# Discovering Modern Set Theory. I: The Basics

## Winfried Just and Martin Weese

### Topics covered in Volume I:

Introduction

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

1. Pairs, relations, and functions

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

2. Partial order relations

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

3. Cardinality

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.

4. Induction

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

5. Formal languages and models

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

6. Power and limitations of the axiomatic method

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.

7. The axioms

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

8. Classes

Legal 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 choice

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.

10. The ordinals

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

11. The cardinals

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.

12. Pictures of the universe

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.

The book can be ordered online from the AMS Bookstore. You may also write to the American Mathematical Society, P.O. Box 5904, Boston, MA 02206-5904. For credit card orders, fax (401) 331-3842 or call toll free 800-321-4267 in the U.S. and Canada, (401) 455-4000 worldwide.

Publisher's Info:
• 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