Previously, we promised a discussion about “infinite sets, larger or smaller”. No doubt that such a description is counterintuitive: infinity is a supreme object, so it seems impossible for it to have any degrees of size. However, we will see that there is no trick or ambiguity involved: we use the common meaning of “size” which applies to finite quantities and will conclude that in the case of infinity, there must exist a hierarchy. The key point is not that the notion of “size” for a set (formally called cardinality) is tricky, but the very concept of infinity itself.
But let us start with the mathematical context and include a bit of history. The history of infinity is a subject which is worth studying in detail, since it is one of the most important concepts in the evolution of ideas, scientifically, philosophically, abstractly speaking alike. In mathematics at least, infinity is closely connected to sets, since, as we will soon explain, the discrete and the continuous sets alike naturally lead to the idea of infinity. In what follows, we will highlight some key points, and should you want to learn more, please leave a comment, or send us an email at contact[at]poligon-edu.xyz. We have prepared a custom course to detail all such notions and can suggest further reading or discussion topics.
Actually and Potentially
It should not surprise anyone that ancient civilizations quickly discovered the notion of infinity and found applications for it. The immensity of the Heavens and stars above was a great start, as most philosophers of antiquity thought they were infinite, as well as the Universe. One of the first concrete examples referred to the origins of life, and philosophers such as Thales from Miletus (sixth century BC) and the poet Hesiod (seventh century BC) wrote about air or water deemed infinite, since any life form needs them, hence in order to give life to all forms on this planet, as well as the entire biosphere that ever existed and will exist, air and water must be infinite. They were followed by Anaximander of Miletus (sixth century BC), which applies an abstract philosophical technique turning the infinite attribute of the substances into a noun. Hence, Anaximander writes about the infinite itself, as a noun, calling it apeiron.
Some centuries later we find the first major advance in the understanding of infinity with the help of Aristotle (fourth century BC). He proposes a categorization of infinity into infinity in division and infinity in addition. The first category means the endless possibility of dividing, i.e., of “cutting” an object into smaller and smaller pieces, whereas the second notion corresponds to the usual understanding of infinity, that is through unending addition of quantities or features or parts, i.e., unending enlargement.
Another categorization that is also attributed to Aristotle is that of potential and actual infinity. The first meaning corresponds to the object we use nowadays in calculus or mathematical analysis and is fundamental for defining limits. We say that a sequence of numbers goes to infinity if whenever one tries to set an upper bound for it, one can still find at least one more term that goes beyond the said bound. This means that this infinity is a tendency, not a realization or a finished process. On the other hand, the actual infinity is one which “really exists”, in a “final” form, hence we can point it out. Typical examples include sets of numbers, such as the non-negative integers, {0, 1, 2, 3, …} or the integers {…, -2, -1, 0, 1, 2, …}. We avoid here more in-depth discussions regarding the definition or construction of such sets, which obviously cannot be a finite process one can algorithmically apply; for the purposes of this article, it suffices to know that there is a difference between infinity understood in mathematical analysis, which is a tendency, a potential, whereas that for sets is usually actual.
Correspondence
An important point of the previous article was to show a mathematical method through which one can prove that two sets have the same number of elements. When their elements could be paired completely, without any leftovers or without using an element more than once, we say that the two sets have the same cardinality (number of elements). We can keep this description intact for infinite sets as well! For example, we can pair non-negative integers with their doubles and thus show that the set {0, 1, 2, 3, 4, …} has the same cardinality with the set {0, 2, 4, 6, 8, …}, which is counterintuitive, since it appears that the second set contains only “half” of those in the first set!
The contemporary history of infinity is connected to one mathematician more than any other: the Russian-German Georg Cantor (1845-1918). His biography, as well as his works, are worth studying separately, for many reasons[1]. One such reason is a natural question that arises: how can one reach the topic of infinity in itself and equally important, what kind of problems could lead to the conclusion that infinity has a hierarchy? The answers are surprising: Cantor did not set out to study set theory or infinity and only reached them coming from problems that in appearance are not connected to such topics, problems which combined calculus and trigonometry! His remarkable idea was to organize the solutions to such problems into sets and try to find connections between the problems by studying the connections between their corresponding sets. He then noticed that although two problems lead to infinite sets, the correspondence could not be established, since one of the sets seemed to be “smaller” than the other.
Further, Cantor’s problems went full blast, not only in a mathematical sense. Growing up and educated in a religious family, Cantor associated infinity with God, like most thinkers. Therefore, a hierarchy of infinity (“smaller” or “larger” infinities) seemed not only mathematically unsound, but heretical! Personal, family, and health issues amplified this crisis and Cantor’s life ended in psychological torment. He did manage, however, to prove the mathematical existence of hierarchies among infinite sets, but as was probably expected, he did not get attention from the mathematical community, but mockery. The lack of appreciation was balanced at least in part by illustrious contemporaries such as David Hilbert and Richard Dedekind, who continued his research and set the rigorous foundations for set theory and infinity.
Hilbert’s Hotel and Cantor’s Diagonal
We close the article with two highlights of abstract thought from David Hilbert and Georg Cantor, both showing surprises that infinity has in store.
It is said that David Hilbert had thought of an argument that became quite popular for understanding infinity, which he explained to whoever wanted to grasp Cantor’s ideas. Its elegance and simplicity are indeed remarkable: assume there exists a hotel with an infinite number of rooms, all of which are taken. One night, a tourist comes to the reception desk and asks for a room. The receptionist agrees to host the tourist and frees room 1 for him, without evicting any of the other tenants! How is that possible? Each tenant is asked to move to the next room: those in room 1 move to room 2, those in room 2 move to room 3 and so on. Since the hotel has an infinite number of rooms, there is always a “next room” and furthermore, the procedure frees room 1 for the new tourist.
The second abstractly elegant argument belongs to Cantor himself and shows that the real numbers are “more” than the integers, for example. The technique uses the proof by contradiction method or reductio ad absurdum as it is known.
Assume that the real numbers are “as many” as the positive integers. For simplicity, let us only consider from the reals those which are smaller than 1, hence which have a decimal for of 0.something. We can make a correspondence to the positive integers by using a list, since the positive integers are used for counting. Hence, we list the reals: the first, the second, the third and so on. Cantor then proposes to define a new real number like so: its first decimal digit is taken from the first number in the list, its second decimal digit is taken from the second number and so on. Finally, we obtain a number which collects the decimal digits diagonally from the list, but which is not found in the list, due to the method we defined it through! The conclusion is that if we tried to list the reals exhaustively, one can always find a new one, which means that there are “more” reals than positive integers.
[1] We recommend Cantor’s biography written by Joseph W. Dauben and titled Georg Cantor: His Mathematics and Philosophy of the Infinite, published in 1979.