Set theory of von Neumann-Bernays-Gödel

The set theory of von Neumann-Bernays-Gödel , shortened in NBG or theory of the classes , is a axiomatic Théorie primarily equivalent to the theory ZFC of Zermelo-Fraenkel with Axiome of the choice (and with the same possible alternatives), but whose expressive capacity is richer. It can be stated in a finished number of axioms , and thus without diagram, contrary to ZFC (see Schéma of axioms of comprehension and Schéma of axioms of replacement). This is not possible that thanks to a modification of the language of the theory, which makes it possible to speak directly about class, a notion in addition useful in Set theory and which appeared already, in a rather abstract way, in the writings of Georg Cantor as of before 1900.

The theory of the classes was introduced in 1925 by John von Neumann, was re-examined and simplified successively by Paul Bernays about 1930, then by Kurt Gödel, for his proof of relative coherence of the Axiome of the choice and the Hypothèse of continuous the by the constructible ones, at the time of conferences with Princeton in 1937-1938 (published in 1940).

Classes

Classes in set theory

The set theories like the theory of Zermelo (in its modern D-stating) and its extensions (Zermelo-Fraenkel, ZFC, etc) are stated in the language of the calculation of the first order predicates levelling built on the only symbol of the membership. In other words, the statements use only variables, which indicate Ensemble S, the equality and the membership, which are relations between these units; they combine relationships to the logical symbols, connectors and quantifiers. In particular the only basic objects of the theory are the units.

These set theories do not have variables for the classes, which are collections of units well defined but “too large” to be whole without leading to paradoxes. Thus the collection of the units which do not belong to themselves cannot be a unit, according to the Paradoxe of Russell. It is however well defined: it is a class. One can thus speak about class in these theories only through the predicates of the language which define them (as that is exposed in the article on the classes).

Classes like primitive objects

Another solution is to add variables for the classes, one has two types of basic objects now, the units and the classes, and it is completely possible to axiomatize such a theory in order not to modify the demonstrable statements of the set theory of origin, those which thus do not call upon these new variables of class. Theory NBG is such a theory. There exists about it in fact as many alternatives as for the set theory in the source language, according to the axioms which one will or not choose to add.

Classes and predicates

Once added to the language of the variables of classes, it is very simple to axiomatize NBG starting from ZFC: it is enough to add a diagram of axioms, the diagram of comprehension for the classes , which associates a class with any predicate (with a variable) of the set theory. The predicates with several variables can be represented thanks to the couples of Wiener-Kuratowski. There is not any more but D-to state all the axioms of theory ZFC by restricting the variables with the units. The diagrams of axioms, which use an axiom by predicate of the set theory, can then be represented by only one axiom, thanks to the concept of class.

There is however a small difficulty: as the language of the theory was extended, this one makes it possible to define new predicates. The classes thus should be forced to be defined by the old predicates, those which do not use a variable of class. In fact it is enough that they do not use a quantifier on the classes, the not quantified variables are simple parameters, to which ultimement purely ensemblists could be substituted only predicates.

Thus one obtains a theory of the classes very similar to ZFC but which finiment is still not axiomatized: the diagram of replacement of ZFC can be replaced by only one axiom, but a new diagram was introduced. However it is now possible to reduce this diagram to a finished number of cases, by mimant in the classes the inductive construction of the formulas of ZFC: to each rule of construction, of finished number, corresponds an axiom of existence. As the formulas with several free variables are represented using finished lists of variables built by the couples of Wiener-Kuratowski, it is necessary to add some axioms to manage those. In addition to that would lead to a stronger theory, it is essential that the predicates are not all the predicates of the new language so that this construction is possible.

This process is not particular with the set theory, and was adapted to other theories with diagrams of axioms, for example within the framework of the arithmetic one.

Axiomatization of NBG

Classes and units

Theory NBG has two types of objects, the units and the classes. Intuitively all the units are classes but certain classes, called clean classes , are not units. There are two possibilities to represent this situation. Bernays uses a calculation of the first order predicate, but with two kinds of basic objects, the units and the classes. Gödel uses the calculation of the first order predicates, with only one kind of object, here the classes, by defining the overall concept by a predicate on the classes. This predicate can be defined starting from the membership: the units are the classes which belong at least to a class, it is thus not need to introduce a new symbol of predicate. The difference between the two approaches is not very essential. In first case one can to write that a class belongs to a class, such statement has not direction, but, when the left member is a class which defines a unit, one can make him correspond a correct statement. In the second case one can write such a statement, but it is always false if the left member is not a unit. The approach followed by Bernays perhaps more immediate but is introduced a certain redundancy since a unit has two representations, one as a unit, the other as a class.

