Cardinal number

This article describes cardinal numbers in mathematics. For cardinals in linguistics, see Names of numbers in English.

Aleph-0, the smallest infinite cardinal

In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" For finite sets, the cardinality is given by a natural number, which is simply the number of elements in the set. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an There are also transfinite cardinal numbers that describe the sizes of infinite sets. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite.

Cardinality is defined in terms of bijective functions. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Two sets have the same cardinal number if and only if there is a bijection between them. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

There is a transfinite sequence of cardinal numbers:

$0, 1, 2, 3, \cdots, n, \cdots ; \aleph_0, \aleph_1, \aleph_2, \cdots, \aleph_{\alpha}, \cdots.$

This sequence starts with the natural numbers (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every The aleph numbers are indexed by ordinal numbers. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. If the axiom of choice fails, the situation is more complicated, with additional infinite cardinals that are not alephs.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Analysis has its beginnings in the rigorous formulation of Calculus.

1 History
2 Motivation
3 Formal definition
4 Cardinal arithmetic
5 The continuum hypothesis
6 See also
7 References
8 External links

History

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia.

Cantor first established cardinality as an instrument to compare finite sets; e. g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality, namely three.

Cantor identified the fact that one-to-one correspondence is the way to tell that two sets have the same size, called "cardinality", in the case of finite sets. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Using this one-to-one correspondence, he applied the concept to infinite sets; e. g. the set of natural numbers N = {0, 1, 2, 3, . . . }. He called these cardinal numbers transfinite cardinal numbers, and defined all sets having a one-to-one correspondence with N to be denumerable (countably infinite) sets. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite.

Naming this cardinal number $\aleph_0$ , aleph-null, Cantor proved that any unbounded subset of N has the same cardinality as N, even if this might appear at first to run contrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (which implies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers is also denumerable. In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or Each algebraic number z may be encoded as a finite sequence of integers which are the coefficients in the polynomial equation of which it is the solution, i. e. the ordered n-tuple $(a_0, a_1, ..., a_n),\; a_i \in \mathbb{Z},$ together with a pair of rationals $(b 0, b 1)$ such that z is the unique root of the polynomial with coefficients $(a 0, a 1,. . . , a n)$ that lies in the interval $(b 0, b 1)$ .

In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbers has cardinality greater than that of N. His original presentation used a complex argument with nested intervals, but in an 1891 paper he proved the same result using his ingenious but simple diagonal argument. Georg Cantor 's first uncountability proof demonstrates that the set of all Real numbers is uncountable. In Mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers I n such that Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 This new cardinal number, called the cardinality of the continuum, was termed c by Cantor. In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of

Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ( $\aleph_0$ , aleph-null) and that for every cardinal number, there is a next-larger cardinal $(\aleph_1, \aleph_2, \aleph_3, \cdots)$ .

His continuum hypothesis is the proposition that c is the same as $\aleph_1$ , but this has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved under the standard assumptions. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite

Motivation

In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included: 0, 1, 2, . In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an . . . They may be identified with the natural numbers beginning with 0. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The counting numbers are exactly what can be defined formally as the finite cardinal numbers. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Infinite cardinals only occur in higher-level mathematics and logic.

More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e. g. 3 describes the position of 'c' in the sequence <'a','b','c','d',. . . >, and we can construct the set {a,b,c} which has 3 elements. However when dealing with infinite sets it is essential to distinguish between the two — the two notions are in fact different for infinite sets. In Set theory, an infinite set is a set that is not a Finite set. Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the cardinal numbers described here.

The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.

A set Y is at least as big as, or greater than or equal to a set X if there is an injective (one-to-one) mapping from the elements of X to the elements of Y. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. A one-to-one mapping identifies each element of the set X with a unique element of the set Y. This is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size we would observe that there is a mapping:

1 → a

2 → b

3 → c

which is one-to-one, and hence conclude that Y has cardinality greater than or equal to X. Note the element d has no element mapping to it, but this is permitted as we only require a one-to-one mapping, and not necessarily a one-to-one and onto mapping. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every The advantage of this notion is that it can be extended to infinite sets.

We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property By the Schroeder-Bernstein theorem, this is equivalent to there being both a one-to-one mapping from X to Y and a one-to-one mapping from Y to X. In set theory, the Cantor–Bernstein–Schroeder Theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states We then write | X | = | Y |. The cardinal number of X itself is often defined as the least ordinal a with | a | = | X |. This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. The Von Neumann cardinal assignment is a Cardinal assignment which uses Ordinal numbers For a Well-ordered set U, we define its In Mathematics, the well-ordering principle states that every non-empty set of positive integers contains a smallest element It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.

