Cahiers du Centre de Logique, vol. 10

(épuisé / out of print)

References

M. R. HOLMES, Elementary Set Theory with a Universal Set
volume 10 of the Cahiers du Centre de logique, Academia-Bruylant, Louvain-la-Neuve (Belgium), 1998, 242 pages
ISBN 2-87209-488-1

This Cahier can be ordered from the publisher Academia-L'Harmattan.

Summary

This book presents an alternative approach to the foundations of mathematics, a variant on Quine's set theory "New Foundations" (usually abbreviated NF) which avoids the drawbacks of that system. R. B. Jensen's system NFU, in which extensionality is weakened to allow atoms, is used. To this system the axioms of Infinity and Choice are adjoined. The author introduces an additional strong infinity axiom, natural in the context of NFU, which is related to large cardinal hypotheses in the usual set theory (open questions remain as to its precise strength).


The book takes the form of an elementary set theory text, roughly parallel in structure to Halmos's classic "Naive set theory", with some additional topics and some more advanced material at the end. Two of the advanced chapters discuss ways to interpret the usual set theory ZFC in the system of the book.
The author hopes to convince the reader that the system of this book is at least as natural and mathematically fluent as the usual set theory ZFC (and, in fact, not so very different from the usual approach). This is the main purpose of the book.


The axiom scheme of stratified comprehension used in New Foundations has been criticized as a "syntactical trick"; we carefully address that criticism in the early chapters of this book, in which a small number of natural constructions for sets and relations are introduced, from which stratified comprehension is developed as a meta-theorem.


The book could be used as a first introduction to set theory (though the author does not recommend this), as an introduction to doing set theory in systems like NF, or as an introduction to the subject of alternative foundations of mathematics (in conjunction with materials on other nonstandard approaches).
A chapter of philosophical reflection is included, for which the author hopes that he may be forgiven. The author first discusses the notion of "set" independently of any particular theory (drawing conclusions similar but not identical to proposals of the philosopher David Lewis), then attempts an intuitive motivation of set theory with stratified comprehension.

Table of contents

1.

Introduction: Why Save the Universe?

13.

The Real Numbers

2.

The Set Concept

14.

The Axiom of Choice

3.

Boolean Operations on Sets

15.

Ordinal Numbers

4.

Building Finite Structures

16.

Cardinal Numbers

5.

The Theory of Relations

17.

Three Theorems

6.

Sentences and Sets

18.

Sets of Real Numbers

7.

Stratified Comprehension

19.

Strongly Cantorian Sets

8.

Philosophical Interlude

20.

Well-Founded Extensional Relations

9.

Equivalence and Order

21.

The Structure of the Transfinite

10.

Introducing Functions

22.

Stratified Lambda-Calculus

11.

Operations on Functions

23.

Acknowledgements and Notes

12.

The Natural Numbers

 

 

       

See here for an on line errata slip — A corrected text is published on line here.

       

17 l 16 l 15 l 14 l 13 l 12 l 11 l 9 l 8 l 7 l 6 l 5 l 4 l 3 l 2 l 1

 

 

September 15, 2009