Amphipolis

In Set theory, a together , intuitively indicates a collection of objects (which one calls elements of the unit), “a multitude which can be included/understood like a whole” , as stated it, the creator of this theory, the mathematician Georg Cantor: “Unter einer “Menge” verstehen to wir jede Zusammenfassung M von bestimmten wohlunterscheidbaren Objekten M unserer Anschauung oder unseres Denkens (welche die “Elemente” von M awkward werden) zu einem Ganzen” . This was particularly innovator, being possibly infinite units (and they are the latter which interests Cantor).

What is obviously concerned to the first chief in the overall concept, it is the relation of membership : an element belongs to a unit. They are the properties of this relation which one axiomatizes in axiomatic Théorie of the units, and it is rather remarkable that one can be satisfied some for a theory which can potentially formalize the Mathématiques (what was not yet clearly at the time of Cantor). However the object of this article is rather to give an intuitive approach of the overall concept, such as it is described in the article naive Théorie of the units.

Units, elements and membership

A together can be seen like a kind of virtual bag surrounding its elements, which models well the Venn diagrams . Often (it is not always possible), one tries typographically to distinguish it from his elements, for example by using a capital Latin letter, for example “ E ” or “ has ”, to represent the unit, and of tiny, such as “ X ” or “ N ”, for its elements.

The elements can be of any nature: Number S, geometrical points, right, functions, other units… One thus gives readily examples of whole apart from the mathematical world. For example: Monday is an element of the whole of the days of the Semaine ; a Bibliothèque is a whole of books, etc

The same object can be element of several units: 4 is an element of the whole of the Nombre S entireties, like of the whole of the even numbers (inevitably whole). These the last two units are infinite , they have an infinity of elements.

The membership of an element, noted for example X , with a unit, noted for example has , is written: X ∈ has .

This statement can be read:

• “ X belongs to has ”,
• “ X is element of has ”,
• “ X is in has ”,
• “ has has as an element X ”,
• “ has has X ”,
• or sometimes “ has contains X ” (there is ambiguity however in this last case, has contains X can mean that X is a Sous-ensemble of has, i.e. X is a unit and that all its elements belong to has, which is very different from “ X belongs to has ”).

The symbol “∈”, drift of the Greek letter ε ( Epsilon ) introduced by Giuseppe Peano as of 1889. For Peano “ X ε has ” is read “ X is a has ”, for example “ X ε NR ” is read “ X is an entirety”. The ε returns to initial of the word “is” (in Latin, language of the article of Peano of 1889!), in French, or Italian (“E”). I am . --> Bertrand Russell off takes again the notations of Peano in 1903 in the Principles Mathematics , work which will take part in their diffusion, and where the out-of-date round form of epsilon is used: “ϵ ”, of use in the Anglo-Saxon mathematical edition.

As often for the relations, one bars this symbol to indicate his negation, the not-membership of an object to a unit:

“ Z ∉ has ” means “ Z does not belong to has ”.

Equality of two units

In mathematics, and not only in mathematics besides, one considers that two objects are equal, when they have the same properties, that one cannot thus distinguish them one from the other (it is the definition of the equality of Leibniz). To say when two objects are equal, i.e. when two expressions, indicate in fact the same object, it is thus to give information on what are these objects. In set theory one decides that a unit is completely characterized by its elements, his extension , whereas it can have several definitions. For example, it is not necessary to distinguish the unit from the entireties different from themselves and the unit from the entireties higher than all the prime numbers: these two units are all the two vacuums, therefore equal (they have the same elements well), even if they have different definitions, and are empty for very different reasons.

One will thus say that two units has and B is equal, one will note it as usual has = B , when they have the same elements exactly. This property is known under the name of extensionnality:

(Extensionalité)       HAS = B   if and only if   ∀ X ( X ∈ has ⇔ X ∈ B )

where “⇔” indicates logical equivalence. Two units which have the same elements are well identical : all that can be known as one can be known as the different one. If we represent ourselves the two whole like labelled bags each one by their name, if they are equal, then it is acted in fact of a single bag with two labels. In opposite direction, the properties of a unit do not depend absolutely on the nature or the shape of the bag, only of its contents.

