Book contents
- Frontmatter
- Contents
- Preface
- Synopsis
- 1 Introduction
- 2 Logical foundations
- 3 Avoiding Russell's paradox
- 4 Further axioms
- 5 Relations and order
- 6 Ordinal numbers and the Axiom of Infinity
- 7 Infinite arithmetic
- 8 Cardinal numbers
- 9 The Axiom of Choice and the Continuum Hypothesis
- 10 Models
- 11 From Gödel to Cohen
- A Peano Arithmetic
- B Zermelo–Fraenkel set theory
- C Gödel's Incompleteness Theorems
- Bibliography
- Index
B - Zermelo–Fraenkel set theory
Published online by Cambridge University Press: 05 August 2014
- Frontmatter
- Contents
- Preface
- Synopsis
- 1 Introduction
- 2 Logical foundations
- 3 Avoiding Russell's paradox
- 4 Further axioms
- 5 Relations and order
- 6 Ordinal numbers and the Axiom of Infinity
- 7 Infinite arithmetic
- 8 Cardinal numbers
- 9 The Axiom of Choice and the Continuum Hypothesis
- 10 Models
- 11 From Gödel to Cohen
- A Peano Arithmetic
- B Zermelo–Fraenkel set theory
- C Gödel's Incompleteness Theorems
- Bibliography
- Index
Summary
Zermelo-Fraenkel set theory (ZF) is a first-order theory with one primitive binary relation ∈ and no primitive operators together with the following nonlogical axioms. Here the axioms are given in a semi-colloquial form making use of some of the notation and terminology which is discussed in more detail in the main text.
Axiom 1: Axiom of Extensionality
For all sets a and b, a =o b if and only if a = b.
Axiom 2: Axiom of Pairing
For all sets a and b, {a,b} is a set.
Axiom 3: Axiom of Unions
For all sets a, ∪a is a set.
Axiom 4: Axiom of Powers
For all sets a, P(a) is a set.
Axiom 5: Axiom Schema of Replacement
If a predicate ø(x,y) induces a function then for all sets a, {y : x ∈ a and ø(x,y)} is a set.
Axiom 6: Axiom of Regularity
If a ≠ ø then there exists an x ∈ a such that x ∩ a = ø.
Axiom 7: Axiom of Infinity
ω is a set.
The Axiom of Pairing is redundant (i.e. it is a consequence of the other axioms).
Neither provable nor disprovable in ZF is the following, which is also assumed by most mathematicians.
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- The Logic of Infinity , pp. 417 - 418Publisher: Cambridge University PressPrint publication year: 2014