Cardinal Number - History

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: 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-null, Cantor proved that any unbounded subset of N has the same cardinality as N, even if this might appear at first glance 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 (a₀, a₁, ..., a_n), a_i ∈ Z together with a pair of rationals (b₀, b₁) such that z is the unique root of the polynomial with coefficients (a₀, a₁, ..., a_n) that lies in the interval (b₀, b₁).

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 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-null) and that for every cardinal number, there is a next-larger cardinal

His continuum hypothesis is the proposition that is the same as . This hypothesis 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.