Infinity has puzzled thinkers since the time of the ancient Greeks. Some see it as something beyond our comprehension. Writing in the 13th century, the Christian theologian, Thomas Aquinas ruled out infinity, saying “no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited …”. The great Italian scientist, Galileo (1564-1642) thought deeply about infinity but went away confused. It wasn’t until the 19th century that the German mathematician Georg Cantor (1845-1918), the founder of Set Theory, “tamed” infinity, becoming the first person to take a systematic look at the illusive concept. He devoted much of his professional life proving that there are not one, but many types of infinity, some infinitely bigger than others, an insight that shook the very foundations of mathematics and set it soaring in new directions.
A Brief Bio of Georg Cantor
Cantor was born in 1845 Saint Petersburg, Russia, where he lived until he was eleven. Thereafter, the family moved to Germany. At the young age of 34, Cantor was made full professor at the University of Halle, renowned for its program in philosophy and the natural sciences. But Cantor had bigger ambitions and wanted to move to a more prestigious university, such as Berlin or Gottingen. Ironically, his efforts were largely thwarted by his mentor, Leopold Kronecker, who fundamentally disagreed with the thrust of Cantor’s work. Indeed, Cantor’s work differed so dramatically from anything before it that it polarized the views of the leading mathematicians of the day. Some loved it. The renowned mathematician David Hilbert believed Cantor had created a new mathematical paradise and the philosopher-mathematician Bertrand Russell viewed Cantor as one of the period’s intellectual giants. But Cantor also has many detractors who considered him a “quack,” with the French genius Henri Poincaré calling Cantor’s work “a grave mathematical malady.” Consumed by the voices of his critics and obsessed at times by perceived conspiracies, Cantor became depressed and mentally unstable. He was admitted to a mental hospital in Halle in 1884, at the age of 39 and spent much of the last few decades of his life there until his death in 1918 at the age of 73.
Infinite Number of Infinities
Cantor’s first ten papers were on Number Theory, after which he turned his attention to Calculus (or Analysis as it had become known by this time). But his main legacy is that he was the first mathematician to really understand the meaning of infinity and to give it mathematical precision.
For millennia, infinity had caused problems. How big does a number has to be for us to call it infinity? What if you add another small number to that? Where is the limit? Cantor showed that the concept of infinity contains many counterintuitive surprises. For example, you might intuitively think that there are half as many even numbers as there are all numbers. Cantor showed that this is isn’t correct; one can line up both sets of numbers to infinity so that each number has its pair; for example 1 can be paired with 2, 2 with 4, 3 with 6, and n with 2n and let n tends to infinity, which proves these two sets of numbers are actually the same size!
This clearly also applies to other subsets of the natural numbers, such as the squares 1, 4, 9, 16, 25, etc, and even the set of negative numbers and integers. In fact, Cantor realized that he could, in the same way, pair up all the fractions (or rational numbers) with all the whole numbers, thus showing that rational numbers were also the same sort of infinity as the natural numbers, despite the intuitive feeling that there must be more fractions than whole numbers.
To demonstrate this assertion, Cantor came up with an ingenious way to compare all whole numbers with all fractions. It starts with a table containing all the fractions. This table has infinitely many rows and columns (a finite version is shown below). For example, the nth column consists of a list of all the fractions 1/n, 2/n, 3/n, ….
The trick is to wend a snake diagonally through the fractions in the table as shown in the illustration. Then, in order to pair the whole numbers with all the fractions, what we do is work our way along the path, pairing 1 with 1/1, 2 with 2/1, 3 with 1/3, 4 with 1/3. The number 9, for example, gets paired with 2/3, which is the ninth fraction we meet as the snake slithers through the table of fractions. Since the snake covers the entire table, every fraction will be paired with some whole number.
Cantor’s trick is a beautiful and surprising idea, a remarkable way to match up all fractions with whole numbers and show they have the same order of magnitude! It’s one of those things in mathematics that are so simple that it slips right under our noses.
That’s not all. Cantor went on to prove that there are infinitely many types of infinities. For example, he showed there are an infinite number of irrational numbers (numbers like pi) in between each and every rational number. The chaotic decimals of irrational numbers simply fill the “spaces” between the patterns of the rational numbers.
Cantor himself was genuinely surprised by his discovery, and said, “I see it, but I do not believe it.” Being an ardent Christian, he used the letters of the Hebrew alphabet to denote these new infinite numbers, where the first letter of the Hebrew alphabet (called aleph-zero) representing the smallest infinity. Cantor was likely aware of the spiritual significance in the Jewish Kabbalah, where it hints at the idea of a new beginning.
In summary, Cantor did not tame infinity as I jokingly alluded to earlier, but he did chased it, only to open a Pandora box by demonstrating that infinity isn’t just endless; it exists in different sizes. What relevance, you might ask, has this insight for the real world? This is a huge subject for another time, but I just like to briefly say that there are two ways of looking at the question. First, that there is a hierarchy of infinities implies that new technologies like AI using Large Language Models (LLM) hold exciting possibilities; like Cantor’s infinities, these tools allow us to create new spaces of thought, realms of possibility that extend human creativity into previously unimaginable dimensions. A simple example would be a scientist using AI to model infinite variations of a theory or a writer collaborating with AI to explore uncharted narrative directions. These aren’t just outputs—they’re new opportunities for exploration. Yet, at the same time, Cantor’s insights also warn us against unthinking use of AI. The reality is: there’s no way to guarantee the factuality of what is generated using these technologies just as there are always new infinities within a given one, including those which are contaminated by fake or fabricated data. To put if more bluntly, all computer-generated “creativity is hallucination to some extent. A rigorous proof that this is the case can be found in a recent paper three machine-learning experts from the National University of Singapore. Interestingly, the proof is derived from an application of Cantor’s diagonalization argument. Interested readers can visit one of the co-author’s website for more details (https://home.ziwei-xu.com/).