Editing Cardinal number (section)

== History ==

The notion of cardinality, as now understood, was formulated by [[Georg Cantor]], the originator of [[set theory]], in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are not ''equal'', but have the ''same cardinality'', namely three. This is established by the existence of a [[bijection]] (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.

Cantor applied his concept of bijection to infinite sets<ref>{{harvnb|Dauben|1990|loc=pg. 54}}</ref> (for example the set of natural numbers '''N''' = {0, 1, 2, 3, ...}). Thus, he called all sets having a bijection with '''N''' [[Countable set|''denumerable (countably infinite) sets'']], which all share the same cardinal number. This cardinal number is called <math>\aleph_0</math>, [[Aleph number|aleph-null]]. He called the cardinal numbers of infinite sets [[transfinite cardinal numbers]].

Cantor proved that any [[Bounded set|unbounded subset]] of '''N''' has the same cardinality as '''N''', even though this might appear to run contrary to intuition. He also proved that the set of all [[ordered pair]]s of natural numbers is denumerable; this implies that the set of all [[rational number]]s is also denumerable, since every rational can be represented by a pair of integers. He later proved that the set of all real [[algebraic number]]s is also denumerable. Each real algebraic number ''z'' may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (''a''<sub>0</sub>, ''a''<sub>1</sub>, ..., ''a<sub>n</sub>''), ''a<sub>i</sub>'' ∈ '''Z''' together with a pair of rationals (''b''<sub>0</sub>, ''b''<sub>1</sub>) such that ''z'' is the unique root of the polynomial with coefficients (''a''<sub>0</sub>, ''a''<sub>1</sub>, ..., ''a<sub>n</sub>'') that lies in the interval (''b''<sub>0</sub>, ''b''<sub>1</sub>).

In his 1874 paper "[[On a Property of the Collection of All Real Algebraic Numbers]]", 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 proof used an argument with [[nested intervals]], but in an 1891 paper, he proved the same result using his ingenious and much simpler [[Cantor's diagonal argument|diagonal argument]]. The new cardinal number of the set of real numbers is called the [[cardinality of the continuum]] and Cantor used the symbol <math>\mathfrak{c}</math> for it.

Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (<math>\aleph_0</math>, aleph-null), and that for every cardinal number there is a next-larger cardinal

:<math>(\aleph_1, \aleph_2, \aleph_3, \ldots).</math>

His [[continuum hypothesis]] is the proposition that the cardinality <math>\mathfrak{c}</math> of the set of real numbers is the same as <math>\aleph_1</math>. This hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them. This was shown in 1963 by [[Paul Cohen (mathematician)|Paul Cohen]], complementing earlier work by [[Kurt Gödel]] in 1940.