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B - Zermelo–Fraenkel set theory

Published online by Cambridge University Press:  05 August 2014

Barnaby Sheppard
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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 - 418
Publisher: Cambridge University Press
Print publication year: 2014

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