KRISHI SEKSARIA and TIM HIRE
“To infinity and beyond” - this classic line used by Disney’s ‘Buzz Lightyear’ character is known by children throughout the world, but is it mathematically true? Is there anything beyond infinity? What is infinity? To gain an understanding about such a mind-numbingly humongous concept (infinite in size!), we must first consider the original roots of infinity in the expansive history of mathematics to discover what the concept really represents.
Our journey begins in the 5th Century BCE where, in Greece, the philosopher Anaximander coined the Latin term aperion (which has connotations of ‘indefinite’ or ‘undefined’ in modern-day translations) to represent his belief that the indefinite was the source of all things, perhaps stemming from the Ancient Greek’s fascination with the seemingly endless number of stars. However, such an abstract postulation generated conflict amongst the Greeks, as such an idea could not be rationalised within the boundaries of their current finite mathematics.
These reservations can be seen in the writings of the Greek mathematician Euclid (300 BC), who wrote “prime numbers are more than any assigned multitude of prime numbers”, thereby avoiding the idea of the indefinite and infinite. This fitted with Aristotle’s differentiation 50 years previously between the potential infinity and actual infinity, which was collectively deemed by the Greek mathematicians to be impossible.
A further distinction of the various ‘segments’ of infinity took place independently by Indian mathematicians, as recorded in the Surya Prajnapti text, which classed the concept in three different separate sets - numerable, innumerable and infinite. Such a distinction perhaps reflects the collective abhorrence towards the unrationalisable felt by ancient mathematicians across the globe.
The so-called ‘dark ages’ of Mathematics followed, as much of the work collected, or stolen, by the Greeks was lost during the burning of the Great Library of Alexandria. Mathematicians subsequently because reclusive and secretive as society in general lost interest in making advances in the field.
Finally, after centuries of stagnation, progress in the realm of the infinite began once more. Galileo is thought to be the first mathematician or philosopher to account for infinity in his philosophy of life and bear the consequences associated with this uncertainty. Interest surrounding the subject was revitalised leading to the most well-known part of infinity being developed - the symbol ∞. The looping symbol, first coined by the English mathematician John Wallis in 1655, embodies the endlessness of infinity. Wallis used the symbol in his notation for infinite sequences, an idea that eventually provided the groundwork for Calculus to be developed.
Differentiation, a famous mathematical process developed simultaneously by Isaac Newton and Gottfried Leibniz for finding the rate of increase of a function, was the by-product of increased understanding about the concept of infinity. Both mathematicians contemplated what would happen to functions if their input values were infinitesimal. Thus, calculus was born.
As the modern world of mathematics developed, as did the uses for a concept of something deemed unconceptualisable by the mathematicians of old. Today, engineers around the world use integration, an aspect of calculus, to calculate the length of power lines required to connect two substations, and to consider the flight trajectory of space vehicles (amongst numerous other examples), a method that would not exist without historical developments of calculus.
However, things were not all plain sailing on the path to infinity. Jumping back in time to 1874, issues associated with infinity began to arise as Georg Cantor provided a proof that there is an infinite number of different sized infinities[1] in what became known as the diagonal proof. To understand infinity further, we must take a look at the different types of infinities and how to compare them.
To begin, we must understand infinity. Many students are unable to conceptualise infinity. In short, infinity is NOT a number, it’s a concept. In calculus, taking a limit at infinity does not involve treating it as a number but rather exploring what happens to a particular function as the input becomes larger. Infinity is not a number, hence you cannot perform arithmetic or other numerical operations on it. For example, infinity plus infinity will not give you twice the original infinity (unless they are the exact same size).
Thus, mathematicians have to carefully consider the ‘size’ of infinity they are working with. There are 2 main types of infinity: the countable infinities and the uncountable infinities, both of these are infinite but the uncountable infinities are much larger. But the question that arises is how we actually measure the sizes of these infinities?
The idea of infinity comes up a lot in the context of set theory, specifically, how we measure and compare the sizes of sets. One such way is to compare the cardinality of the sets ie. the number of elements in the sets. Two sets can be shown as having the same number of elements if you can find a bijection, which is a one to one function that maps the elements from one set to the other set leaving no element unpaired. In short, if you can pair up each element from one set with a unique element of the other set with no elements left over, you can conclude that both the sets are equal in cardinality.
Different ‘sizes’ of infinity also means that problems with infinity are very counterintuitive. For example: the set of all even positive integers = {2, 4, 6, 8…} has the same number of elements (same cardinality) as that of the set of all whole numbers = {1, 2, 3, 4…}. This is because if you pair up the two sets (using a bijection), every even number pairs up with the number at half its value in the set of whole numbers, this bijection can be represented by the function f(x) = x/2. Intuitively this seems incorrect as the set of whole numbers obviously contains the set of even numbers. However, you have got to remember that both the sets have infinite elements ie. there will always be an even number to pair with a whole number.
There are a lot of other ways to measure and compare the sizes of infinite sets such as set containment and the Lebesgue measure, which use a unique characteristic of infinite sets to compare them, however, this is a different topic entirely!
Infinities are generally infinitely larger or smaller than other infinities, for example, the cardinality of the set of integers is infinitely less than the cardinality of the set of the real numbers. However, not all infinities are infinitely far apart. In some cases, specific infinities are very close to other ones. An example of this is the Euler-Mascheroni constant (represented by 𝛾) is the difference between two infinities (more specifically, the sum to infinity of the harmonic series and the natural log of infinity, expressed as limits). Surprisingly, it evaluates to 0.577… ! In this rare case, these two infinities are less than 1 apart. This mind-boggling fact exemplifies the interesting nature of the concept of infinity.
Of course, there is infinitely more to talk about the concept of infinity, but that would take infinitely more storage than what exists. We recommend the TED Talk linked below for more information on the subject.
[1] https://www.ted.com/talks/dennis_wildfogel_how_big_is_infinity?language=en