Preface
Introduction
1.Sets and Relations and Operations among them
1.Set algebra and the set-builder
2.Russell's paradox
3.Infinite unions and intersections
4.Ordered couples and Cartesian products
5.Relations and functions
6.Sets of sets, power set, arbitrary Cartesian product
7.Structures
8.Partial orders and orders
2.Cardinal Numbers and Finite Sets
1.Cardinal numbers, +, and ≤
2.Natural numbers and finite sets
3.Multiplication and exponentiation
4.Definition by induction
5.Axiom of infinity, Peano axioms, Dedekind infinite sets
3.The Number Systems
1.Introductory remarks
2.Construction and characterization up to isomorphism of the integers, rationals, and reals
4.More on Cardinal Numbers
1.The Cantor-Bernstein Theorem
2.Infinite sums and products of cardinals
3.Different kinds of infinity
4.N0, 2N0, and 22N0 - the simplest infinite cardinals
5.Orders and Order Types
1.Ordered sums and products
2.Order types
6.Axiomatic Set Theory
1.A formalized language
2.The axioms of set theory
3.On Chapter 1
4.On Chapters 2-5
7.Well-orderings, Cardinals, and Ordinals
1.Well-orders
2.Von Neumann ordinals
3.The well-ordering theorem
4.Defining A and TpA
5.Easy consequences for cardinals of the Well-ordering Principle
6.A harder consequence and its corollaries
7.The Continuum hypothesis
8.The Axiom of Regularity
1.Partial universes and the axiom of regularity
2.Consequences of regularity
3.Avoiding replacement
9.Logic and Formalized Theories
1.Language and grammar
2.Truth and tables
3.Formal proofs
4.Substitution for predicate symbols and relativization
5.Three forms of ZFC
10.Independence Proofs
1.Relativization of axioms to partial universes.Consistency of adding Reg.
2.Three other consistency results
3.Informal note on the set theories NB and M
11.More on Cardinals and Ordinals
1.Character of cofinality
2.More on ordinal arithmetic
Appendix
Proofs of some results in Chapter 9
Bibliography
Other References
Recommendations for more advanced reading
Index
Solutions to Selected Problems
