Cardinal number

Definitions

Classify units

Two units are known as equipotent if there exists a Bijection one on the other. The relation being reflexive, symmetrical and transitive on the class of the Together S, each Classe of equivalence is called cardinal number or simply cardinal .

If there exists a injection of a unit in a unit B then the cardinal of has is known as smaller than the cardinal of B , which notes $\ mathrm {card} (A) \ the \ mathrm {card} (B)$ . In this case, there exists an injection of any equipotent whole with has in any equipotent whole with B . Moreover, the Théorème of Cantor-Bernstein makes it possible to show that if two cardinals are both smaller one than the other, then they are equal. This relation is thus a Relation of order on the cardinals.

The Empty set and the whole whole of of the form $\ left \ {1, \ dowries, N \ right \}$ form whole of cardinals two to two différents.
A unit is known as finished if it is equipotent with the one of these units, infinite in the contrary case. Any finished cardinal is lower than any infinite cardinal.

One notes $\ mathrm {card} (\ emptyset) = 0 \$ and $\ \ mathrm {card} \ left (\ left \ {1, \ dowries, N \ right \} \ right) = n$ , so that the order on the cardinals prolongs the order on the natural entireties.

Ordinal

In the axiomatic Set theory of Zermelo-Fraenkel (ZF), the addition of the Axiome of the choice (giving theory ZFC) makes it possible to define the cardinal of the whole like smallest ordinal Nombre which is equipotent for him. A cardinal number is then ordinal which is not equipotent with any of its elements.

Apart from the assumption of the axiom of the choice, it can be judicious to be limited to the units for which such ordinal equipotent exists.

General properties

f

is a function of

A

B

, then

\ mathrm {card} (F (A)) \ the \ mathrm {card} (A)

Fundamental theorem

A unit E is never equipotent with the whole of its parts

\ mathfrak P (E)

, although it is injected inside by the whole of the singletons of E , which makes it possible to be written:

\ mathrm {card} (E) < \ mathrm {card} (\ mathfrak P (E))

It is the Théorème of Cantor.

This result justifies the fact that there exist infinite different cardinals. It gives even a process of construction of an infinity of them by Itération.

The infinite cardinals are represented by means of the Hebraic letter aleph $\ \ aleph$ . The smallest infinite cardinal is $\ \ aleph_0$ . It is the cardinal of the unit $\ \ mathbb N$ of the natural entireties, which is also indicated as an ordinal number by $\ omega$ . The immediately higher cardinal is $\ \ aleph_1$ , etc Generally, an unspecified cardinal is written $\ \ aleph_ \ alpha$ where $\ alpha$ is ordinal.

Finished cardinal

The cardinal of a finished unit thus corresponds simply to the number of elements which it contains. For example,

\ mathrm {card} (\ {1, 2,5 \}) = 3

Properties

Toute part of a finished unit is finished.
If has and B is two parts of a finished unit, then $\ mathrm {card} (has \ cup B) = \ mathrm {card} (A) + \ mathrm {card} (B) - \ mathrm {card} (has \ course B)$ .
If has is part of a unit finished E then $\ \ mathrm {card} (E-A) = \ mathrm {card} (E) - \ mathrm {card} (A)$

Operations ensemblists

Are E and F two units finished with E of cardinal K and F of cardinal N .

the disjoined sum $E \ sqcup F$ is finished of cardinal

\ mathrm {card} (E \ sqcup F) = n+k

the Cartesian Produit $E \ times F$ is finished of cardinal

\ mathrm {card} (E \ times F) = nk

the whole of the applications of E in F , sometimes noted $\ mathrm {Appl} (E, F)$ is finished of cardinal

\ \ mathrm {card} (\ mathrm {Appl} (E, F)) = n^k

with the convention 0⁰=1 if E and F is both vacuums.

