Cardinal number

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Aleph-0, the smallest infinite cardinal

In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. For finite sets the cardinality is given by a natural number, being simply the number of elements in the set. There are also transfinite cardinal numbers to describe the sizes of infinite sets. On one hand, a proper subset A of an infinite set S may have the same cardinality as S. On the other hand, perhaps also counterintuitively, not all infinite sets have the same cardinality. There is a formal characterization that explains how some infinite sets have cardinalities that are strictly smaller than other infinite sets.

The cardinal numbers are: $0, 1, 2, 3, \cdots, n, \cdots ; \aleph_0, \aleph_1, \aleph_2, \cdots, \aleph_{\alpha}, \cdots$ . That is, they are the natural numbers (finite cardinals) followed by the aleph numbers (infinite cardinals). The aleph numbers are indexed by ordinal numbers. The natural numbers and aleph numbers are subclasses of the ordinal numbers. If the axiom of choice fails, the situation becomes more complicated — there are additional infinite cardinals which are not alephs.

Concepts of cardinality are embedded in most branches of mathematics and are essential to their study. 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.

History

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884.

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.

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 $(b 0, b 1)$ 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. This new cardinal number, called the cardinality of the continuum, was termed c by Cantor.

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.

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.

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.

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:

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.

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.

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:

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. 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.

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).

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 |.

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.

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.

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.

Cardinal addition

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}.

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

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

Addition is associative (κ + μ) + ν = κ + (μ + ν).

Addition is commutative κ + μ = μ + κ.

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.

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

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.

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.

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.

κ⁰ = 1 (in particular 0⁰ = 1), see empty function.

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. 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.

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.

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.

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 2^| X |. The continuum hypothesis is independent from the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice ( ZFC).

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