összeszámlálhatatlanság

Összeszámlálhatatlanság (in English, uncountability) is a concept in set theory, a branch of mathematical logic. A set is said to be összeszámlálhatatlan if it cannot be put into a one-to-one correspondence with the set of natural numbers. This means that there is no way to pair each element of the set with a unique natural number, and vice versa.

The concept of összeszámlálhatatlanság was introduced by Georg Cantor in the late 19th century. He demonstrated that the set of real numbers is összeszámlálhatatlan, while the set of natural numbers is countable. This discovery led to a revolution in mathematics, as it challenged the then-prevailing belief that all infinite sets could be put into a one-to-one correspondence with the set of natural numbers.

Cantor's work also led to the development of cardinal numbers, which are used to classify sets according to their size. The cardinality of the set of natural numbers is denoted by aleph-null (ℵ₀), while the cardinality of the set of real numbers is denoted by the continuum (c). It is known that ℵ₀ < c, meaning that the set of real numbers is strictly larger than the set of natural numbers.

The concept of összeszámlálhatatlanság has applications in various areas of mathematics, including measure theory, topology, and functional analysis. It is also relevant in computer science, where it is used to analyze the complexity of algorithms and data structures.