Editing Cardinal number (section)

== 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 set|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 that has exactly the right size. For example, 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 set]]s, it is essential to distinguish between the two, since 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 refined notions.

A set ''Y'' is at least as big as a set ''X'' if there is an [[injective function|injective]] [[map (mathematics)|mapping]] from the elements of ''X'' to the elements of ''Y''. An injective 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 injective, and hence conclude that ''Y'' has cardinality greater than or equal to ''X''. The element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a [[bijective]] 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 [[Set (mathematics)|sets]] ''X'' and ''Y'' are said to have the same ''cardinality'' if there exists a [[bijection]] between ''X'' and ''Y''.  By the [[Cantor–Bernstein–Schroeder theorem|Schroeder–Bernstein theorem]], this is equivalent to there being ''both'' an injective mapping from ''X'' to ''Y'', ''and'' an injective 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''|.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Cardinal Number|url=https://mathworld.wolfram.com/CardinalNumber.html|access-date=2020-09-06|website=mathworld.wolfram.com|language=en}}</ref> 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]].  Supposing there is an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives.  It is 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
: ...
With this assignment, 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 that has a proper subset of the same cardinality (i.e., a [[Dedekind-infinite set]]); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.

When considering these large objects, one 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 does not; 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 [[Ordinal number|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 number]]s 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''.