The language of the theory

The adopted point of view is the second, the only primitive objects are the classes , the only variables of the language are variables of class. Another primitive notion of the theory is the concept of membership , noted “∈”. Like the set theory ZFC, theory NBG is a first order levelling logical theory, with the membership like primitive nonlogical symbol. Theory NBG is thus, like ZFC, expressed in Calcul of the predicates first order egalitarian

is a together any class which belongs to at least a class:

“ X is a unit” means: ∃ C X ∈ C .
The classes which are not units are called clean classes :

“ X is a clean class” if and only if: ∀ C X ∉ C .
Any class is thus either a unit, or a clean class, and no class is both at the same time (at least if the theory is coherent).
The existence of at least a clean class will be a consequence of the axioms of comprehension for the classes: one shows that the classes of the X which do not belong to themselves cannot be a unit, it is the argument of the Paradoxe of Russell. As soon as there exists a clean class, by definition of those, it cannot exist of class of the clean classes just like of class of all the classes .
The existence of at least a unit will be also a consequence of the axioms (axiom of the empty set for example). It will be seen that one can deduce from the axioms of this theory the existence of a class of all the units, that one will note “V”, i.e. that one will have for all X :

X ∈ V if ∃ C X ∈ C

One can define the predicate “being a unit”, one can thus define quantifications relativized in the whole in the usual way, for a formula Φ :

∀ X ^V Φ and an abbreviation of ∀ X ∃ '' C '' '' X '' ∈ '' C '') → '' Φ '' and means “for any unit X , Φ ”:
∃ X ^V Φ and an abbreviation of ∃ X ∃ '' C '' '' X '' ∈ '' C '') ∧ '' Φ '' and means “it exists a unit X such as Φ ”.
One thus will translate the axioms of ZFC by relativizing them with the units.

Axioms of theory NBG

The theory NBG takes again the axioms of ZFC , modified to take account of the classes, to which she adds axioms connecting class and predicate.
Extensionality

Axiom of extensionnality

If two classes have the same elements, then they are identical .
i.e.:

∀ has , B '' X '' ('' X '' ∈ '' has '' ↔ '' X '' ∈ '' B '') → '' has '' = '' B ''
The axiom has as a particular case the axiom of extensionnality for the units (one can of course suppose that has and B is units, in any use of this statement, X can only indicate one unit, it is useless to specify it). Generalization with the classes goes from oneself. It is pointed out that, as in any levelling theory, the equality satisfied, in addition to reflexivity, the diagram of axioms according to, which states that if has = B , any property of has is a property of B :

∀ C₁,…, C_n ∀ has , B → ('' Φ has C₁… C_n '' → '' Φ B C₁… C_n '')
who has as a particular case reciprocal axiom of extensionnality.

Axioms ensemblists
For these axioms, one takes again their formulation in ZFC, by relativizing the quantifications with the units (it is not inevitably necessary to relativize all the quantifications with the units, some are it already implicitly, but this does not have great importance). Thanks to the classes the diagrams of axioms become axioms.

Axiome of the empty set

the axiom of the empty set does not result directly any more from comprehension, as in ZFC, since if one supposes in first order logic that the model of interpretation is always nonempty, it can contain very well here only one class (the empty class). One thus needs the axiom:
There exists a unit which does not have elements.
i.e.:
∃ has ^V ∀ X X ∉ has .
Just like in ZFC, the unicity of this unit rises from the axiom of extensionnality, and one can speak about the empty set, and use the usual notation ∅.
the empty set is also the empty class, but the existence of the class empties could have been shown by comprehension on the classes (see the continuation) without specific axiom. It will be seen that, by an axiom of passage to complementary, the existence of the class of all the units will result from the existence of the empty class.

Axiom of the pair

If X and is units there, then there exists a unit of which they are the single elements.
i.e.:
∀ X ^V ∀ there ^V ∃ p ^V ∀ Z ∈ '' p '' ↔ ('' Z '' = '' X '' ∨ '' Z '' = '' there '').

