Infinity

The term infinity denotes the negation or cancellation of finiteness, less precisely also its opposite. Its mathematical symbol is the infinity sign . Theoretically, the term "infinite" describes an object (such as a sphere) or process with no end or conclusion, but possibly with a beginning or start. So in geometry, a ray or a circular path would be described as infinite.

The concept of "infinity" was clarified above all in mathematics, essentially initiated by the work of Bernard Bolzano, Georg Cantor and Richard Dedekind, which led to set theory and in particular to the theory of infinite sets and transfinite cardinal numbers. In particular, it also experienced for the first time an extension by properties that cannot be derived from the above negative definition.

Methodical approaches

Infinity can only be developed abstractly in the humanities or natural sciences and is applied to objects and concepts that have no spatial and/or temporal limits.

In theology and some philosophical conceptions (such as natural theology), infinity is one of God's attributes, while creation is per se finite or transient. The nature of the infinite is especially a topic of metaphysics as well as mysticism, for example in the Kabbalah under the name En Sof or with Christian mystics such as Nikolaus von Kues and Meister Eckhart.

In philosophy, since Aristotle, there have been two conceptions of the infinite: the "actual infinite" and the "potential infinite". Accordingly, scholasticism distinguishes between the potentially infinite ("indefinite"), which can be multiplied without end, and the actual infinite ("infinite"), which positively excludes any limit. In the narrow and proper sense, therefore, only God has actual infinity. It is the limitless fullness of being, but not to be misunderstood in a pantheistic sense.

Hegel coined the term "bad infinity" (Encyclopedia § 93 f.), by which he understands in a dialectical way a demarcation from finitude.

In astronomy, given the depth and vastness of the starry sky, the idea of an infinitely extended space was often developed.

The concept of infinity is also known in relation to time, where the term eternity is used. While higher mathematics often operates with the abstract term "infinite", in theoretical physics the phenomenon of singularity is more important - for example in connection with the terms big bang (beginning of the visible universe) and black hole. A singularity is a point in spacetime at which mass is concentrated in an expansionless point with infinite density.

Besides the infinite expansion to ever increasing sizes, the term is also used for the infinite divisibility, the infinitely fine, whose limit is zero, but does not reach zero. The negation of the infinitely fine and its paradoxes resulted in the original Greek "atomic theory" of the "indivisible".

Infinity in mathematics

In mathematics, "infinity" gives its name to the axiom of infinity in set theory. Usually, however, the adjective infinite is used to characterize some mathematical concepts in more detail; as a rule, this characterization is complementary to the term finite.