Zermelo-Fraenkel axioms

The Zermelo-Fraenkel axioms form one of several possible formulations of axiomatic set theory.

The axioms

There are eight Zermelo-Fraenkel (ZF) axioms; for the meaning of the symbols, see Logic symbols. The numbering of these axioms varies from author to author.

• 1. Axiom of extensionality: If X and Y have the same elements, then X=Y

∀x∀y[∀z(z∈x ≡ z∈y) → x=y]

• 2. Axiom of pairing: For any a and b there exists a set {a, b} that contains exactly a and b

∀x∀y∃z∀w(w∈z ≡ w=x ∨ w=y)

• 3. Axiom schema of separation: If φ is a property with parameter p, then for any X and p there exists a set Y that contains all those elements u∈X that have the property φ; that is, the set Y={u∈X | φ(u, p)}

∀u1…∀uk[∀w∃v∀r(r∈v ≡ r∈w & ψx,û[r,û])]

• 4. Axiom of union: For any set X there exists a set Y = ∪ X, the union of all elements of X

∀x∃y∀z[z∈y ≡ ∃w(w∈x & z∈w)]

• 5. Axiom of power set: For any X there exists a set Y=P(X), the set of all subsets of X

∀x∃y∀z[z∈y ≡ ∀w(w∈z → w∈x)]

• 6. Axiom of infinity: There exists an infinite set

∃x[∅∈x & ∀y(y∈x → ∪{y,{y}}∈x)]

• 7. Axiom schema of replacement: If f is a function, then for any X there exists a set Y, denoted F(X) such that F(X)={f(x) | x∈X}

∀u1…∀uk[∀x∃!yφ(x,y,û) →

∀w∃v∀r(r∈v ≡ ∃s(s∈w & φx,y,û[s,r,û]))]

• 8. Axiom of regularity: Every nonempty set has an ∈-minimal element

∀x[x≠∅ → ∃y(y∈x & ∀z(z∈x → ¬(z∈y)))]

If to these is added the axiom of choice, the theory is designated as the ZFC theory:

• 9. Axiom of choice: Every family of nonempty sets has a choice function

For further discussion of these axioms, see the bibliography and the linked articles.