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Published online by Cambridge University Press: 05 August 2014
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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