The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. Hilbert's paradox of the Grand Hotel is a mathematical Paradox about Infinite sets presented by German mathematician David Hilbert (1862–1943 Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It's possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 ↔ 2

2 ↔ 3

3 ↔ 4

. . .

n ↔ n+1

. . .

In this way we can see that the set {1,2,3,. . . } has the same cardinality as the set {2,3,4,. . . } since a bijection between the first and the second has been shown. This motivates the definition of an infinite set being any set which has a proper subset of the same cardinality; in this case {2,3,4,. . . } is a proper subset of {1,2,3,. . . }.

When considering these large objects, we might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it doesn't; by considering the above example we can see that if some object "one greater than infinity" exists, then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, called ordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.

It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. In Mathematics, the real numbers may be described informally in several different ways This can be visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite In more recent times mathematicians have been describing the properties of larger and larger cardinals.

Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.

Formal definition

Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijection between X and α. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. This definition is known as the von Neumann cardinal assignment. The Von Neumann cardinal assignment is a Cardinal assignment which uses Ordinal numbers For a Well-ordered set U, we define its If the axiom of choice is not assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the set of all sets which are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems. The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work (this is a trick due to Dana Scott: it works because the collection of objects with any given rank is a set). In mathematical Set theory, the rank of a set is defined inductively as the smallest Ordinal number greater than the rank of any member of the set Dana Stewart Scott (born 1932 is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie

Formally, the order among cardinal numbers is defined as follows: | X | ≤ | Y | means that there exists an injective function from X to Y. The Cantor–Bernstein–Schroeder theorem states that if | X | ≤ | Y | and | Y | ≤ | X | then | X | = | Y |. In set theory, the Cantor–Bernstein–Schroeder Theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states The axiom of choice is equivalent to the statement that given two sets X and Y, either | X | ≤ | Y | or | Y | ≤ | X |. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

A set X is Dedekind-infinite if there exists a proper subset Y of X with | X | = | Y |, and Dedekind-finite if such a subset doesn't exist. In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. The finite cardinals are just the natural numbers, i. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an e. , a set X is finite if and only if | X | = | n | = n for some natural number n. Any other set is infinite. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal $\aleph_0$ (aleph-0, where aleph is the first letter in the Hebrew alphabet, represented $\aleph$ ) of the set of natural numbers is the smallest infinite cardinal, i. The Hebrew alphabet (אָלֶף-בֵּית עִבְרִי alephbet ’ivri) consists of 22 letters used for writing the Hebrew language. e. that any infinite set has a subset of cardinality $\aleph_0.$ The next larger cardinal is denoted by $\aleph_1$ and so on. For every ordinal α there is a cardinal number $\aleph_{\alpha},$ and this list exhausts all infinite cardinal numbers.

Cardinal arithmetic

We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.

Successor cardinal

For more details on this topic, see Successor cardinal. In the theory of Cardinal numbers, we can define a successor operation similar to that in the Ordinal numbers This coincides with the ordinal successor operation for

If the axiom of choice holds, every cardinal κ has a successor κ⁺ > κ, and there are no cardinals between κ and its successor. For finite cardinals, the successor is simply κ+1. For infinite cardinals, the successor cardinal differs from the successor ordinal. When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one

Cardinal addition

If X and Y are disjoint, addition is given by the union of X and Y. In Mathematics, two sets are said to be disjoint if they have no element in common In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets If the two sets are not already disjoint, then they can be replaced by disjoint sets, i. e. replace X by X×{0} and Y by Y×{1}.

$|X| + |Y| = | X \cup Y|.$

Zero is an additive identity κ + 0 = 0 + κ = κ.

Addition is associative (κ + μ) + ν = κ + (μ + ν). In Mathematics, associativity is a property that a Binary operation can have

Addition is commutative κ + μ = μ + κ. In Mathematics, commutativity is the ability to change the order of something without changing the end result

Addition is non-decreasing in both arguments:

$(\kappa \le \mu) \rightarrow ((\kappa + \nu \le \mu + \nu) \mbox{ and } (\nu + \kappa \le \nu + \mu)).$

If the axiom of choice holds, addition of infinite cardinal numbers is easy. If either $κ$ or $μ$ is infinite, then

κ + μ = max{κ,μ}.

