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, AcademiaBruylant, LouvainlaNeuve
(Belgium), 1998, 242 pages
ISBN 2872094881
This Cahier can be ordered from the publisher AcademiaL'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 metatheorem.
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. 
WellFounded Extensional Relations 
9. 
Equivalence and Order 
21. 
The Structure of the Transfinite 
10. 
Introducing Functions 
22. 
Stratified LambdaCalculus 
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. 





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