infinies

Infinies, or infinities, is a term used to describe quantities that have no finite bound. In mathematics, infinity is not a number but a concept that describes unbounded growth, unending processes, or the size of certain infinite sets. There are two broad interpretations: potential infinity, describing an ongoing process without limit, and actual infinity, describing a completed infinite total.

In set theory, infinities occur in different sizes. Cantor showed that infinite sets can have different cardinalities. The set of natural numbers is infinite and countable, meaning its elements can be listed in a sequence. Some infinite sets, such as the real numbers, are uncountable and have strictly larger cardinalities. The cardinality of the real numbers is 2^aleph-null, often called the continuum.

Ordinal numbers describe infinite order types. The first infinite ordinal is omega, representing the order type of the natural numbers. Beyond finite, ordinals can encode transfinite sequences. The continuum hypothesis posits that there is no set whose cardinality lies strictly between aleph-null and the continuum; its status is independent of standard set theory (ZFC).

In calculus and analysis, infinity appears as a limit: x approaches infinity, a sequence diverges, or a series converges to a finite value despite having infinitely many terms. The extended real number line includes symbols +infinity and -infinity to describe unbounded limits.

Infinity has applications across mathematics and physics and raises philosophical questions about actual versus potential infinity and the nature of mathematical existence.