Cardinal number (math)

Cardinal numbers ( Latin cardo "door hinge", "pivot point") are a generalization of natural numbers in mathematics to describe the power , also called cardinality, of sets .

The thickness of a finite set is a natural number - the number of elements in the set. The mathematician Georg Cantor described how this concept can be generalized to infinite sets within set theory and how one can calculate with infinite cardinal numbers.

Infinite quantities can have different thicknesses. These are denoted by the symbol ( Aleph , the first letter of the Hebrew alphabet) and an index (initially an integer). The power of the natural numbers , the smallest infinity, is in this notation . ${\displaystyle \aleph }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \aleph _{0}}$

A natural number can be used for two purposes: on the one hand to describe the number of elements of a finite set, and on the other hand to indicate the position of an element in a finitely ordered set. While these two concepts agree for finite sets, one has to distinguish them for infinite sets. The description of the position in an ordered set leads to the concept of the ordinal number , while the specification of the size leads to cardinal numbers, which are described here.

definition

Two sets and are called equal if there is a bijection from to ; one then writes or . Uniformity is an equivalence relation on the class of all sets. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \left\vert X\right\vert =\left\vert Y\right\vert }$ ${\displaystyle X\sim Y}$ ${\displaystyle \sim }$

Cardinal numbers as real classes

The equivalence class of the set with respect to the relation of equality is called the cardinal number . ${\displaystyle X}$ ${\displaystyle \left\vert X\right\vert }$

The problem with this definition is that the cardinal numbers themselves are not sets, but real classes. (With the exception of ). ${\displaystyle \left\vert \emptyset \right\vert }$

This problem can be circumvented by not designating the entire equivalence class with, but choosing an element from it, one choosing a representative system, so to speak. In order to do this formally correctly, one uses the theory of ordinal numbers , which one must have defined beforehand with this approach: ${\displaystyle \left\vert X\right\vert }$

Cardinal numbers as a special ordinal number

Each amount is equipotent to a well-ordered amount (as long as one to the axiom of choice equivalent well-ordering theorem requires). To hear an ordinal. can be chosen so that this ordinal number is as small as possible, since the ordinal numbers themselves are well ordered; then is a starting number . One can equate the cardinal number with this smallest ordinal number. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle \left\vert A\right\vert }$

With this set-theoretic handle, the cardinality of a set is itself a set. Immediately follows the comparability principle that the cardinal numbers are totally ordered, because as a subset of the ordinal numbers they are even well ordered. This cannot be proven without the axiom of choice.

motivation

Cardinal numbers are clearly used to compare the size of sets without having to refer to the appearance of their elements. This is easy for finite sets. You just count the number of elements. It takes a little more work to compare the power of infinite sets.

In the following, the terms are at most equally powerful and less powerful:

If there is a bijection of onto a subset of , then at most is called equal to . Then you write . ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \left\vert A\right\vert \leq \left\vert B\right\vert }$

If there is a bijection from onto a subset of , but no bijection from to exists, then is called less powerful than and more powerful than . Then you write . ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle \left\vert A\right\vert <\left\vert B\right\vert }$

These terms are explained in more detail in the article Power .

For example, for finite sets it is true that real subsets are less powerful than the entire set, whereas the article Hilbert's Hotel uses an example to illustrate that infinite sets have real subsets that are equal to them.

When examining these large sets, the question arises as to whether ordered sets of equal power necessarily have matching orders. It turns out that this is not the case for infinite sets, e.g. B. the ordinary order of natural numbers differs from the ordered set . The crowd is equal too . Such is a bijection, but in contrast to there is one greatest element. If one takes into account the order of sets, one arrives at ordinal numbers . The ordinal of is called and that of is . ${\displaystyle \mathbb {N} =\{0<1<2<3<\dotsb \}}$ ${\displaystyle A:=\{0<1<2<3<\dotsb <0^{\prime }\}}$ ${\displaystyle A}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle f\colon 0\mapsto 1,1\mapsto 2,2\mapsto 3,\dots ,0^{\prime }\mapsto 0}$ ${\displaystyle A}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \omega }$ ${\displaystyle A}$ ${\displaystyle \omega +1}$

properties

In the article cardinality it is shown that the cardinal numbers are totally ordered .

