite sets. Set theory without AC. The Axiom of Determinacy. The Banach-Tarski Paradox.
10. The ordinalsDefinition of ordinals. Ordinal arithmetic. Cantor's Normal Form Theorem. Goodstein's Theorem.
11. The cardinalsDefinition of cardinals as initial ordinals. Cofinality. Theorems of Hessenberg, Hausdorff, König, Tarski and Bukovsky-Hechler (with proof); and Silver, Easton and Shelah (without proof) on cardinal arithmetic. Inaccessible cardinals.
12. Pictures of the universeThe cumulative hierarchy. The models $V_\omega$, $V_{\omega + \omega}$, and $V_\kappa$ for strongly inaccessible $\kappa$. The constructible universe L. Equiconsistency of ZF and ZFC (sketch of proof). Absoluteness between V and L. Consistency of GCH (sketch of proof). Equiconsistency of weakly and strongly inaccesible cardinals.
Also available on this website: Information about Volume II, including table of contents, and topics covered.
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