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Axiomatic Set Theory

Paul Bernays
Publisher: 
Dover Publications
Publication Date: 
1991
Number of Pages: 
234
Format: 
Paperback
Price: 
14.95
ISBN: 
9780486666372
Category: 
Monograph
[Reviewed by
Allen Stenger
, on
02/17/2016
]

This book is a valuable historical exposition of the development of axiomatic set theory, but it is no longer useful as a textbook, because of the antiquated language. The present book is a Dover 1991 unaltered reprint of the 1968 second edition from North-Holland, which in turn was based on the 1958 first edition. The second edition added a bibliography of more recent works but seems to have no other changes.

Paul Bernays (1888–1977) was one of the pioneers of axiomatic set theory (he also worked in mathematical logic and foundations), and this book represented his take on how set theory should be developed. The book includes a valuable historical introduction to set theory (33 pages) by Abraham Fraenkel.

The axioms presented here are not the widely-used Zermelo–Fraenkel (ZF) axioms but the closely related von Neumann–Bernays–Gödel (NBG) axioms. NBG differs from ZF primarily in allowing classes as well as sets. This was the first book-length exposition of the NBG theory. Most of the book deals not with axiomatics per se but with the elements of set theory, in particular ordinals, cardinals, and transfinite recursion. The coverage is reasonable for an introductory book, although it necessarily omits newer topics such as forcing, filters, and ultrafilters.

The book is hard to follow because the notation has changed a lot since it was written (happily there is a thorough index of notation in the back) and because there are a lot of untranslated German terms, ranging from Aussonderungstheorem to Zahlenklasse.

Although this is not a bad book, time has passed and there are better books today. An oldie but goodie, that’s actually the same age, is Patrick Suppes’s Axiomatic Set Theory (Dover, 1972 reprint of the Van Nostrand 1960 edition). It has a much clearer exposition, and is a true textbook with lots of exercises. A more modern book, that I have not seen but that is well-regarded, is Hrbacek & Jech’s Introduction to Set Theory. It is a ZF-axiomatic development, but the axioms are introduced as needed and are scattered through the book, with a recapitulation at the end. If you are interested in the machinery of set theory without the axiomatics, Halmos’s Naïve Set Theory is a good start.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

 

PREFACE
  PART I. HISTORICAL INTRODUCTION
  1. INTRODUCTORY REMARKS
  2. ZERMELO'S SYSTEM. EQUALITY AND EXTENSIONALITY
  3. "CONSTRUCTIVE" AXIOMS OF "GENERAL" SET THEORY"
  4. THE AXIOM OF CHOICE
  5. AXIOMS OF INFINITY AND OF RESTRICTION
  6. DEVELOPMENT OF SET-THEORY FROM THE AXIOMS OF Z
  7. "REMARKS ON THE AXIOM SYSTEMS OF VON NEUMANN, BERNAYS, GÖDEL"
  PART II. AXIOMATIC SET THEORY
  INTRODUCTION
  CHAPTER I. THE FRAME OF LOGIC AND CLASS THEORY
  1. Predicate Calculus; Class Terms and Descriptions; Explicit Definitions
  2. Equality and Extensionality. Application to Descriptions
  3. Class Formalism. Class Operations
  4. Functionality and Mappings
  CHAPTER II. THE START OF GENERAL SET THEORY
  1. The Axioms of General Set Theory
  2. Aussonderungstheorem. Intersection
  3. Sum Theorem. Theorem of Replacement
  4. Functional Sets. One-to-one Correspondences
  CHAPTER III. ORDINALS; NATURAL NUMBERS; FINITE SETS
  1. Fundaments of the Theory of Ordinals
  2. Existential Statements on Ordinals. Limit Numbers
  3. Fundaments of Number Theory
  4. Iteration. Primitive Recursion
  5. Finite Sets and Classes
  CHAPTER IV. TRANSFINITE RECURSION
  1. The General Recursion Theorem
  2. The Schema of Transfinite Recursion
  3. Generated Numeration
  CHAPTER V. POWER; ORDER; WELLORDER
  1. Comparison of Powers
  2. Order and Partial Order
  3. Wellorder
  CHAPTER VI. THE COMPLETING AXIOMS
  1. The Potency Axiom
  2. The Axiom of Choice
  3. The Numeration Theorem. First Concepts of Cardinal Arithmetic
  4. Zorn's Lemma and Related Principles
  5. Axiom of Infinity. Denumerability
  CHAPTER VII. ANALYSIS; CARDINAL ARITHMETIC; ABSTRACT THEORIES
  1. Theory of Real Numbers
  2. Some Topics of Ordinal Arithmetic
  3. Cardinal Operations
  4. Formal Laws on Cardinals
  5. Abstract Theories
  CHAPTER VIII. FURTHER STRENGTHENING OF THE AXIOM SYSTEM
  1. A Strengthening of the Axiom of Choice
  2. The Fundierungsaxiom
  3. A one-to-one Correspondence between the Class of Ordinals and the Class of all Sets
  INDEX OF AUTHORS (PART I)
  INDEX OF SYMBOLS (PART II)
    Predicates
    Functors and Operators
    Primitive Symbols
  INDEX OF MATTERS (PART II)
  LIST OF ATOMS (PART II)
  BIBLIOGRAPHY (PART I AND II)
 
 
 

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