What is cardinal number

Power of the set, cardinal number (cardinal) of the set Discrete Mathematics....

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set cardinality , the cardinal number of a set (lat. cardinalis ← cardo “the main circumstance; basis; heart”) is a characteristic of sets (including infinite ones), generalizing the concept of the number (number) of elements of a finite set.

This concept is based on natural ideas about comparing sets:

1. Any two sets, between whose elements a one-to-one correspondence (bijection) can be established, contain the same number of elements (have the same cardinality, are equivalent). nine0038
2. Conversely, equivalent sets must allow such a one-to-one correspondence.
3. Part of the set does not exceed the full set in cardinality (that is, in the number of elements).

Before the construction of the theory of cardinality of sets, sets differed in terms of features: empty/non-empty and finite/infinite, and finite sets also differed in the number of elements. Infinite sets could not be compared.

Cardinality of sets allows infinite sets to be compared. For example, countable sets are the "smallest" infinite sets. nine0003

The cardinality of a set is denoted by . Sometimes there are notations , and .

block diagram of the algorithm for determining the cardinality of a set

In set theory, the cardinal number or cardinal is called an object that characterizes the cardinality of a set. The cardinal number of any set A is denoted as | A | or Card A .

For a finite set A, the cardinal number |A| is a natural number, which means the number of elements of this set. For infinite sets, the cardinal number is a generalization of the concept of the number of elements. nine0003

Although the cardinal numbers of infinite sets are not reflected in natural numbers, they can be compared. Let A and B be infinite sets, then the following four cases are logically possible:

1. A ~ B and | A |=| B |.
2. There is a one-to-one correspondence between the set A and some own subset B' of the set B . Then they say that the cardinality of the set A is not greater than the cardinality of the set B and write | A |≤| B |.
3. The set A is equivalent to some subset of the set B , and vice versa, the set B is equivalent to some subset of the set A , that is, A ~ B' ⊆ B0242 and B ~ A' ⊆ A . According to the Cantor-Bernstein theorem, in this case, A ~ B is fulfilled, i.e. | A |=| B |.
4. There is no one-to-one correspondence between set A and any subset of set B , and there is also no one-to-one correspondence between set B and any subset of set A . It follows from this that the cardinalities of the sets A and B are not comparable.

Thus, the cardinalities of any two sets A and B are always comparable. That is, for cardinal numbers | A | and | B | arbitrary sets A and B , one of the three relations holds: | nine0241 A |=|B|, | A |≤| B | or | B |≤| A |. If | A |≤| B |, but the set A is not equivalent to the set B , then | A |<| B |.

Definition

Assuming the axiom of choice to be true, the cardinality of a set is formally defined as the smallest ordinal number , under which a bijective correspondence can be established between and. This definition is also called the von Neumann distribution of cardinal numbers. nine0003

If the axiom of choice is not accepted, then a different approach is required. The very first definition of the cardinality of a set (it is implicit in Cantor's work and explicitly stated in Frege, as well as in Principia Mathematica) is the class of all sets that are equivalent in cardinality. In axiomatic systems based on the theory of ZFC, such a definition is inapplicable, since for non-empty such a collection is too large to fit the definition of a set. More precisely, if , then there is an injective mapping of the universal set into , under which each set goes to , whence, by virtue of the axiom of the size restriction, it follows that is a proper class. This definition can be used in type theory and "new foundations" ^[en] , as well as in related axiomatic systems. In the case of ZFC, the definition can be used by restricting the collection to the least rank equivalent sets (this trick, proposed by Dana Scott, works because the collection of objects that have a given rank is a set).

