Home > Uncountable set

The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable as well, for instance the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of natural numbers.

Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers. The statement that R is the smallest uncountable set (in the sense that its cardinal number is the smallest uncountable cardinal number) is the continuum hypothesis; this hypothesis is independent from the ordinary axioms of set theory.

The Cantor set is an uncountable subset of R. The Cantor set is a fractalA fractal is a geometric object which is "broken up" in a radical way. The term fractal was coined in 1975 by Benoit Mandelbrot, from the Latin fractus or "broken", in order to call attention to such objects. They are in a number of major aspects differen and has Hausdorff dimensionIn mathematics, Hausdorff dimension is a non-negative real number associated to any metric space. It is named after the mathematician Felix Hausdorff. Intuitively, the dimension of a set, for example a subset of Euclidean space is the number of independen greater than zero but less than one. (R has dimension one.) This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.