numerabili

Numerabili is a term used in mathematics to describe sets that are countable. In this sense, a numerabile set can be put into a one-to-one correspondence with a subset of the natural numbers. In some contexts, numerabile is used for any finite or countably infinite set, while a related term, denumerabile, is reserved for sets that are countably infinite. The exact terminology can vary by author, but the core idea is that such sets have a listable, non-gapped structure in terms of natural numbers.

Formally, a set S is numerabile if there exists a bijection between S and a subset of the natural numbers N. If the subset is finite, S is finite; if the subset is infinite, S is countably infinite. Thus numerabile often corresponds to “at most countable,” with denumerabile highlighting the infinite case. An equivalent common criterion is that S is numerabile if S is finite or there exists a bijection between S and N.

Examples of numerabile sets include the natural numbers N, the integers Z, and the rational numbers Q. Each of these can be listed in a sequence without omissions. In contrast, the real numbers R are not numerabile; Cantor’s diagonal argument shows that no listing of all real numbers can be completed, making R uncountable.

Key properties include that every subset of a numerabile set is numerabile, and under standard mathematical assumptions, a countable union of numerable sets is numerable as well. Numerability is a foundational concept in discussions of cardinality and in areas such as analysis and measure theory.