Just like in ZFC, the unicity of this whole for X and given rises there from the axiom of extensionnality, and one can thus speak well about the Paire X , there and one can introduce the notation { X , there }.
One is satisfied to recall the axioms according to, whose statements are formalized in the language of the theory of the classes exactly in the same way.

Axiom of the meeting .

If X is a unit, then there exists at least a whole which is the meeting of the elements of X , i.e. to which belong all the elements of the elements of X and them only .

Axiom of the whole of the parts

If X is a unit, then there exists a unit containing the subsets of X and them only .

Axiom of comprehension for the units or axiom of separation

In ZFC, it acts of a diagram of axioms, in NBG, it is reduced to the only following axiom:
For any class C and any unit X , there exists a unit has which gathers the elements of X which also belong to C .
i.e.:
∀ C ∀ X ^V ∃ has ^V∀ Z ∈ '' has '' ↔ ('' Z '' ∈ '' X '' ∧ '' Z '' ∈ '' C '')
In this context, where comprehension in fact is managed on the level of the classes (following section), one will speak rather about axiom of separation.

Axiome of infinite the

There exists a unit to which the empty set belongs, and which is closed by the operation which with the unit X associates the unit X ∪ { X }.
∃ has ^V ∈ '' has '' ∧ ∀ '' X '' ('' X '' ∈ '' has '' → '' X '' ∪ {'' X ''} ∈ '' has '').
One can speak well about singleton and binary meeting ensemblist in the presence of the axiom of the pair, of the axiom of the meeting, and the axiom of extensionnality (for unicity).

Axiome of replacement still says also axiom of substitution

Là, which is a diagram of axioms in ZFC, is brought back to only one axiom in NBG. But it is necessary for that to have defined the couples. One takes the usual definition of Wiener-Kuratowski, for two units X and one has there by definition:

( X , there ) =

the use of this notation requires the axiom of the pair, and the axiom of extensionnality. One easily defines the predicate “being a couple”, the first and the second projections of a couple. One can thus say that a class is functional:
C functional signifie ∀ Z ∈ '' C '' → ∃ '' X '' ^V∃ '' there '' ^V '' Z '' = ('' X '', '' there '') ∧ ∀ X ∀ there ∀ there' ∈ '' C '' ∧ ('' X '', '' there' '') ∈ '' C '' → '' there '' = ''' ''
One can there now state the axiom:
Is a functional class C , then for any unit X , it exists a unit has gathering the second components of the couples of C whose first component belongs to X .
i.e.:

∀ C ( C functional → ∀ X ^V ∃ has ^V ∀ Z ^V ∈ '' has '' ↔ ∃ '' T '' ('' T '' ∈ '' X '' ∧ ('' T '', '' Z '') ∈ '' C ''))

Axiome of foundation says also axiom of regularity

Any not-vacuum unit has at least an element with which it does not have a common element .
or:
∀ X ^V ≠ ∅ → ∃ '' has '' ('' has '' ∈ '' X '' ∧ '' has '' ∩ '' X '' = ∅)
state Them uses notations introduced for clearness (∅, ∩) but is translated without evil into statements of the theory of the classes.

Axiom of the choice

The version of Zermelo is simplest to formalize:

For any family of units disjoined not-vacuums two to two, there exists a unit containing an element and only one of each whole of the family.
i.e.:
∀ U ^V {∀ Z '' Z '' ∈ '' U '' → '' Z '' ≠ ∅ ∧ ∀ '' Z' '' (['' Z' '' ∈ '' U '' ∧ '' Z' '' ≠ '' Z '' → Z ∩ Z' = ∅)] → ∃ C ^V ∀ Z '' Z '' ∈ '' U '' → ∃! '' X '' ('' X '' ∈ '' Z '' ∩ '' C '')}

where ∃! X Φx mean ∃ X ∧ ∀ '' there '' ('' Φy '' → '' there '' = '' X '') (there exists single a X such as Φx ).
There still, the symbols “∅” and “∩” are eliminated easily.