Thus a unit is completely determined by its elements. When a unit is finished, it is thus possible to define it by giving the list of its elements, that one notes traditionally between accodances. For examples the unit to which belong elements 2,3, and 5 and only these elements {2, 3,5} is noted. The unit is defined in extension .

But one cannot proceed thus in any general information, one could not define an infinite unit thus. Even if some artifices of notation which resemble the notation in extension are possible, to see below, the most general way to define a unit is to give an index property of the elements of this unit. For example, one could define the whole of the prime numbers by an index property of those, be different from 1 and to have for only dividers 1 and itself. One speaks about definition in comprehension . The unit {2, 3,5} can be defined in comprehension like the whole of all the prime numbers lower than 6. The definition in extension of the finished units can be seen like a simple particular case of definition in comprehension: for example the unit {2, 3,5} is characterized by the property, for an integer, to be equal to 2 or 3 or 5.

Finished together

When one speaks about finished units, it is in an intuitive direction, without to have really defined this concept. A unit is finished when one can count his elements using entireties all smaller than a given entirety.

The finished units can be defined in extension , by the list of their elements, and described like such; one places the list of the elements of a whole between accodances, as one already saw for the unit {2, 3,5}. For example, the whole of the days of the week can be represented by {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.

Let us note that the notation of a whole in extension is not single: the same unit can be noted in extension in way different.

• the order of the elements is of no importance, for example {1,2} = {2,1}.
• the repetition of elements between the accodances does not modify the unit:

always with the same example, {1,2,2} = {1,1,1,2} = {1,2}. Because of the property of extensionnality, it is not question of distinguishing from the whole by the number of repetitions of the same element to these units: an element belongs or does not belong to a unit, it cannot belong to a unit one, two, or three times… One could impose that the notation is done without repetitions, it would be enough malcommode as soon as intervene of the variables: one could not note a whole in extension without having to suppose that its elements are distinct.

It can happen that one has overall need “with repetition”, in the finished case, he more precisely acts, of continuations finished with the order of the elements near, one then defines the concept of Multiensemble finished (which can be defined starting from the concept of finished continuation).

The units reduced to only one element are called singleton S. For example the unit which contains for only element 0 is called “singleton 0” and is noted {0}. The units which have two elements exactly are called even, the pair of elements 1 and 2, noted {1,2}, should not be confused with the couple (1,2), which has a given order.

When one axiomatizes the set theory the pairs (and let us singletons) play a particular part, to see the article Axiome of the pair.

By extensionnality, there is one whole without elements, the empty set, which one notes ∅ or {}.

Definition of a whole in comprehension

A unit can be defined in comprehension , i.e. one defines it by an index property among the elements of a given unit. Thus the whole of the even natural entireties is clearly defined by comprehension, by the property “being even” among the natural entireties. One can use the notation of a whole in comprehension , for example for the whole of the even natural entireties, one will write ($\backslash \; mathbb\; N$ indicating the whole of the natural entireties):

$\backslash \; \{X\; \backslash \; in\; \backslash \; mathbb\; NR\; \backslash \; mid\; X\; \backslash \; \backslash \; rm\; even\; \backslash \}$.

One will in the same way define the whole of the relative entireties ranging between -7 and 23 ($\backslash \; mathbb\; Z$ indicating the whole of the relative entireties):

$\backslash \; \{X\; \backslash \; in\; \backslash \; mathbb\; Z\; \backslash \; mid\; -7\; \backslash \; Leq\; X\; \backslash \; Leq\; 23\; \backslash \}$.

the whole of the nonnull perfect squares:

$\backslash \; left\; \backslash \; \{X\; \backslash \; in\; \backslash \; mathbb\; NR\; \backslash \; mid\; \backslash \; exists\; there\; \backslash \; in\; \backslash \; mathbb\; NR\; \backslash \; (there\; \backslash \; geq\; 1\; \backslash \; \{\backslash \; rm\; and\}\; \backslash \; X\; =\; y^2)\; \backslash \; right\; \backslash \}$.