The formal order among cardinal numbers is introduced as follows: means that the set can be injectively mapped to . According to the Cantor-Bernstein theorem, it follows from the pair of inequalities and that . The axiom of choice is equivalent to the statement that for any sets and , at least one of the inequalities or . nine0003

A set is called Dedekind infinite ^[en] if it has a proper subset such that . Otherwise, the set is called Dedekind finite. Finite cardinal numbers coincide with ordinary natural numbers - in other words, the set is finite if and only if for some natural . All other sets are infinite. Subject to the axiom of choice, one can prove that the Dedekind definitions coincide with the standard ones. In addition, it can be proved that the cardinality of the set of natural numbers (alef-zero, or aleph-0 - the name is derived from the first letter of the Hebrew alphabet) is the smallest infinitely large cardinal number, that is, in any infinite set there is a subset of cardinality. The cardinal number next in order is denoted, and so on, the number of alephs is infinite. Any ordinal number corresponds to a cardinal number, and in this way any infinitely large cardinal number can be described. nine0003

Related definitions

• The cardinality of the set of natural numbers is denoted by the symbol ("aleph-zero"). A set is called infinite if its cardinality is (not less than the cardinality of the set of natural numbers), so countable sets are the "smallest" of infinite sets. The next cardinal numbers in ascending order are denoted (where the index runs through all ordinal numbers). Among the cardinal numbers, there is no largest: for any set of cardinal numbers, there is a cardinal number that is greater than all elements of this set. nine0038

• Sets that are equivalent to the set of all real numbers are said to have the cardinality of the continuum, and the cardinality of such sets is denoted by the symbol . The assumption that , is called the continuum hypothesis.

• For powers, as in the case of finite sets, there are concepts: "equality", "greater", "less". That is, for any sets and only one of the three is possible:
  1. , or and are equivalent;
  2. , or is more powerful than , that is, it contains a subset that is equally powerful, but not equally powerful;
  3. , or more powerful - in this case, contains a subset that is equivalent to , but not equally powerful.
  • The situation in which and are not equivalent and in none of them there is a part that is equivalent to the other is impossible. This is indicated by the site https://intellect.icu. This follows from Zermelo's theorem. Otherwise, this would mean the existence of incomparable powers (which is possible in principle if one does not accept the axiom of choice). nine0038
  • The situation in which and is impossible by the Cantor-Bernstein theorem.
• Sets and are called equivalent to if there exists a one-to-one mapping of a set onto a set .

Examples

• A set is called finite if it is equivalent to a segment of the natural series for some non-negative integer . The number expresses the number of elements in the finite set. For , the set contains no elements (the empty set). If , then there is no injective mapping from to (the Dirichlet principle), and hence there is no bijection between them. Therefore, the sets and have different cardinalities. nine0038

• A set is called countable if it is equivalent to the set of all natural numbers . The countable sets are:
  • Set for any natural . Correspondence: .
  • Set of . Correspondence: .
  • The set of integers . Correspondence is obtained by comparing the terms of the series with its partial sums (the terms of the series are taken without taking into account the sign).
  • Set of pairs of natural numbers .
  • The set of rational numbers is mapped injectively into a set (an irreducible fraction of the form corresponds to a pair of numbers ). Therefore, the set of rational numbers is at most countable. But since it contains the set of natural numbers, it is not less than countable. By the Cantor-Bernstein theorem, it is countable. nine0038

• Infinite sets not equivalent to the set are called uncountable. According to Cantor's theorem, the set of infinite sequences composed of the digits 0 and 1 is uncountable. The cardinality of this set is called the continuum.

• The cardinality of the set of real numbers is equal to the continuum.

Properties

• Two finite sets are equivalent if and only if they consist of the same number of elements. That is, for a finite set, the concept of power coincides with the usual concept of quantity. nine0038
• For infinite sets, the cardinality of the set can be the same as the cardinality of its own subset, for example .
• Moreover, a set is infinite if and only if it contains an equally powerful proper (that is, not coinciding with the main set) subset.
• Any infinite set is equivalent to the set of all its finite subsets.
• Cantor's theorem The set of all subsets of the set A has more power than A , or .
  • In particular, there exists a set more powerful than any given one.
• Using the Cantor square, one can also prove the following useful statement: The Cartesian product of an infinite set A with itself is equivalent to A .
• Cartesian product power:

• Inclusion-exclusion formula for two and three sets: nine0004

• Power of the symmetric difference of two and three sets:

