przeliczalnego

Przeliczalnego is the genitive singular form of the Polish adjective przeliczalny, which in mathematical contexts is used to describe sets that are countable. The term przeliczalny refers to a property of being enumerable in a finite or countably infinite way, and it is contrasted with uncountable (nieprzeliczalny) in Polish mathematical literature.

In set theory, a set is przeliczalny (countable) if there exists a bijection between the set and the natural numbers. This includes finite sets and sets that are in bijection with N, such as the set of natural numbers themselves, the set of even numbers, the set of integers, and the set of rational numbers. By definition, the real numbers are not przeliczalny; they form an uncountable set.

Key characteristics include that any subset of a przeliczalny zbiór is przeliczalny, and the union of a finite or countable family of przeliczalne zbiory (countable sets) is itself przeliczalny under standard assumptions in set theory. These concepts are central to cardinality and the study of different sizes of infinity.

Etymologically, przeliczalnego derives from przeliczyć (to count) with the suffix -alny, and the form shows grammatical inflection appropriate for genitive constructions in Polish sentences, such as “zbiór przeliczalnego typu” or similar phrases.

See also: przeliczalny zbiór, liczby przeliczalne, nieprzeliczalny.