*Some admonitions.*

*Some history. Role of set theory as foundation of mathematics.
Russell's Paradox. Naive vs. axiomatic set theory.
*

**PART ONE: NOT ENTIRELY NAIVE SET THEORY**

*Also Cartesian products, equivalence relations, equivalence classes.
Meant as a review of selected prerequisites.*

*Partial order, linear order,wellorder, and wellfounded relations.
Order types, operations on order types.*

*Equipotency, 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.*

*Induction 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**

*Formal languages, terms, formulas, proofs. Models. Theories,
consistency, independence. Gödel's Completeness Theorem.
Compactness Theorem.*

*Complete 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.*

*Axioms of ZFC. Models for some of the axioms.
Axiom schemata and nonconstructive axioms are discussed.*

*Legal and illegal ways of referring to proper classes
in ZFC. Definitions of some classes that will be later used.*

*Zorn's Lemma, Zermelo's Theorem,
Hausdorff Maximum Principle, DC,
Dedekind finite sets. Set theory without AC. The Axiom of Determinacy.
The Banach-Tarski Paradox.*

*Definition of ordinals. Ordinal arithmetic.
Cantor's Normal Form Theorem. Goodstein's Theorem.*

*Definition 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.*

*The 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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**Publisher:**American Mathematical Society**Distributor:**American Mathematical Society**Series:**Graduate Studies in Mathematics, ISSN: 1065-7339**Volume:**8**Publication Year:**1996**ISBN:**0-8218-0266-6**Paging:**210 pp.**Binding:**Hardcover**List Price:**$36**Institutional Member Price:**$29**Individual Member Price:**$29**Itemcode:**GSM/8