Infinity

Potential infinity is used to refer to processes which can in principle be continued forever, or to objects which can in principle be enlarged forever. For example, the sequence 2, 4, 6, 8, 10, 12, ... is potentially infinite: it is clear how to extend it beyond all bounds. In mathematics, if a function grows beyond all bounds when the argument approaches a certain value, we say that the limit is infinity (written as ∞); this is also an example of potential infinity. The concept of potential infinity is generally accepted and does not pose any problems.

By contrast, it was the subject of much debate whether a complete and existing entity can have infinite size, which one would label actual infinity. In mathematics, actually infinite sets were first considered by Georg Cantor around 1873 and met with much resistance. Cantor went ahead and realized that infinite sets can even have different sizes, distinguished between countably infinite and uncountable sets, and developed his theory of cardinal numbers based on this observation. His view prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the surreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no boundaries. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might operate on a similar principle; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point.

Another question is whether the mathematical conception of infinity has any relation to the religious concept of God. This question was addressed by both Cantor, with his concept of the Absolute Infinite which he equated with God, and Kurt Gödel with his "ontological proof" of the existence of an entity he related to God.

See also: Infinitesimal