A set is called finite if there is a natural number such that it has exactly elements. So that means that either is empty, if , or that there is a bijection of onto the set . A set is called infinite if there is no such natural number. A set is called countably infinite if there is a bijection of onto the set of natural numbers , i.e. i.e. if their thickness is. A set is called countable if it is finite or countably infinite. The thickness of the real numbers is denoted by (thickness of the continuum). ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle n=0}$ ${\displaystyle M}$ ${\displaystyle \{1,\dots ,n\}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \aleph _{0}}$ ${\displaystyle {\mathfrak {c}}}$

One can show the following:

• The infinite sets are precisely those sets that are equal to a real subset (see Dedekind-infinite ).
• Cantor's diagonal proof shows that for every set the set of all its subsets has a greater thickness, i.e. H. . It follows that there is no greatest cardinal number. For finite sets , reason for the alternative spelling for the power set: . Equal sets have equal power sets; H. the assignment for infinite sets is independent of the particular choice of this set for a given thickness - this is true for finite sets anyway. ${\displaystyle M}$ ${\displaystyle {\mathcal {P}}(M)}$ ${\displaystyle |{\mathcal {P}}(M)|>|M|}$
  ${\displaystyle |{\mathcal {P}}(M)|=2^{|M|}}$ ${\displaystyle {\mathcal {P}}(M)=2^{M}}$
  ${\displaystyle |M|\mapsto 2^{|M|}:=|2^{M}|=|{\mathcal {P}}(M)|}$ ${\displaystyle M}$

• The set of real numbers is equipotent to the power set of the natural numbers: . ${\displaystyle {\mathfrak {c}}\equiv |\mathbb {R} |=|2^{\mathbb {N} }|\equiv 2^{\aleph _{0}}}$
• It is also true that the cardinal number is the smallest infinite cardinal number. The next higher cardinal number is referred to by definition . Assuming the continuum hypothesis is ; however, it is certainly true even without the continuum hypothesis . For every ordinal there is a -th infinite cardinal number , and every infinite cardinal number is reached in this way. Since the ordinals form a real class , the cardinal class is also real . ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{1}}$ ${\displaystyle \aleph _{1}=\left\vert \mathbb {R} \right\vert }$ ${\displaystyle \aleph _{1}\leq \left\vert \mathbb {R} \right\vert }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \aleph _{\alpha }}$

Note that without the axiom of choice, sets cannot necessarily be well ordered , and the equation of cardinal numbers with certain ordinals given in the section on definition cannot be derived. One can then nevertheless define cardinal numbers as equivalence classes of sets of equal power. These are then only partially ordered , since different cardinal numbers no longer have to be comparable (this requirement is equivalent to the axiom of choice). But you can also examine the cardinality of sets without using cardinal numbers at all.

Arithmetic operations

→ Main article : Cardinal number arithmetic

If and are disjoint sets, then one defines ${\displaystyle X}$ ${\displaystyle Y}$

• ${\displaystyle |X|+|Y|:=|X\cup Y|}$
• ${\displaystyle |X|\cdot |Y|:=|X\times Y|}$
• ${\displaystyle |X|^{|Y|}:=|X^{Y}|}$.