Comprehension for the classes
One could state a diagram of axioms of comprehension for the classes , namely that, any predicate of the language of the theory whose quantifiers are restricted with the units determines a class (the diagram is stated, like theorem, at the end of the section).
The predicate can contain parameters, which do not indicate necessarily units. It would be about a diagram of axioms, an axiom for each formula. However this diagram can be axiomatized finiment. Primarily, thanks to the classes, it is possible to rebuild all the formulas (of which the quantifiers are restricted with the units!) while bringing back them, primarily, with a finished number of particular cases. There is an axiom for the relation of membership (atomic formulas), the axioms for each logical operator (negation, conjunction, existential quantification), and of the axioms which make it possible to permute in a homogeneous way the elements of a class of tuples, in order to be able to make pass any component in tail, and thus to restrict itself with this case for the existential quantification. The preceding axioms make it possible well to define, like usually in set theory, the couples (and thus tuples) of Wiener-Kuratowski:

( X , there ) =
It is necessary to make a choice for the definition of the tuples. One agrees in the continuation that by definition:

( X ) = X ;
for N > 1, ( X ₀,…, X _n-1, X _n) = (( X ₀,…, X _n-1), X _n).

The first four axioms correspond to constructions of the formulas of the set theory: atomic formulas (membership, equality from of deduced), negation, conjunction and existential quantification.

There exists a class containing exactly all the couples of whole such as the first belongs to the second .

∃ has ∀ X ^V ∀ ^V ∈ has there ↔ '' X '' ∈ '' there ''
One can do, by extensionnality, of a specific axiom for the levelling atomic formulas.

For any class, there exists a complementary class gathering the units not belonging to this class .

∀ C ∃ D ∀ X ^V ( X ∈ D ↔ X ∉ C )
Note:: One thus shows the existence of the class of all the units, noted V, which is the class complementary to the empty class.

the intersection of two classes is a class .

∀ C ∀ D ∃ NR ∀ X ^V ∈ '' NR '' ↔ ('' X '' ∈ '' C '' ∧ '' X '' ∈ '' D '')
One can use the usual sign of intersection for the classes. For example the axiom of separation can be reexpressed by saying that for any class C , and for any unit X , X ∩ C is a unit. One deduces from this axiom and the preceding axiom, by law of Morgan, that the meeting of two classes is a class.

the field of a class of couples is a class, i.e. which there exists a class gathering the first components of a class of couples .

∀ C ∃ E ∀ X ^V ∈ '' E '' ↔ ∃ '' there '' ^V ('' X '', '' there '') ∈ '' C ''
By passages to complementary, one has a similar axiom for the universal quantifier.
There are now axioms to manage on the level of the classes the variables of the predicates. One needs, to build predicates with several variables, to add “useless” variables, thus ( Φx ∧ Ψy ), formula with two free variables, is obtained by conjunction of two formulas to a free variable. It is the use of the axiom which follows.

For any class, the couples whose first component is element of this class form themselves a class .

∀ C ∃ D ∀ X ^V ∀ there ^V ∈ '' D '' ↔ '' X '' ∈ '' C ''
The adequate axiom makes it possible to represent the existential quantification on the last component of the tuples of the corresponding class. Permutations are useful to manage the order of these components. They are the object of the last two axioms.

For any class of triplets, the triplets obtained by circular shift of those form a class .

∀ C ∃ D ∀ X ^V ∀ there ^V ∀ Z ^V ∈ '' D '' ↔ ('' Z '', '' X '', '' there '') ∈ '' C ''

For any class of triplets, the triplets obtained by transposition of the first two components form a class .

∀ C ∃ D ∀ X ^V ∀ there ^V ∀ Z ^V ∈ '' D '' ↔ ('' there '', '' X '', '' Z '') ∈ '' C ''
One deduces from these seven axioms the diagram of comprehension for the classes, which is thus a diagram of theorems.
Proposal (diagram of comprehension for the classes). For any formula Φ of which all the quantifications are relativized with the units, and whose free variables are among x₁,…, x_n, C₁,…, C_p, one has :

