Infinity

Infinity refers to the concept of limitlessness and unboundedness in size, number or extent. One distinguishes between potential infinity and actual 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 thing can have infinite size, which one would label actual infinity. In mathematics, actually infinite sets were first considered by Georg Cantor around 1900 and met with much resistance. Cantor went ahead and realized that infinite sets can even have different sizes, 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.

An intriguing question is whether actual infinity exists in our physical universe. Are there infinitely many stars, does the universe have infinite volume? This is an important opeen question of cosmology. One should note however that even if the universe were finite, it does not necessarily have boundaries: the two-dimensional surface of the Earth is finite, yet lacks boundaries; similarly, the three-dimensional universe could be finite without boundaries. Just like on Earth, extending a straight line forever in such a universe will eventually revisit the starting point.

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