. . . . . . 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

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Graeme McRae.