Zermelo-Fraenkel axioms

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

The axioms

There are eight Zermelo-Fraenkel (ZF) axioms;[1] for the meaning of the symbols, see Logic symbols. The numbering of these axioms varies from author to author.

Note: This article contains logic symbols. Without proper rendering support, you may see question marks, boxes, or other symbols instead of logic symbols.
  • 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 uX that have the property φ; that is, the set Y={uX | 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) | xX}
∀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:[2]

  • 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.

References

  1. Thomas J Jech (1978). Set theory. Academic Press. ISBN 0123819504. 
  2. Bell, John L. (Spring 2009 Edition). Edward N. Zalta, editor:The Axiom of Choice. The Stanford Encyclopedia of Philosophy.