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1993 | Buch | 2. Auflage

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The Joy of Sets

Fundamentals of Contemporary Set Theory

verfasst von: Keith Devlin

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

This book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. The book is written in a fairly easy-going style, with minimal formalism. In Chapter 1, the basic principles of set theory are developed in a 'naive' manner. Here the notions of 'set', 'union', 'intersection', 'power set', 'rela tion', 'function', etc., are defined and discussed. One assumption in writing Chapter 1 has been that, whereas the reader may have met all of these 1 concepts before and be familiar with their usage, she may not have con sidered the various notions as forming part of the continuous development of a pure subject (namely, set theory). Consequently, the presentation is at the same time rigorous and fast.

Inhaltsverzeichnis

Frontmatter

1. Naive Set Theory

Abstract

Zermelo—Fraenkel set theory, which forms the main topic of the book, is a rigorous theory, based on a precise set of axioms. However, it is possible to develop the theory of sets considerably without any knowledge of those axioms. Indeed, the axioms can only be fully understood after the theory has been investigated to some extent. This state of affairs is to be expected. The concept of a’ set of objects’ is a very intuitive one, and, with care, considerable, sound progress may be made on the basis of this intuition alone. Then, by analyzing the nature of the’ set’ concept on the basis of that initial progress, the axioms may be ‘discovered’ in a perfectly natural manner.

Keith Devlin

2. The Zermelo—Fraenkel Axioms

Abstract

In this chapter, I develop an axiomatic framework for set theory. For the most part, the axioms will be simple existence assertions about sets, and it may be argued that they are all self-evident ‘truths’ about sets. But why axiomatize set theory in the first place? Well, for one thing, it is well known that set theory provides a unified framework for the whole of pure mathematics, and surely if anything deserves to be put on a sound basis it is such a foundational subject. “But surely,” you say, “the concept of a set is so simple that nothing further need be said. We simply regard any collection of objects as a single entity in its own right, and that provides us with our set theory.” Alas, nothing could be further from the truth. Certainly, the idea of being able to regard any collection of objects as a single entity forms the very core of set theory. But a great deal more needs to be said about this.

Keith Devlin

3. Ordinal and Cardinal Numbers

Abstract

The concept of an ordinal number (or ordinal) was introduced in Section 1.7, where an ordinal was defined to be a woset (X, <) such that

$$ a = \{ x \in X|x < a\} $$

for every a ∈ X. We saw that any two ordinals are either identical or else nonisomorphic (as ordered sets), and that, if X, Y are nonidentical ordinals, then either X ∈ Y or Y ∈ X. We also noted that, if (Y, <) is an ordinal, then the ordering < is just ⊂ on X, which (in the case of ordinals) is just ∈ on X. (This justifies my referring simply to X, Y above)

Keith Devlin

4. Topics in Pure Set Theory

Abstract

In this chapter, we take a look at a number of topics in pure set theory. Some of the proofs are fairly complex, and at first reading, can be glossed over, or even ignored, without affecting the ability to follow the remainder of the book. The various sections in the chapter are all largely independent of each other.

Keith Devlin

5. The Axiom of Constructibility

Abstract

Before reading this chapter, the reader should go back and reread Sections 2.2 and 2.3 of Chapter 2, where we developed the concept of the set-theoretic hierarchy, 〉V_α ∣ α ∈ On〉.

Keith Devlin

6. Independence Proofs in Set Theory

Abstract

The following statements are known to be undecidable in the system ZFC. (Though they are all decidable in constructible set theory, by the way.)

Keith Devlin

7. Non-Well-Founded Set Theory

Abstract

The approach to set theory that has motivated and dominated the study presented so far in this book has essentially been one of synthesis: from an initial set of axioms, we build a framework of sets that can be used to provide a foundation for all of mathematics. By starting with pure sets provided by the Zermelo—Fraenkel axioms, and progressively adding more and more structure, we may obtain all of the usual structures of mathematics. And then, of course, we may make use of those mathematical structures to model various aspects of the world we live in. In this way, set theory may be used to provide ways to model ‘mathematical’ aspects of our world.

Keith Devlin

Backmatter

Titel: The Joy of Sets
verfasst von: Keith Devlin
Verlag: Springer New York
Electronic ISBN: 978-1-4612-0903-4
Print ISBN: 978-1-4612-6941-0
DOI: https://doi.org/10.1007/978-1-4612-0903-4

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Naive Set Theory

2. The Zermelo—Fraenkel Axioms

3. Ordinal and Cardinal Numbers

4. Topics in Pure Set Theory

5. The Axiom of Constructibility

6. Independence Proofs in Set Theory

7. Non-Well-Founded Set Theory

Backmatter