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 Math Help > Sets, Set theory, Number systems > Sets > ZF Set Theory

. . . . . . an article on ZF set theory will be forthcoming.  In the mean time, here is a link to Wikipedia

### ZF Axioms

1. Axiom of extensionality: Two sets are equal (are the same set) if they have the same elements.

2. Axiom of regularity (also called the Axiom of foundation): Every non-empty set x contains a member y such that x and y are disjoint sets.

3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): If z is a set, and Ø is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property.

4. Axiom of pairing: If x and y are sets, then there exists a set which contains x and y as elements.

5. Axiom of union: For any set F there is a set A containing every set that is a member of some member of F.

6. Axiom schema of collection: If the domain of a function f is a set, and f(x) is a set for any x in that domain, then the range of f is a subclass of a set, subject to a restriction needed to avoid paradoxes.

7. Axiom of infinity: There exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω.

8. Axiom of power set: For any set x, there is a set y which is a superset of the power set of x. The power set of x is the class whose members are all of the subsets of x.

### Axiom of Choice

9. Well-ordering theorem: For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R.

Axiom 9 depends on and is dependent on (i.e. is equivalent to) the axiom of choice.

### Internet references

Wikipedia: Zermelo–Fraenkel set theory

### Related pages in this website

Axiom of Choice -- Zermelo–Fraenkel set theory with the axiom of choice (ZFC set theory), Well-ordering theorem

The webmaster and author of this Math Help site is Graeme McRae.