Subtraction

If the axiom of choice holds and given an infinite cardinal σ and a cardinal μ, there will be a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.

Cardinal multiplication

The product of cardinals comes from the cartesian product. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

$|X|\cdot|Y| = |X \times Y|$

κ·0 = 0·κ = 0.

κ·μ = 0 $\rightarrow$ (κ = 0 or μ = 0).

One is a multiplicative identity κ·1 = 1·κ = κ.

Multiplication is associative (κ·μ)·ν = κ·(μ·ν).

Multiplication is commutative κ·μ = μ·κ. In Mathematics, commutativity is the ability to change the order of something without changing the end result

Multiplication is non-decreasing in both arguments: κ ≤ μ $\rightarrow$ (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).

Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

If the axiom of choice holds, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then

$\kappa\cdot\mu = \max\{\kappa, \mu\}.$

Division

If the axiom of choice holds and given an infinite cardinal π and a non-zero cardinal μ, there will be a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π.

Cardinal exponentiation

Exponentiation is given by

$|X|^{|Y|} = \left|X^Y\right|$

where X^Y is the set of all functions from Y to X. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

κ⁰ = 1 (in particular 0⁰ = 1), see empty function. In Mathematics, an empty function is a function whose domain is the Empty set.

If 1 ≤ μ, then 0^μ = 0.

1^μ = 1.

κ¹ = κ.

κ^{μ + ν} = κ^μ·κ^ν.

κ^μ·ν = (κ^μ)^ν.

(κ·μ)^ν = κ^ν·μ^ν.

If κ and μ are both finite and greater than 1, and ν is infinite, then κ^ν = μ^ν.

If κ is infinite and μ is finite and non-zero, then κ^μ = κ.

Exponentiation is non-decreasing in both arguments:

(1 ≤ ν and κ ≤ μ) $\rightarrow$ (ν^κ ≤ ν^μ) and

(κ ≤ μ) $\rightarrow$ (κ^ν ≤ μ^ν).

Note that 2^| X | is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^| X | > | X | for any set X. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2^κ). In fact, the class of cardinals is a proper class. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

If the axiom of choice holds and 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:

Max (κ, 2^μ) ≤ κ^μ ≤ Max (2^κ, 2^μ).

Using König's theorem, one can prove κ < κ^cf(κ) and κ < cf(2^κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ. For other uses see König's theorem. In Set theory, König's theorem (named after the Hungarian mathematician Gyula König In Mathematics, especially in Order theory, the cofinality cf( A) of a Partially ordered set A is the least of the cardinalities

Neither roots nor logarithms can be defined uniquely for infinite cardinals.

The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2^μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess. ^[1]^[2]^[3]

The continuum hypothesis

The continuum hypothesis (CH) states that there are no cardinals strictly between $\aleph_0$ and $2^{\aleph_0}.$ The latter cardinal number is also often denoted by c; it is the cardinality of the continuum (the set of real numbers). In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of In Mathematics, the real numbers may be described informally in several different ways In this case $2^{\aleph_0} = \aleph_1.$ The generalized continuum hypothesis (GCH) states that for every infinite set X, there are no cardinals strictly between | X | and 2^| X |. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite The continuum hypothesis is independent from the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC). Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common

References

Hahn, Hans, Infinity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting In the mathematical field of Set theory, a large cardinal property is a certain kind of property of Transfinite Cardinal numbers Cardinals with such properties Nominal numbers are numerals used for identification only The Numerical value is irrelevant and they do not indicate quantity rank or any other measurement In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. A serial number is a unique Number assigned for Identification which varies from its Successor or Predecessor by a fixed discrete Integer In Set theory, Cantor's paradox is the Theorem that there is no greatest Cardinal number, so that the collection of "infinite sizes" is itself In Mathematics, the Infinite Cardinal numbers are represented by the Hebrew letter \aleph ( aleph) indexed with a subscript that runs New York: Simon and Schuster, 1956.
Halmos, Paul, Naive set theory. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician See also Naive set theory for the mathematical topic Naive Set Theory is a Mathematics textbook by Paul Halmos Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

^ Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315, Springer-Verlag.
^ Eduard Cech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
^ D. A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

External links

Eric W. Weisstein, Cardinal Number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Cardinality at ProvenMath formal proofs of the basic theorems on cardinality.

Dictionary

-noun

A number used to denote quantity; a counting number.
(mathematics) A generalized kind of number used to denote the size of a set, including infinite sets.
(grammar) A word that expresses a countable quantity; a cardinal numeral.

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