The most beautiful inequation in mathematics

It is likely that part of this entry will not have the rigour and precision that a mathematician would demand, because how do you rigorously define the beauty, elegance or depth of an idea? And even more, how to convince someone who is not a mathematician or does not have a certain background in mathematics that there is beauty or elegance in it? At first sight, it is something subjective – there’s no accounting for taste – and it will be a philosopher of mathematics, I imagine, who will come closest to answering whether this has any prospect of ever being resolved. But the fact is that mathematicians use this kind of terminology to refer to some of the works of the greatest authors who have preceded us (and also those of today). Famous is the case of Paul Erdös and “The Book”: “You don’t have to believe in God, but you should believe in The Book”, referring to the book in which the most “beautiful” proofs would be included.

Trying to clarify what all this means is at least a difficult task, but we can find studies in this direction, such as the special issue “Aesthetics in Mathematics” in the journal Philosophia Mathematica, Oxford -the only journal devoted entirely to the philosophy of mathematics-, the book La poesía de los números by Antonio J. Durán (RBA, Barcelona 2011), or meetings in which mathematicians, philosophers and historians take part to try to elucidate these questions and put some order into them, such as the “Mathematical Depth Workshop” held in 2014 at the University of California in Irvine –even in this Blog there have been several entriess: Maths is so lovely!, Physics is so lovely! o The volcano is sublime–. In this meeting, well-known present-day names such as Andrew Wiles or Terence Tao and, of course, classics of this discipline such as Gauss and Euler are mentioned.

And here we are getting closer to the subject of this post. Euler, of whom there is no need to say anything that is not already known, is famous, among many other things, for having come up with one of the most “beautiful” equations, equalities or theorems in mathematics. The famous equality \(e^{i\pi}+1=0\). The reason for this qualification is obvious (or not): it is a simple equality—two addends in the first member, one element in the second; it involves the constants, or some of the most important constants in mathematics: \(e, i, π, 1, 0\); and it connects the real with the imaginary. I don’t recall meeting anyone who did not appreciate its “beauty”, and it is in fact more than well known for it. In the book Euler’s Pioneering Equation: The most beautiful theorem in mathematics by Robin Wilson (Open University & Oxford) this issue is discussed. But the fact is that at some point, and bearing in mind the title of the aforementioned book, I began to think of inequalities, not equalities, that might be worthy of being on a par with Euler’s. Something surprising that involved somehow fundamental constants, whose proof was “elegant”, and whose repercussions were “striking” and “profound”.

The choice I bring for this entry is the one known as Cantor’s Theorem. This blog has already written about this brilliant author -see, for example, Darwin and Wallace, Dedekind and Cantor, Cantor and madness, Cantor, discoverer of the antinomies of set theory–, and the IMUS itself held a conference in 2016 – “The year of Cantor”- in which his life and work were made known. I think there is little doubt that we are talking about a rare, uniquely creative and brilliant person. A genius. Anyone who knows the man and his work in any detail will probably not be surprised by the choice. In its simplified version, this theorem can be written as follows:

$$2^{\aleph_{0}}>\aleph_{0}.$$

Comparisons are odious, but let us dare.

Of course, the inequation is simple, with a single term on each side of it, and nothing more; it involves one of the most surprising and important constants in modern mathematics, \(\aleph_{0}\), the first transfinite cardinal – infinity made number – and connects the finite (\(2\)) with the infinite (\(\aleph_{0}, 2^{\aleph_{0}}\)). On the other hand, this “beautiful” theorem is not nearly as well known as Euler’s theorem. There is no doubt that a good understanding of its meaning – both its parts, its proof, what it implies, and more philosophical reflections, such as, for example, whether we are willing to accept it or not – escapes many. But the case of Euler’s formula is similar: for example, we may be used to \(\pi\), but it has been anything but a simple number to understand, from its tender infancy without a self-awareness in ancient Greece to its maturity in 19th century Germany; that is, more than 2000 years of struggle. And yet, there are still people who don’t believe in it (things of infinity…). Despite the differences, something similar could be said of \(e\) or \(i\).

Now, what about the “depth” of Cantor’s Theorem, of its “shocking” consequences? Is it that big a deal? Let’s see: \(\aleph_{0}\) is the cardinal of the totality of natural numbers, that is, the number of natural numbers existing. Without knowing what the quantity of the first member represents –which is actually the number of subsets that we can consider within the natural numbers, without filter or restrictions–, what this result is telling us about this quantity is that it is greater than \(\aleph_{0}\), which is itself an infinite quantity. Therefore, the conclusion is… there are different sizes of infinity!!!

