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Zermelo-Fraenkel Set Theory

In 1908 Ernst Zermelo proposed the axiomatic set theory now called Zermelo set theory. One of its drawback was that it did not allow to construct arbitrary ordinal numbers.  In 1922  Abraham Fraenkel and Thoralf Skolem independently gave impetus to formulation of the Axiom of Replacement. This axiom together with Zermelo set theory gives what we call the Zermelo-Fraenkel set theory, abbreviated as ZF [1], p. 31-96.

Similarly as the naïve set theory also ZF depends on a single primitive non-defined ontological notion, that of set, and a single ontological assumption, that all mathematical objects are sets. The second underlying item is a single primitive dyadic relation, the set membership: means that is an element of ¹ .  In other words, we have two primitive (i.e. undefined) concept in axiomatization of set theory:

• the concept of belonging
• the concept of equality

Note that set equality and the concept of belonging are two different things. Set equality is transitive, as follows from the axioms below. Unlike set equality, belonging is not transitive: point is an element of a straight line, , line is an element of a bundle of lines , but is not element of .

The axioms then govern how sets behave and interact. The axiom of the Zermelo-Fraenkel set theory  (ZF) are:

(Z1) Axiom of Existence:

Axiom Z1 ensures the existence of at least one set.

(Z2) Axiom of Extensionality:   
Note that this axiom of extension does not discribe only a logically necessary property of equality of two sets but touches in a non-trivial way also the concept of  belonging. For instance, if and are integers, write whenever divides . The axiom says that if and have the same divisors then and are equal.   But this is not true, take and . Thsu even it is not for the first moment apparent, the axiom of extensionality provides a non-trivial pronouncement about belonging.

It defines that two sets are equal if and only if they contain the same elements. Thus .

Define the notion of a subset as follows: if   and are two sets, then is a subset of ,  write  , if every element of   is also an element of . Then the Axiom of Extensionability can be expressed also in the following way:

() Axiom of Extensionality: Two sets are equal if and only if   and .

Let denote a formula in first-order theory that contains as a free variable:
(Z3) Axiom of Separation (Subset Axiom or Axiom of Comprehension):
That is, if is a set and is a condition on sets, there exists a set, in the common set-builder noatation denoted as , whose elements are precisely the elements for is true.  

(Z4) Unordered Pair Axiom:
There exists a set containing given two elements (sets).

(Z5) Union Axiom:
If is a set, then there exists a set containing every element of each .  This set is called the union and denoted .

(Z6) Powerset Axiom:
If   is a set, then there exists a set , usually denoted as with the property that if and only if any element is also an element of .

Axioms Z1 - Z6 are also called axioms of transformations and with their help  all of finite mathematics can be constructed.

The next axiom ensures the existence of an infinite set. In some axiom systems it is replaced by a weaker axiom Z1.  In what follows , where is some fixed set.

(Z7) Axiom of Infinity:
Another formulation: There exists a non-empty set with the property that, for any , there is some such that   but .

Axiom of infinity guarantees the existence of a set with elements , , , ...

Here denotes a function which takes any object as argument and yields as its value the singleton set with that object as its only member . Thus the set which existence is ensured by the Axiom of Infinity is the set .

By the way, the set of non-negative integers can be modeled using this set when is interpreted as , as , as , etc. In this case is the successor function which maps any non-negative integer to its successor value.

Another possibility for is .  

Let be a predicate in 2 variables not depending on . The next axiom of replacement scheme describes how new sets can be defined from exiting sets using relationship   that defines as a function fo
(ZF8) Axiom of Replacement: ,
or in words: given any set such that for any , there is a unique (denoted as ) such that holds for and , there is a set B which we get by replacing every element of with the object for which holds.

Notation is an abbreviation for .

Axiom of replacement the construction that if   is a set, and there is a way to assign each set to some set , then the image of   under this assignment is a set.

(Z9) Axiom of Regularity (Axiom of Foundation): .

The Axiom of Regularity, in combination with the Unordered Pair Axiom rules out every set that contains itself as a member. To see this remember that the Unordered Pair Axiom says that for every element of a set, there must be the singleton that contains it and nothing else. But the Axiom of Regularity rules this out, i.e. there is no set that is a member of itself.

(Z10) Axiom of Choice: If is a function² with non-empty domain and for each , is a non-empty set then there is a function also with domain such that for each we have .

If we define  , then the Axiom of choice can be formally define as

Axiom of Choice can be formulated in many ways.  .  The general idea behind this axiom is that for any collection of sets, you can choose one member from each of them. Here the values of function give the initial collection of sets, and the function is the selector function that chooses a single element from each of the sets that are the values of .

The above axioms without axiom Z8 form the so-called Zermelo set theory, abbreviated as Z. In this system it is possible to develop the almost all parts of the classical mathematics, as arithmetic, analysis, functional analysis, etc. However it is impossible to develop within Z the theory of cardinals greater than .  From this reason Fraenkel proposed to add a new axiom Z8.  The axioms(Z1)-(Z9)  form a foundation for what is now called Zermelo-Fraenkel set theory, or ZF. This set of axioms along with the Axiom of Choice is often denoted ZFC.  Since Axiom of Choice is non-constructive, it is customary to state explicitly when a mathematical result is proved using it.

In 1957, Richard Montague proved  [2]  that ZF (and consequently also ZFC) cannot be be finitely axiomatized, that is its construction requires at least one axiom schema.

ZFC is believed to be consistent ³. A.Abian and S.Lamacchia proved  [3] that if in a theory of sets every set is a power set and if the Powerset Axiom is valid then the Axioms of Extensionality, Union and of Choice are valid. They also proved [4]

• The Axioms of Extensionality, Replacement, Union and of Choice form a consistent system of axioms.
• The Axiom of Infinity is independent of Axioms of Extensionality, Replacement, Powerset, Union and of Choice.
• The Powerset Axiom is independent of Axioms of Axioms of Extensionality, Powerset, Union and of Choice.
• The Axiom of Infinity is independent of Axioms of Extensionality, Replacement, Union and of Choice.

Abian previously proved [5] that  the Axioms of Extensionality, Powerset, Union and of Choice form a consistent and independent system of axioms.

As mentioned above ZFC requires an infinite number of first-order axioms. Therefore the question whether it is possible to develop a set theory with a finite number of axioms is very natural. The question was answered in the affirmative by J. von Neumann in 1925.  However von Neumann used functions rather than sets as his primitive notion. The first-order version which is widely known now is due to the reworking in the late 1930's by Bernays with a contribution of Gödel. It is called von Neumann-Bernays-Gödel set theory, abbreviated to NBG set theory.

For the origin of notation used in set theory visit .

Notes

1	The symbol has its origin in a notation introduced by Peano. He derived it from Greek . He wrote in 1889: The sign signifies is.

References

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