This property justifies the more current notation

F^E

the unit $\ mathfrak P (E)$ of the left E , which is identified with the whole of the applications of E in $\ left \ {0,1 \ right \}$ , is thus finished and cardinal

\ mathrm {card} (\ mathfrak P (E)) = 2^k

the whole of the correspondences of E in F , usually noted $\ mathrm {Corr} (E, F)$ , is thus identified with $\ mathfrak P (E \ times F)$ is finished of cardinal

\ \ mathrm {card} (\ mathrm {Corr} (E, F)) = 2^ {nk}

the whole of the functions of E in F , often noted $\ mathcal {F} (E, F)$ , is a subset of the precedent, cardinal

\ mathrm {card} (\ mathcal {F} (E, F)) = (n+1) ^k

the whole of the injections of E in F , usually noted $\ mathrm {Inj} (E, F)$ , is empty if $\ mathrm {card} (E) < \ mathrm {card} (F)$ . In the contrary case,

\ mathrm {card} (\ mathrm {Inj} (E, F))= \ frac {N!}{(n-k)! }

the whole of the Surjection S of E in F , usually noted $\ mathrm {Surj} (E, F)$ , is empty if $\ mathrm {card} (E) > \ mathrm {card} (F)$ . In the contrary case,

\ \ mathrm {card} (\ mathrm {Surj} (E, F)) = \ sum_ {I = 0} ^ {N} (- 1) ^ {I} \ frac {N!}{I! (N - I)! } (N - I) ^ {K}

the whole of the bijections of E in F , usually noted $\ mathrm {Bij} (E, F)$ , is empty if $\ mathrm {card} (E) \ neq \ mathrm {card} (F)$ . In the contrary case:

\ \ mathrm {card} (\ mathrm {Bij} (E, F)) = N!

Infinite cardinal

A

is infinite then

\ mathrm {card} (\ mathfrak P (A))

is noted

2^ {\ mathrm {card} (A)}

by analogy with the finished case.

Examples

the cardinal of the whole of the real numbers is the same one as that of the whole of the parts of $\ N$ :

\ mathrm {card} (\ R) = 2^ {\ aleph_0} > \ aleph_0 = \ mathrm {card} (\ NR)

This cardinal being equal to that of

\ mathbb R

, one also notes it

\ mathfrak c

, known as cardinal of the continuous one.

However, contrary to the intuition first, the whole of natural entireties and rational is equipotent.

\ mathrm {card} (\ mathbb {NR}) = \ mathrm {card} (\ mathbb {Q})

Properties

a unit has is infinite if and only if $\ mathrm {card} (A) = \ mathrm {card} (\ cup \ {has \})$ .
If $A$ is infinite and if $\ mathfrak F (A)$ indicates the whole of the finished parts of $E$ , then $\ mathrm {card} (A) = \ mathrm {card} (\ mathfrak F (A))$ .
If $A$ is infinite and $B$ not vacuum, then $\ mathrm {card} (has \ cup B) = \ mathrm {card} (has \ times B) = \ max (\ mathrm {card} (A), \ mathrm {card} (B))$ .
If $B$ is included in $A$ infinite with $\ mathrm {card} (B) < \ mathrm {card} (A)$ , then $\ \ mathrm {card} (A-B) = \ mathrm {card} (A)$ .
If $A$ is infinite, then $\ mathrm {card} (has \ times A) = \ mathrm {card} (A)$
If $A$ is infinite and if $2 \ the \ mathrm {card} (B) \ the \ mathrm {card} (A)$ , then $\ \ mathrm {card} (B^A) = 2^ {\ mathrm {card} (A)}$ where $B^A$ indicates the whole of the functions of $A$ in $B$ . -->

Inaccessible cardinal

The accessibility is the possibility of reaching ordinal or a cardinal given starting from ordinal more the petits.
ordinal a

\ alpha

is known as cofinal with ordinal a

\ beta

lower if there exists a strictly increasing application

f

\ beta

\ alpha

such as

\ alpha

is the limit of

f

to the following direction:

\ forall \ gamma \ in \ alpha, \ exists \ delta \ in \ beta, \ gamma \ the F (\ delta)

For example, $\ aleph_0$ is not cofinal with any ordinal strictly more small, since an ordinal lower than $\ aleph_0$ is an entirety $n = \ {0,1,…, n-1 \}$ and that a strictly increasing application definite on $\ {0,1,…, n-1 \}$ is limited. The cardinal $\ aleph_0$ is known as then regular , it is the case of all the cardinals successors.