Whoever has a background in these matters, without necessarily having studied Cantor’s work, will not be surprised. Whoever does not, will be shocked right now. The subject of infinity has also been a problem from the beginning, as attractive as it is difficult. To have been able to give it a numerical treatment, and to show that it is not a single idea, so to speak, but that it hides multiple facets, with different sizes, must be regarded as a revolution in all fields. Cantor discussed these issues with mathematicians, philosophers and theologians, and he was getting into everyone’s territory. In that sense, Cantor’s theorem is at least on a par with Euler’s theorem.

Before analysing his proof, we should make some subtleties. Cantor had already proved in 1873, in a letter to Dedekind, that the cardinality of the interval \((0, 1)\), which is the same as that of \(\mathbb{R}\), is greater than that of \(\mathbb{N}\) (I use modern language for convenience), a proof published in 1874 in On a property of the collection of all real algebraic numbers; this would already show that there are different sizes of infinity. On the other hand, in 1891 he gives a simplified demonstration of this fact in his article On an Elementary Question of Variety Theory, which is where his famous diagonal procedure appears (actually the first to use it was Du Bois-Reymond). This is the demonstration we are going to choose.

Cantor proceeds by reductio ad absurdum. Considering the real numbers of the interval \((0, 1)\) expressed in the binary system, and supposing that they can be numbered, we could have them arranged as follows (in his notation):

$$
\begin{eqnarray*}
E_{1} &=& (a_{1,1}, a_{1,2}, \ldots, a_{1,\nu}, \ldots)\\
E_{2} &=& (a_{2,1}, a_{2,2}, \ldots, a_{2,\nu}, \ldots)\\
&\cdots &\\
E_{\mu} &=& (a_{\mu,1}, a_{\mu,2}, \ldots, a_{\mu,\nu}, \ldots)\\
&\cdots &
\end{eqnarray*}
$$

where he identifies each of them with a decimal number, e.g. \(E_{1} = (a_{1,1}, a_{1,2}, \ldots, a_{1,\nu}, \ldots) \equiv 0.a_{1,1}, a_{1,2}, \ldots, a_{1,\nu}, \ldots\)

Each and every entry of each of these numbers takes only two possible values (\(m\) or \(n\), in Cantor notation, but think of 0 and 1 if you like). Now if we consider \(E_{0}=(b_{1},b_{2},b_{3},\ldots)\), where \(b_{1}\neq a_{1,1}, b_{2}\neq a_{2,2}, b_{3}\neq a_{3,3}\), etc, that is, if we traverse the diagonal carefully, we obtain a real number which, on the one hand, is in the interval \((0,1)\), but which, on the other hand, is not in that list, since each of its digits differs from that of at least one of all those that are. But in that list they are all supposed to be there, which is a contradiction.

Now, to give this theorem the form it takes in the presented inequation, let us turn to his 1895 work, Contributions to the Foundation of the Theory of Transfinite Sets. In section 4, The raising of powers, he introduces –based on what he calls “covering of the set \(N\) with elements of the set \(M\)”– the definition of the power \(a^{b}\) in which each of these values would be the cardinality of a set -\(a\) that of \(M\), \(b\) that of \(N\)-, generalizing what would happen in the finite case. As Cantor says, it is easy to be convinced that, following this notation, the cardinal of the continuum (that is, that of \((0,1)\), which is the same as that of \(\mathbb{R}\)) can be represented in the form \(2^{\aleph_{0}}\) (in fact, this is quite obvious by simply thinking of the decimal representation of numbers in base 2). The steps are not immediate, but all these ingredients together eventually lead us to \(2^{\aleph_{0}}>\aleph_{0}\).

Initially, Cantor did not put the question in this way, nor did he speak of the parts of a set (what we previously described as the totality of the subsets that we can make with the natural numbers, such as \(\{1,10\}\) or \(\{25, 2, 46\}\)), but the approach was made using the terminology of functions.

After proving the theorem we have presented, Cantor comments as follows: “This proof is worthy of attention not only for its great simplicity, but also particularly for the reason that the principle it follows can be extended without further ado to the more general proposition that the powers of well-defined sets do not have a maximum, or, in other words, that to any given set \(L\) can be attached another \(M\) which is of higher power than \(L\)”.

Therefore, to the characteristics we have already pointed out of the theorem, we can add another very notable one which is that of its capacity for generalisation and application (something that also happens with Euler’s, which allows us to prove the transcendence of \(\pi\) given that \(e^{x}\) cannot be rational if \(x\) is a non-zero algebraic complex –as Lindemann proved in 1882–, and that \(e^{i\pi}= -1\), which is a rational number).

This theorem, with all its parts, is in my particular edition of “The Book”. Some will share my choice, many others will not. And there’s no accounting for taste, or maybe there is.

References

Bermúdez, C. G., Georg Cantor. Sistemas de números y conjuntos, Universidad de A Coruña (2009).

Ferreirós, J. Georg Cantor. Fundamentos para una teoría general de conjuntos. Escritos y correspondencia selecta, Barcelona: Crítica (2006).