The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1), 
stands for exists, 
means for all, 
stands for "is an element of," 
for the empty set, 
for implies, 
for AND, 
for OR, and 
for "is equivalent to."
1. Axiom of Extensionality: If 
and 
have the same elements, then 
.
 |
(1)
|
2. Axiom of the Unordered Pair: For any 
and 
there exists a set 
that contains exactly 
and 
. (also called Axiom of Pairing)
 |
(2)
|
3. Axiom of Subsets: If 
is a property (with parameter 
), then for any 
and 
there exists a set 
that contains all those 
that have the property 
. (also called Axiom of Separation or Axiom of Comprehension)
 |
(3)
|
4. Axiom of the Sum Set: For any 
there exists a set 
, the union of all elements of 
. (also called Axiom of Union)
 |
(4)
|
5. Axiom of the Power Set: For any 
there exists a set 
, the set of all subsets of 
.
 |
(5)
|
6. Axiom of Infinity: There exists an infinite set.
![exists S[emptyset in S ^ ( forall x in S)[x union {x} in S]].](/images/equations/Zermelo-FraenkelAxioms/NumberedEquation6.svg) |
(6)
|
7. Axiom of Replacement: If 
is a function, then for any 
there exists a set ![Y=F[X]={F(x):x in X}](/images/equations/Zermelo-FraenkelAxioms/Inline32.svg)
.
![forall x forall y forall z[phi(x,y,p) ^ phi(x,z,p)=>y=z]
=> forall X exists Y forall y[y in Y=( exists x in X)phi(x,y,p)].](/images/equations/Zermelo-FraenkelAxioms/NumberedEquation7.svg) |
(7)
|
8. Axiom of Foundation: Every nonempty set has an 
-minimal
element. (also called Axiom of Regularity)
![forall S[S!=emptyset=>( exists x in S)S intersection x=emptyset].](/images/equations/Zermelo-FraenkelAxioms/NumberedEquation8.svg) |
(8)
|
9. Axiom of Choice: Every family of nonempty sets has a choice function.
 |
(9)
|
The system of axioms 1-8 is called Zermelo-Fraenkel set theory, denoted "ZF." The system of axioms 1-8 minus the axiom of replacement (i.e., axioms 1-6 plus 8) is called Zermelo set theory, denoted "Z." The set of axioms 1-9 with the axiom of choice is usually denoted "ZFC."
Unfortunately, there seems to be some disagreement in the literature about just what axioms constitute "Zermelo set theory."
Mendelson (1997) does not include the axioms
of choice or foundation in Zermelo set
theory, but does include the axiom of replacement.
Enderton (1977) includes the axioms of choice and
foundation, but does not include the
axiom of replacement. Itô includes an
Axiom of the empty set, which can be gotten
from (6) and (3), via 
and 
.
Abian (1969) proved consistency and independence of four of the Zermelo-Fraenkel axioms.