On the other hand, the cardinal $\ aleph_ {\ Omega}$ is cofinal with $\ omega$ by means of the $f application: N \ in \ Omega \ mapsto \ aleph_n$ .
This cardinal $\ aleph_ {\ Omega}$ is known as then singular .

By noting $\ mathrm {cf} (\ alpha)$ smallest ordinal for which $\ alpha$ is cofinal, one obtains $\ mathrm {cf} (\ Omega) = \ mathrm {cf} (\ aleph_ {\ Omega}) = \ omega$ .

The cardinals are classified then as follows:

those of the form $\ aleph_ {\ alpha+1}$ , subscripted by ordinal a $\ alpha+1$ successor of ordinal a $\ alpha$ ;
those of the form $\ aleph_ \ alpha$ , subscripted by ordinal a $\ alpha$ limit and which are singular;
those of the form $\ aleph_ \ alpha$ , subscripted by ordinal a $\ alpha$ limit and which are regular.

This last type of cardinal is described as slightly inaccessible because they cannot be conceived starting from more small cardinals. One distinguishes among them the strongly inaccessible cardinals who check moreover

\ mathrm {card} (X) < \ aleph_ {\ alpha} \ Longrightarrow 2^ {\ mathrm {card} (X)} < \ aleph_ {\ alpha}

. The existence of such cardinals cannot result from the axioms of the set theory ZFC.
The first two types of cardinals are qualified contrary to accessible , because conceivable starting from cardinals smaller than them.

Assumption of the continuous one

The inequality

\ mathrm {card} (\ mathbb {NR}) = \ aleph_0 < \ mathrm {card} (\ mathbb {R}) = 2^ {\ aleph_0}

shown above makes it possible to write

\ aleph_1 \ the 2^ {\ aleph_0}

since

\ aleph_1

is the smallest cardinal strictly higher than

\ aleph_0

The Hypothèse of continuous the affirms the equality $\ aleph_1 = 2^ {\ aleph_0}$ . It is shown that this property is Indécidable in ZFC. By extension, the assumption generalized of continuous states that, for very ordinal $\ alpha$ , one has $\ aleph_ {\ alpha+1} = 2^ {\ aleph_ {\ alpha}}$ .

The following results are obtained by admitting like axiom the assumption generalized of the continuous one.

the Axiome of the choice is demonstrable.
There is equivalence between the concepts of cardinals slightly inaccessible and strongly inaccessible.
By noting $\ aleph_ {\ alpha} ^ {\ aleph_ {\ beta}}$ the whole of the functions of $\ aleph_ {\ beta}$ in $\ aleph_ {\ alpha}$ , it comes

$\ mathrm {card} (\ aleph_ {\ alpha} ^ {\ aleph_ {\ beta}}) = \ aleph_ {\ alpha}$ if $\ aleph_ {\ beta} < {\ rm cf} (\ aleph_ {\ alpha})$ ;
$\ mathrm {card} (\ aleph_ {\ alpha} ^ {\ aleph_ {\ beta}}) = \ aleph_ {\ alpha+1}$ if ${\ rm cf} (\ aleph_ {\ alpha}) \ the \ aleph_ {\ beta} \ the \ aleph_ {\ alpha}$ ;
$\ mathrm {card} (\ aleph_ {\ alpha} ^ {\ aleph_ {\ beta}}) = \ aleph_ {\ beta+1}$ if $\ aleph_ {\ alpha} \ the \ aleph_ {\ beta}$ .

See too

Simple: Cardinal number

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