∀ C ₁… ∀ C _p ∃ D ∀ X ₁^V… ∀ X _n^V…, x_n '') ∈ '' D '' ↔ '' Φ ''.
Proof. One shows this result, for any formula Φ of which all the quantifications are relativized with the units, and for any continuation finished of variables (x₁,…, x_n) container (with the direction ensemblist) the free variables of Φ , by induction on the construction of Φ . In calculation of the first order predicates, any formula can be written with ¬, ∧, ∃, and thus, once the result shown for the atomic formulas, it follows directly by the axioms of complementary, the intersection, and the field. In this last case one uses the selected definition of the tuples; well-sure it is essential that the quantification is relativized with the units to be able to use the axiom. It thus remains to show the result for the atomic formulas.
First of all one can, by using the extensionnality to eliminate the equalities. One also can, by susbstituant an adequate formula, to eliminate the occurrences from the parameters C_i on the left of the membership, and the formulas of the form X ∈ X . There remain two types of atomic formulas to treat, those of the form x_i ∈ x_j , where I ≠ J and those of the form x_i ∈ C , where C and one of the parameters C_k . For x_i ∈ x_j , the result results directly from the axiom for the membership if I = 1, J = N = 2. For x_i ∈ C , it is obvious if I = N = 1. In the other cases it is thus necessary to possibly be able to reverse the order of the variables (first type of formulas), and to add “useless” variables at the head, in tail, and, for the first type of formulas, between the two variables. It is the object of the following lemma.
Lemma. One shows in NBG, using the 4 axioms of useless addition of variable, circular shift, transposition, and existence of the field, that:

∀ C ∃ D ∀ X ^V ∀ there ^V ∈ '' D '' ↔ ('' X '', '' there '') ∈ '' C '';
∀ C ∃ D ∀ X ^V ∀ there ^V ∀ Z ^V '' X '', '' there '') ∈ '' D '' ↔ ('' X '', '' there '') ∈ '' C '';
∀ C ∃ D ∀ X ^V ∀ there ^V ∀ Z ^V '' there '') ∈ '' D '' ↔ ('' X '', '' there '') ∈ '' C '';
∀ C ∃ D ∀ X ^V ∀ there ^V ∀ Z ^V '' Z '') ∈ '' D '' ↔ ('' X '', '' there '') ∈ '' C ''.

The result follows, for an atomic formula of the first type considered, by recurrence on the number of variables, by reiterating applications of the statements of the lemma, in the order of stating (the order is not indifferent, one uses the definition of N - uplets starting from the couples). If it is supposed that I < J , one adds all the variables ( x₁ ,…, x_i-1 ) in an application of statement 2 of the lemma, then the variables ( x_i+1 ,…, x_j-1 ) one by one by reiterating statement 3 of the lemma, then the variables ( x_j+1 ,…, x_n ) one by one by statement 4 (makes of it the axiom of useless addition of variable).
For an atomic formula of the second type considered, one uses the axiom of addition of an useless variable, which one at the head puts by the first statement of the lemma, to add in once all the variables which precede x_i , then one supplements in a reiterated way, by the axiom of addition of an useless variable.
The lemma is shown, for the 4 by the axiom of addition of an useless variable. For the 1 one uses the 4, the axiom of transposition, and the axiom of the field. For the 3 one uses the 4 then two axioms of permutations to obtain the good transposition. For the 2 they is the 4 then the circular axiom of shift.

NBG and ZFC
Theory NBG cannot be logically equivalent to theory ZFC, since its language is “richer”, and thus contains statements, which speak about the classes in general, and which cannot be given by ZFC. However NBG and ZFC show exactly the same restricted statements with the language of the set theory . It is said whereas NBG is a conservative extension of ZFC: NBG can show new properties, but those will use necessarily quantifications on the classes, NBG will not show anything for statements again purely ensemblists. For a formulation of NBG with two kinds of objects, they are really the same statements. For a presentation with only one kind of object like that given above, one makes correspond to a statement of ZFC a statement obtained by relativizing each quantifier with the class of the units.
A consequence is the property (weaker) than these theories are équi-coherent : from a contradiction shown in ZFC, one would deduce obviously a contradiction in NBG since its demonstration can reproduce there such as it is, by relativizing the statements with the units, but reciprocally of a contradiction shown in NBG, one would deduce also a demonstration from this contradiction in ZFC, since there is no evil to express this one in the language of ZFC (when a contradiction is demonstrable, all is demonstrable). Theory NBG, although more expressive, is thus not more “risky” that ZFC.
However, although Gödel adopted it for the first significant demonstration of relative coherence, that of the axiom of the choice and the assumption of continuous, the theorists of the units prefer to use the language of theory ZF: while making it possible to speak about a new concept, that of class, one can only complicate this kind of evidence, where one reasons on the theory theory NBG is still called upon, as a theory to formalize mathematics, when one needs clean classes for example in Théorie of the categories.
The change of language is well a need to obtain a finished axiomatization of the set theory. Richard Montague showed into 1961 that one could not find a number finished of axioms equivalent to ZF: ZF is not finiment axiomatisable (of course under the assumption that ZF is coherent).

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