Theory of Sets 1st Edition by E Kamke ISBN 0486601412 9780486601410

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ISBN 10: 0486601412 
ISBN 13: 9780486601410
Author: E Kamke

“Exceptionally well written.” — School Science and Mathematics
“A very fine book.” — Mathematics Teacher
“Of real service to logicians and philosophers who have hitherto had no access to a concise and accurate introduction to the general theory of sets.” — Philosophical Review
This is the clearest and simplest introduction yet written to the theory of sets. Making use of the discoveries of Cantor, Russell, Weierstrass, Zermelo, Bernstein, Dedekind, and other mathematicians, it analyzes concepts and principles and offers innumerable examples. Its emphasis is on fundamentals and the presentation is easily comprehensible to readers with some college algebra. But special subdivisions, such as the theory of sets of points, are considered.
The contents include rudiments (first classifications, subsets, sums, intersection of sets, nonenumerable sets, etc.); arbitrary sets and their cardinal numbers (extensions of number concept, equivalence of sets, sums and products of two and many cardinal numbers, etc.); ordered sets and their order types; and well-ordered sets and their ordinal numbers (addition and multiplication of ordinal numbers, transfinite induction, products and powers of ordinal numbers, well-ordering theorem, well-ordering of cardinal and ordinal numbers, etc.).

Theory of Sets 1st Table of contents:

• Chapter I: Introduction

  • The Concept of a Set
  • Examples of Sets
  • Ways of Describing Sets
  • Equality of Sets
  • Subsets
  • The Null Set
  • Universal Set and Complement
  • Set Operations (Union, Intersection, Difference)
  • Venn Diagrams

• Chapter II: Cardinal Numbers

  • One-to-One Correspondence
  • Equivalent Sets
  • The Concept of a Cardinal Number
  • Finite and Infinite Sets
  • Countable Sets (Denumerable Sets)
  • Examples of Countable Sets (Integers, Rational Numbers)
  • Cantor’s Diagonal Method
  • Uncountable Sets (Real Numbers)
  • Cardinal Numbers of Uncountable Sets
  • Comparison of Cardinal Numbers
  • The Continuum Hypothesis (brief mention)

• Chapter III: Ordinal Numbers

  • Ordered Sets
  • Similar Sets (Order Isomorphic Sets)
  • Order Types and Ordinal Numbers
  • Well-Ordered Sets
  • Ordinal Numbers of Well-Ordered Sets
  • Properties of Ordinal Numbers
  • Transfinite Induction
  • Operations on Ordinal Numbers (Addition, Multiplication)
  • The Principle of Transfinite Recursion

• Chapter IV: Zermelo’s Axiom of Choice and its Applications

  • Statement of the Axiom of Choice
  • Equivalents of the Axiom of Choice (Well-Ordering Theorem, Zorn’s Lemma)
  • Applications in Mathematics (e.g., existence of basis in vector spaces, Hahn-Banach theorem – likely basic applications)

• Chapter V: Paradoxes in Set Theory

  • Russell’s Paradox
  • Cantor’s Paradox
  • The Burali-Forti Paradox
  • Axiomatic Set Theory (brief introduction to the need for it)

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Tags: E Kamke, Theory, Sets