For this last unit, one can adopt another notation, more immediate:

$\backslash \; \{y^2\; \backslash \; mid\; there\; \backslash \; in\; \backslash \; mathbb\; NR\; \backslash \; \{\backslash \; rm\; and\}\; \backslash \; there\; \backslash \; geq\; 1\; \backslash \}$.

There are thus two forms of notation:

{ X ∈ E | P ( X )},

for “the whole of the X of E such as the condition P ( X ) is true,

{ F (X) | X ∈ E and P ( X )}

for the whole of the images by F of the elements X of E satisfactory P .

Here of other examples:

• $\backslash \; \{Z\; \backslash \; in\; \backslash \; mathbb\; C\; \backslash ;\; |\backslash ;\; Z\; =\; \backslash \; bar\; Z\; \backslash \}$ indicates the whole of the real numbers $\backslash \; mathbb\; \{R\}$.
• $\backslash \; \{M\; \backslash \; in\; \backslash \; mathcal\; M\_n\; (\backslash \; mathbb\; K)\; \backslash ;\; |\backslash ;\; \{\}\; ^tM\; =\; M\; \backslash \}$ indicates the whole of the symmetrical matrices.
• $\backslash \; \{X\; \backslash \; in\; \backslash \; mathbb\; Z\; \backslash ;\; |\backslash ;\; X\; \backslash ;\; \backslash \; rm\; \{even\}\; \backslash ;\; \backslash \}$ is the whole of all the even entireties
• $\backslash \; \{2x\; \backslash ;\; |\backslash ;\; X\; \backslash \; in\; \backslash \; mathbb\; Z\; \backslash \}$ is still the whole of all the even entireties.

For each of the two notations, one must restrict comprehension with a unit (the unit on which the function for the second notation is defined). That can appear superfluous: why not take any condition? But if it were the case one could define the unit {X | X ∉ X}, which leads to a contradiction (it is the Paradoxe of Russell). The restriction of comprehension on a known unit protects from this kind from paradoxes (the first notation corresponds directly to the Schéma of axioms of comprehension of the theory of Zermelo, the second from of deduced, by using the definition of a function in set theory, whose graph is a whole of couple S). One does not need this kind of restriction to introduce (as in the preceding paragraph) units finished by the list of their elements, or to introduce whole by operations usual ensemblists, like the meeting, or the Ensemble of the parts of a unit.

One did not say what one understood by “property” or “condition”. In spite of the preceding restriction, one cannot all authorize, under penalty of other paradoxes like the Paradoxe of Richard or the Paradoxe of Berry, which utilizes, for example, “the whole of the definable natural entireties in less than fifteen French words”. It is necessary to specify the language in which one can define these conditions. In particular this language must be defined a priori , and can be wide only using definitions which are either simple abbreviations, or result from evidence of existence and unicity.

Other notations

There exist other convenient notations, in particular for the whole of numbers, and more generally for the completely ordered units.

One can use points of suspension, for notations inspired of the notation in extension for whole of infinite, or finished but nongiven cardinality. For example, the whole of the natural entireties can be noted by: $\backslash \; mathbb\; NR$ = {0,1,2,3,…}. If it is clearly in addition that N indicates a natural entirety, {1, 2,…, N }, even {1,…, N } in general indicates the whole of the entireties equal to or higher than 1 and inferiors or equal to N . In the same way one can write $\backslash \; mathbb\; Z$ = {…, -3,-2,-1,0,1,2,3,…}, or {- N, - n+1,…., n-1, N}. When there is a simple iterative process to generate the elements of the unit, one can risk oneself with notations like {0, 2,4,6,…} for the whole of even natural entireties etc One can of course use these notations for units having “many” elements, {1, 2,…, 1000} rather than to write the first thousand nonnull integers, or {3,5,…, 21} rather than {3,5,7,9,11,13,15,17,19,21}.

All these notations are not systematic, nor universal, and for the last at least, not very rigorous. One can still announce, the notation, rigorous this one, certain real line subsets, the interval S.

By abuse notation, sometimes one does not note the variable in the definition in comprehension, but only the property. Thus one notes a whole while placing between accodances nature, or an index property, objects which belong to him. For example the notation {dogs} indicates the whole of all the dogs; to take a more mathematical example, one could write {sometimes even} for the whole of the even numbers.

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