Book contents
- Frontmatter
- Contents
- Preface
- Part I Introduction to set theory
- Appendix. An axiomatic development of set theory
- Introduction
- A1 The Zermelo–Fraenkel axiom system of set theory
- A2 Definition of concepts; extension of the language
- A3 A sketch of the development. Metatheorems
- A4 A sketch of the development. Definitions of simple operations and properties (continued)
- A5 A sketch of the development. Basic theorems, the introduction of ω and ℝ (continued)
- A6 The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2–7
- A7 The role of the Axiom of Regularity
- A8 Proofs of relative consistency. The method of interpretation
- A9 Proofs of relative consistency. The method of models
- Part II Topics in combinatorial set theory
- Bibliography
- List of symbols
- Name index
- Subject index
A1 - The Zermelo–Fraenkel axiom system of set theory
Published online by Cambridge University Press: 10 May 2010
- Frontmatter
- Contents
- Preface
- Part I Introduction to set theory
- Appendix. An axiomatic development of set theory
- Introduction
- A1 The Zermelo–Fraenkel axiom system of set theory
- A2 Definition of concepts; extension of the language
- A3 A sketch of the development. Metatheorems
- A4 A sketch of the development. Definitions of simple operations and properties (continued)
- A5 A sketch of the development. Basic theorems, the introduction of ω and ℝ (continued)
- A6 The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2–7
- A7 The role of the Axiom of Regularity
- A8 Proofs of relative consistency. The method of interpretation
- A9 Proofs of relative consistency. The method of models
- Part II Topics in combinatorial set theory
- Bibliography
- List of symbols
- Name index
- Subject index
Summary
Before we list the axioms of set theory, we make one more important remark. The fact that in the language we have the only predicate symbols = and ∈, and we do not introduce a one-place predicate symbol to say that something is a set implicitly means that we only discuss sets; other objects are of no concern to us. The attentive reader may have noticed that this practice was tacitly followed even in the first part of the book, after the introduction of good sets. That this will not impose undue restrictions on us will be clear exactly from the fact that the axiomatic development can be carried out in this way.
The today generally accepted Zermelo–Fraenkel axiom system of set theory contains the following axioms.
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- Set Theory , pp. 111 - 113Publisher: Cambridge University PressPrint publication year: 1999