Zermelo-Fraenkel (ZF) is the earliest axiomatic system in set
theory. The first axiomatization was by Zermelo (1908). Fraenkel
(1922) observed a weakness of Zermelo's system and proposed a way to
overcome it. His proposal was reformulated by Skolem (1922) by
introducing a new axiom. This axiomatization is carried out in a
language which includes sets as objects and 
for membership.
Equality is defined externally by the Axiom of Extensionality
which states that two sets are equal if and only if they have the same
elements.
ZF's essential feature is the cumulative hierarchy it proposes (Parsons 1977). The intention is to build up mathematics by starting with the empty set and then construct further sets in a stepwise manner by various defined operators. Hence there are no individual objects (urelements) in the universe of this theory. The cumulative hierarchy works as follows (Tiles 1989).
The Null Set Axiom guarantees that there is a set with no
elements, i.e., the empty set 
. This is the only set whose
existence is explicitly stated. The Pair Set Axiom states the
existence of a set which has a member when the only existing set is
. So the set 
can now be formed now and we
have two objects 
and 
. The application of
the axiom repetitively yields any finite number of sets, each with
only one or two elements. It is the Sum Set Axiom which
states the existence of sets containing any finite number of elements
by defining the union of already existing sets. Thus 
. However it
should be noted that all these sets will be finite because only
finitely many sets can be formed by applying Pair Set and Sum Set
finitely many times. It is the Axiom of Infinity which states
the existence of at least one infinite set, from which other infinite
sets can be formed. The set which the axiom asserts to exist is
.
The cumulative hierarchy is depicted in Figure 1.
Thus, the ZF universe simply starts with the 
and extends
to infinity. It can be noticed that cumulative hierarchy produces all
finite sets and many infinite ones, but it does not produce all
infinite sets (e.g., V).
Figure 1: ZF universe extending in a cumulative hierarchy
While the first five axioms of ZF are quite obvious, the Axiom of
Foundation cannot be considered so. The axiom states that every
set has elements which are minimal (cf. Note 3) with respect to
membership, i.e., no infinite set can contain an infinite sequence of
members 
. Infinite
sets can only contain sets which are formed by a finite number of
iterations of set formation. Hence this axiom forbids the formation
of sets which require an infinity of iterations of an operation to
form sets. It also forbids sets which are members of themselves,
i.e., circular sets. Russell's Paradox is avoided since the
problematic set 
cannot be shown to exist. (This will be
demonstrated shortly.) The Axiom of Separation makes it
possible to collect together all the sets belonging to a set whose
existence has already been guaranteed by the previous axioms and which
satisfy a property 
:
The axiom does not allow to simply collect all the things satisfying a
given meaningful description together into a set, as assumed by Cantor
by his Axiom of Abstraction. It only allows to form subsets of a set
whose existence is already guaranteed. It also forbids the universe
of sets to be considered as a set, hence avoiding the Cantor's Paradox
of the set of all sets. The Axiom of Replacement is a
stronger version of the Axiom of Separation. It allows the use of
functions for the formation of sets but still has the restriction of
the original Axiom of Separation. It should be noted that these two
axioms are in fact not single axioms but axiom schemes. They
become axioms when one substitutes a specific description or
relational expression in the language of ZF instead of the variable
expression 
. Therefore, we say that ZF is not finitely
axiomatizable (cf. Note 4).
The Power Axiom states the existence of the set of all subsets
of a previously defined set. The formal definition of the power
operation, P, is 
. The Power Axiom
is an important axiom, because Cantor's notion of an infinite number
was led by showing that for any set, the cardinality of its power set
must be greater than its cardinality (cf. Note 5).
The Axiom of Choice is not considered as a basic axiom and is explicitly stated when used in a proof. ZF with the Axiom of Choice is known as ZFC.
It should be noted that the informal notion of cumulative hierarchy
summarized above has a formal treatment. The class WF of
well-founded sets is defined recursively in ZF starting with
and iterating the power set operation P where a
rank function 
is defined for 
, the
class of all ordinals (cf. Note 6):
,
,
when 
is
a limit ordinal,
.
This universe of WF is depicted in Figure 2 which bears a resemblance to Figure 1. This is justified by the common acceptance of the statement that the universe of ZF is equivalent to the universe of WF (Kunen 1980).
Let us now recall Russell's Paradox. We let r be the set whose
members are all sets x such that x is not a member of x. Then
for every set x, 
if and only if 
. Substituting r for x, we obtained the contradiction.
With the preceding discussion of WF the explanation is not
difficult. When we are forming a set z by choosing its members, we
do not yet have the object z, and hence cannot use it as a member of
z. The same reasoning shows that certain other sets cannot be
members of z. For example, suppose that 
. Then we cannot
form y until we have formed z. Hence y is not available and
therefore cannot be a member of z. Carrying this analysis a bit
further, we arrive at the following. Sets are formed in ``stages.''
For each stage S, there are certain stages which are before S.
Stages are important because they enable us to form sets. Suppose that
x is a collection of sets and 
is a collection of stages
such that each member of x is formed at a stage which is a member of
. If there is a stage after all of the members of 
,
then we can form x at this stage. Now the question becomes: Given a
collection 
of stages, is there a stage after all of the
members of 
? We would like to have an affirmative answer to
this question. Still, the answer cannot always be ``yes''; if 
is the collection of all stages, then there is no stage after every
stage in 
.
Figure 2: WF universe defined recursively in terms of ordinals
It can be said that ZF and NBG produce essentially equivalent set
theories, since it can be shown that NBG is a conservative
extension of ZF, i.e., for any sentence 
, if ZF 
, then NBG 
(Mendelson 1987). The main
difference between the two is that NBG is finitely axiomatizable,
whereas ZF is not. Still, most of the current research in set theory,
e.g., research on independency and consistency, is being carried out
in ZF. Nevertheless, ZF has its own drawbacks (Barwise 1975). First
of all, it is too weak to decide some questions like the
Continuum Hypothesis (Gödel 1947). Another critical point is
that while the cumulative hierarchy provides a precise formulation of
many mathematical concepts, it may be asked whether it is limiting, in
the sense that it might be omitting some interesting sets one would
like to have around, e.g., circular sets. Clearly, the theory is weak
in applications involving self-reference because circular sets are
prohibited by the Axiom of Foundation.
Strangely enough, ZF is too strong in some ways. Important
differences on the nature of the sets defined in it are occasionally
lost. For example, being a prime number between 6 and 12 is a
different property than being a solution to 
, but
this difference disappears in ZF. Similarly, for an arbitrary Abelian
group 
, all of the following subgroups of G are
considered as equivalent in ZF (Barwise 1975), while the definitions
are increasing in logical complexity (cf. Note 7):
the left coset of G,
for some integer n>0 = the torsion subgroup
of G,
is a divisible subgroup of 
the
divisible part of G.
A desirable property, the Principle of Parsimony, which states
that simple facts should have simple proofs, is quite often violated
in ZF (Barwise 1975). For example, the verification of a trivial fact
like the existence in ZF of 
, the set of all ordered pairs
such that 
and 
, relies on the
Power Set Axiom (cf. Note 8).
It can also be claimed mathematical practice suffers from the fact
that all the mathematical objects are represented as sets in ZF. For
example, while one can construct in ZF something isomorphic to the
real line, the practicing mathematician is not very interested in
this. Representing reals as sets could be considered important from a
theoretical viewpoint, but we should hardly ever worry about the fact
that 
can be determined by the infinite sequence 