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]
- Axiom of extensionality: If X and Y have the same elements, then X=Y
- Axiom of pairing: For any a and b there exists a set {a, b} that contains exactly a and b
- 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)}
- Axiom of union: For any set X there exists a set Y=∪X, the union of all elements of X
- Axiom of power set: For any X there exists a set Y=P(X), the set of all subsets of X
- Axiom of infinity: There exists an infinite set
- 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}
- Axiom of regularity: Every nonempty set has an ∈-minimal element
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 Suppes[2], or Hrbacek.[3]
References
- ↑ Thomas J Jech (1978). Set theory. Academic Press. ISBN 0123819504.
- ↑ Patrick Suppes (1972). Axiomatic set theory, Reprint of D Van Nostrand 1960 ed. Courier-Dover, pp. 14 ff. ISBN 0486616304.
- ↑ Karel Hrbacek, Thomas J. Jech. Introduction to set theory, 3rd ed. CRC Press. ISBN 0824779150.