Here is a Cartesian product and the set of all functions from to . Since the power set of a set (per indicator function for ) can be mapped bijectively to the set of functions , this definition is in agreement with the previous definition for the power of the power sets (in other words , a continuation for ). ${\displaystyle X\times Y}$ ${\displaystyle X^{Y}}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle Z\mapsto I_{Z}}$ ${\displaystyle Z\subseteq X}$ ${\displaystyle X\to \{0,1\}}$ ${\displaystyle |2^{Y}|=2^{|Y|}}$ ${\displaystyle |X|\neq 2}$

One can show that these links for natural numbers agree with the usual arithmetic operations. In addition, applies to all quantities , , : ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$

• Addition and multiplication are associative and commutative .
• Addition and multiplication fulfill the distributive law .
• The power laws and apply . ${\displaystyle |X|^{|Y|+|Z|}=|X|^{|Y|}\cdot |X|^{|Z|}}$ ${\displaystyle |X|^{|Y|\cdot |Z|}=(|X|^{|Y|})^{|Z|}}$
• The addition and multiplication of infinite cardinal numbers is easy (assuming the axiom of choice): is or infinite and in the case of multiplication both sets are not empty, then applies ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle |X|+|Y|=|X|\cdot |Y|=\max\{|X|,|Y|\}}$

No cardinal number except has an opposite number (an element that is inverse with respect to addition ), so the cardinal numbers do not form a group with addition and certainly not a ring . ${\displaystyle 0}$

Notation

The finite cardinal numbers are the natural numbers and are noted accordingly. The aleph notation is usually used for infinite cardinal numbers, i.e. for the first infinite cardinal number, for the second, etc. In general, there is also a cardinal number for every ordinal number . ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{1}}$ ${\displaystyle \alpha }$ ${\displaystyle \aleph _{\alpha }}$

The ordinal numbers actually known are occasionally represented with the help of the Beth function . One of these is significant (note that the aleph has no index here). In mathematics, outside of basic research, sets of quantity occasionally occur (e.g. the power set of , the number of Lebesgue measurable sets, the set of all - not necessarily continuous - functions from to or the like), higher numbers usually not. ${\displaystyle \beth _{1}=\aleph ={\mathfrak {c}}=2^{\aleph _{0}}=|\mathbb {R} |}$ ${\displaystyle \beth _{2}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

The respective use as a cardinal number can be recognized by the spelling. The von-Neumann model applies accordingly (note the lack of power lines), but the former is used for the ordinal number, the mean for the cardinal number, and the latter for the otherwise used set of natural numbers. ${\displaystyle \omega =\aleph _{0}=\mathbb {N} }$

Continuum hypothesis

→ Main article : Continuum hypothesis

The generalized continuum hypothesis ( English generalized continuum hypothesis , hence GCH for short) states that for every infinite set there are between the cardinal numbers and no further cardinal numbers. The continuum hypothesis ( English continuum hypothesis , hence CH for short) makes this claim just in case . It is independent of the Zermelo-Fraenkel set theory together with the axiom of choice (ZFC). ${\displaystyle X}$ ${\displaystyle |X|}$ ${\displaystyle 2^{|X|}}$ ${\displaystyle X=\mathbb {N} }$

Individual evidence

1. ↑ Dieter Klaua : Set theory . De Gruyter textbook. de Gruyter, Berlin, New York 1979, ISBN 3-11-007726-4 . Here p. 75, definition 16, part 1, definition 16, part 2
2. ↑ H. König: Design and structural theory of controls for production facilities (=  ISW research and practice . Volume 13 ). Springer-Verlag, Berlin / Heidelberg 1976, ISBN 3-540-07669-7 , pp. 15-17 , doi : 10.1007 / 978-3-642-81027-5_1 . Here: page 21
3. ↑ Тhοmas Stеιnfеld: equal thickness to Mathpedia
4. ↑ In ZFC is the only unreachable cardinal number. In a Grothendieck universe , however, there are unreachable cardinal numbers . ${\displaystyle \aleph _{0}}$

See also

• Large cardinal number
• Ordinal number

literature

• Erich Kamke : Set theory (= Göschen Collection. Vol. 999 / 999a). 7th edition. de Gruyter, Berlin et al. 1971, ISBN 3-11-003911-7 .