infinity

The term “infinity” does not itself have a settled mathematical definition, but instead refers to an intuitive notion of being boundless. The word, Latin in origin (infinitas, “without finish”), corresponded originally to the Greek word \(\alpha\pi\varepsilon\iota\rho\omicron\nu\) (apeiron), meaning without form or limit.

The Greek philosopher Aristotle argued that nothing that actually exists could be limitless, because to exist means to have a particular form, and form implies limits. On his authority mathematicians refused to treat infinity as meaningful until late in the 19th century, when the invention of set theory by Georg Cantor introduced infinite collections with well-defined properties.

In modern mathematics the notion corresponds to infinite sets, including countably infinite and uncountable sets. In certain non-standard conceptions of the real number line there is a notion of infinity as a reciprocal of an infinitessimal, a quantity smaller than any definable real number. It may also be used less formally to indicate a procedure that may be carried out indefinitely, as in the “infinite divisibility” of a continuous line segment into smaller segments. In projective geometry the terms “point at infinity,” “line at infinity,” etc., are used to refer to certain formally defined objects. Finally, in formal number theory and mathematical logic there is a well-defined inference rule called “transfinite (infinite) induction.”