összeszámlálható

In mathematics, the term "összeszámlálható" (Hungarian for "countable") refers to a property of sets that can be placed in a one-to-one correspondence with the natural numbers. This concept is fundamental in set theory and is used to classify the size or cardinality of infinite sets.

A set is called countable if its elements can be enumerated, meaning each element can be assigned a unique natural number. There are two types of countable sets: finite sets and countably infinite sets. Finite sets have a limited number of elements, while countably infinite sets have an infinite number of elements that can still be put into a sequence where each element corresponds to a natural number. For example, the set of all natural numbers (1, 2, 3, ...) is countably infinite.

The concept of countability was introduced to distinguish between different types of infinite sets. A set that is not countable is called uncountable. The real numbers, for instance, form an uncountable set, as demonstrated by Georg Cantor's diagonal argument, which shows that there is no one-to-one correspondence between the real numbers and the natural numbers.

Countable sets play a crucial role in various areas of mathematics, including analysis, topology, and logic. They are essential in defining measurable functions, constructing continuous functions, and understanding the structure of mathematical spaces. The study of countable sets also provides insight into the nature of infinity and the different magnitudes of infinite quantities.