Arithmetic of cardinal numbers

Ordinary arithmetic operations on natural numbers can be generalized to the case of cardinal numbers. It can also be shown that in the case of finite cardinal numbers these operations coincide with the corresponding arithmetic operations on numbers. In addition, operations on cardinal numbers retain many of the properties of ordinary arithmetic operations. nine0003

The next cardinal number

Subject to the axiom of choice, for each cardinal number it is possible to determine the number following it, and there are no other cardinal numbers between and. If of course, then the next cardinal number is the same as . In the case of infinite, the next cardinal number is different from the next ordinal number.

Addition of cardinal numbers

If the sets and have no common elements, then the sum of cardinalities is determined by the cardinality of their union. If there are common elements, the original sets can be replaced by non-intersecting sets of the same cardinality — for example, by replacing with , and with . nine0003

Neutrality of zero regarding the addition:

Association:

CommetTativity:

Monotony (failure) addition to both arguments:

The sum of two infinite cardinal numbers can be easily calculated with the observation of the axiom of the choice . If one of the numbers or is infinite, then

Subtraction

Subject to the axiom of choice for any infinite cardinal number and arbitrary cardinal number, the existence of , for which , is equivalent to the inequality . This is unique (and coincides with ) if and only if . nine0003

Multiplication of fundamental numbers

The work of two cardinal numbers is expressed through the decartion of the sets:

Octos of zero:

Unit relative to multiplication:

9000 (non-decreasing) multiplication with respect to both arguments:

Distributivity of multiplication with respect to addition:

By analogy with addition, the product of two infinite cardinal numbers can be easily calculated if the axiom of choice is observed. If the numbers and are nonzero and at least one of them is infinite, then

Division

Subject to the axiom of choice, for any pair of cardinal numbers and , where is infinite and not equal to zero, the existence of , for which , is equivalent to the inequality . This is unique (and coincides with ) if and only if . nine0003

Exponentiation of cardinal numbers

Exponentiation is defined as follows:

,

where denotes the set of all functions from to .

(in particular), see the empty function

Monotony:

We note that it is the power of Bulean and, therefore, for any set (see Cantor's Diagonal Method). This implies that among the cardinal numbers there is no largest (since for any cardinal number, a larger number can be specified). In fact, the class of all cardinal numbers is proper (although in some axiomatizations of the theory, the set of this cannot be proved - such, for example, is the system of "New Foundations" ^[en]).

All subsequent statements in this section rely on the axiom of choice.

If and are finite numbers greater than 1, and is an infinite cardinal number, then If the cardinal number is infinite, and finitely different from zero, then .

If and , and at least one of them is infinite, then

.

Using Koenig's theorem, we can prove that for any infinite cardinal number the following inequalities hold:

,

where denotes confinality .

Extracting roots

Subject to the axiom of choice, for any infinite cardinal and finite cardinal there exists a cardinal number such that , and .

Logarithms

Subject to the axiom of choice, a cardinal number that satisfies the condition , given infinite and finite , does not always exist. If such exists, then it is infinite and less than , and any finite cardinal number will also satisfy the equality . nine0003

The logarithm of an infinite cardinal number is the smallest cardinal number that satisfies the condition. Despite the fact that the logarithms of infinitely large cardinal numbers lack some of the properties that are characteristic of the logarithms of positive real numbers, they turn out to be useful in certain areas of mathematics - in particular, in the study of cardinal invariants of topological spaces.

Continuum hypothesis

According to the continuum hypothesis, there are no other cardinal numbers between and. The cardinal number is also denoted and represents the cardinality of the continuum (that is, the set of real numbers). In this case . The generalized continuum hypothesis denies the existence of cardinal numbers strictly between and for any infinite set of . The continuum hypothesis is independent of the standard axiomatization of set theory, i.e. the Zermelo-Fraenkel axiom system combined with the axiom of choice (see Zermelo-Fraenkel set theory). nine0003

See also

• Ordinal number

• TRANSFINITE NUMBER

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