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Encyclopedia > Cardinal number
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. For finite sets, the cardinality is given by a natural number, which is simply the number of elements in the set. There are also transfinite cardinal numbers that describe the sizes of infinite sets. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Number (disambiguation). ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...

Cardinality is defined in terms of bijective functions. 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. 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. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] â€“ January 6, 1918) was a German mathematician. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

There is a transfinite sequence of cardinal numbers:

This sequence starts with the natural numbers (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If the axiom of choice fails, the situation is more complicated, with additional infinite cardinals that are not alephs. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Analysis has its beginnings in the rigorous formulation of calculus. ...

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 (March 3, 1845[1] â€“ January 6, 1918) was a German mathematician. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

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. 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. A bijective function. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In mathematics, a countable set is a set with the same cardinality (i. ...

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. 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 (b0,b1) such that z is the unique root of the polynomial with coefficients (a0,a1,...,an) that lies in the interval (b0,b1). In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, an ordered pair is a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or left projection) and the other as the second coordinate (second entry, right projection). ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...

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. This new cardinal number, called the cardinality of the continuum, was termed c by Cantor. Contrary to what most mathematicians believe, Georg Cantors first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. ... In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such that each set In is an interval of the real line, for n = 1, 2, 3, ... , and that further In + 1 is a subset of In for all n. ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...

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 is a hypothesis about the possible sizes of infinite sets. ...

## 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, .... They may be identified with the natural numbers beginning with 0. The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

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. Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the cardinal numbers described here. In set theory, an infinite set is a set that is not a finite set. ...

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. 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: One-to-one redirects here. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

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. The advantage of this notion is that it can be extended to infinite sets. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

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. 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. 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. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... A bijective function. ... In set theory, the Cantor-Bernstein-Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of the... The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... Sometimes the phrase well-ordering principle (or the axiom of choice) is taken to be synonymous with well-ordering theorem. On other occasions the phrase is taken to mean the proposition that the set of natural numbers {1, 2, 3, ....} is well-ordered, i. ...

The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. 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: Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 â€“ 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ...

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. Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. 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. In more recent times mathematicians have been describing the properties of larger and larger cardinals. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

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. This definition is known as the von Neumann cardinal assignment. 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. 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 mathematics, the axiom of choice, or AC, is an axiom of set theory. ... The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ... In mathematical set theory, the rank of a set is recursively defined 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 Mellon University; he is now retired and lives in Berkeley, California. ...

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 |. The axiom of choice is equivalent to the statement that given two sets X and Y, either | X | ≤ | Y | or | Y | ≤ | X |. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In set theory, the Cantorâ€“Bernsteinâ€“Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst SchrÃ¶der, states that, if there exist injective functions f : A â†’ B and g : B â†’ A between the sets A and B, then there exists a bijective function h : A â†’ B. In terms of... 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. The finite cardinals are just the natural numbers, i.e., a set X is finite if and only if | X | = | n | = n for some natural number n. Any other set is infinite. 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.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. In set theory a set S is Dedekind-infinite if there is a bijective function from S to some proper subset of S, or equivalently if there is an injective function from the natural numbers into S. In the absence of choice, Dedekind-infinite is a stronger condition than merely... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Note: This article contains special characters. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

## Cardinal arithmetic

We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. 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. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...

### Successor cardinal

For more details on this topic, see Successor cardinal.

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. In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. ... 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. ...

If X and Y are disjoint, addition is given by the union of X and Y. 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}. In mathematics, two sets are said to be disjoint if they have no element in common. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

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

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, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

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 μσ. 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. In mathematics, the Cartesian product is a direct product of sets. ...

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

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·μν = κ·(μ·ν).

Multiplication is commutative κ·μ = μ·κ. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Multiplication is non-decreasing in both arguments: κμ $rightarrow$ (κ·νμ·ν and ν·κν·μ).

Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + νκ = μ·κ + ν·κ. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

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

$kappacdotmu = max{kappa, mu}.$

#### 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 μπ. It will be unique (and equal to π) if and only if μ < π.

### Cardinal exponentiation

Exponentiation is given by

$|X|^{|Y|} = left|X^Yright|$

κ0 = 1 (in particular 00 = 1), see empty function.
If 1 ≤ μ, then 0μ = 0.
1μ = 1.
κ1 = κ.
κμ + ν = κμ·κν.
κμ·ν = (κμ)ν.
(κ·μ)ν = κν·μν.
If κ and μ are both finite and greater than 1, and ν is infinite, then κν = μν.
If κ is infinite and μ is finite and non-zero, then κμ = κ.

Exponentiation is non-decreasing in both arguments: In mathematics, an empty or nullary function, is a function whose domain is the empty set. ...

(1 ≤ ν and κ ≤ μ) $rightarrow$ (νκ ≤ νμ) and
(κ ≤ μ) $rightarrow$ (κν ≤ μν).

Note that 2X | is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2X | > | X | for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2κ). In fact, the class of cardinals is a proper class. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

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

Max (κ, 2μ) ≤ κμ ≤ Max (2κ, 2μ).

Using König's theorem, one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ. In set theory, KÃ¶nigs theorem (named after the Hungarian mathematician Julius KÃ¶nig) colloquially states that if the axiom of choice holds and if I is a set and mi and ni are cardinal numbers for every i in I, and then The sum here is the cardinality... In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. ...

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 ≤ 2μ. 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 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 2X |. The continuum hypothesis is independent from the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC). In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation 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 with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers... In English, numbers are pronounced different ways in different regions. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ... Nominal numbers are numbers used for identification only. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... A serial number is a unique number that is one of a series assigned for identification which varies from its successor or predecessor by a fixed discrete integer value. ... Cantors paradox, also known as the paradox of the greatest cardinal, demonstrates that there is no cardinal greater than all other cardinalsâ€”that the class of cardinal numbers is infinite. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...

## References

• Hahn, Hans, Infinity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon and Schuster, 1956.
• Halmos, Paul, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
1. ^ Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315, Springer-Verlag.
2. ^ Eduard Cech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
3. ^ D.A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

Paul Halmos Paul Richard Halmos (March 3, 1916 â€” October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ... Naive Set Theory is a mathematics textbook by Paul Halmos originally published in 1960. ...

Results from FactBites:

 Cardinal number - Wikipedia, the free encyclopedia (2186 words) In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. Cardinality is also an area studied for its own sake as part of set theory, particularly in trying to describe the properties of large cardinals.
 Cardinal number - definition of Cardinal number in Encyclopedia (1886 words) In mathematics, cardinal numbers, or cardinals for short, are numbers used to denote the size of a set. The cardinal numbers were invented by Georg Cantor, when he was developing the set theory now called naive set theory in 1874–1884. The latter cardinal number is also often denoted by c; it is the cardinality of the set of real numbers, or the continuum, whence the name.
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