(Cover Artist: Jada Moncur)
Have you ever wondered how infinity came to be? Infinity is a concept that has been theorized and philosophized for a long time, since the times of the ancient Greeks and Indians. Many cultures and religions have hypothesized a thing that is bigger or grander than all other things, but these concepts aren’t really what mathematical infinity today focuses on. We all might have a sense of what infinity is — it’s something that’s bigger than all numbers. However, the modern definition of infinity only emerged relatively recently with many different variants popping up throughout.
Infinity has also been the subject of mathematical paradoxes and controversies over the years, since it is inherently difficult to grasp the concept of something that never ends. The mathematician Leopold Kronecker was a champion of finitism, which is a philosophical position that does not allow for infinities in math. For this reason, Kronecker is known to have disliked Georg Cantor’s set theory, which heavily focused on infinite sets and different kinds of infinity. This battle was part of a larger war in the early-to-mid 20th century between intuitionism and formalism.
There are a lot of different ways in which infinity can be expressed, depending what definition you use. An example of this concept is Cantor’s theory of countable and uncountable sets. Cantor took the natural numbers as the prototypical countable set and then worked from there. Countable sets (also known as denumerable sets) are defined as sets for which there exist bijections between them and the set of natural numbers. Cantor gave these countable sets a cardinality of aleph-0, the first of the cardinal numbers. By contrast, uncountable sets are sets for which there do not even exist surjections between them and the naturals, and they have a cardinality of aleph-1 or higher. Then, in some sense, uncountable sets are “bigger” than countable sets. A famous example of this is Cantor’s usage of his diagonalization proof to prove that the reals are an uncountable set.
In contrast to the cardinal numbers, Cantor also created a number system called the ordinal numbers. These numbers, denoting using the Greek letter ω, are defined to be larger than any real number, so in that sense they are examples of infinities. However, operations with them are a bit counterintuitive and differ with how we might think operations with infinity are supposed to work. For example, addition of ordinal numbers are not commutative, e.g. ω + 1 > ω = 1 + ω.
The concept of infinitesimals, which are related to infinity, and can be best described as numbers that are closer to zero than any real number, but aren’t equal to 0. Note that these are like the cardinal numbers except smaller than any real number rather than larger. 1 is an example of an infinitesimal. Infinitesimals were used to develop the concept of limits, derivatives, and integrals during the time of Isaac Newton and Gottfried Leibniz. You can see why this approach was appealing since limits, derivatives, and integrals deal with very small things and what could be smaller than an infinitesimal? However, this method was later passed up in favor of a more formalized approach by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass, trading intuition for rigor. It was only with the advent of nonstandard analysis in the late 20th century, already after “standard analysis” had won out, that infinitesimals became a viable approach to formal mathematics.
Two relatively simple concepts of infinity are the affinely extended real numbers and the projectively extended real numbers. The former simply appends a positive and negative infinity element to the existing set of real numbers. This addition changes up the algebra of the real numbers quite a bit, since you can now add, subtract, multiply, and divide by infinity. These numbers also provide a nice model for various fields, such as measure theory. The latter only adds a general, all-encompassing “infinity” term that you approach whether you go increasingly negative or positive. Funnily enough, the projectively extended real numbers allow division by zero. Like the affinely extended real line, the projectively extended real line has its uses, advantages, and disadvantages that make it a useful tool in select cases, such as with map projections.
Yet another infinity-related topic is the surreal numbers, a creation of John Conway. The surreal numbers offer more types of infinity and generalize a lot of other number systems, including real numbers, infinitesimals, and ordinal numbers. Despite seeming like a pure mathematician’s paradise, the surreal numbers actually have several applications, including combinatorial game theory, where they can be used to model and give numerical values to various game states where only using the real numbers will not suffice.
Thus ends this foray into infinity, but just like its definition, the concept itself is endless. There are so many different nuances, nooks, and crannies to infinity that are impossible to cover in a short article. This article doesn't even cover all of the different number systems of infinities or go into too much depth on the ones that were included. Let this article serve as just a first dip into the wonderful world of infinity.
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