*Endless. Boundless. Limitless.*

Infinity is a concept which has existed for thousands of years and that everyone is familiar with, yet if we try and scratch past its surface we realise that there is much more to it than meets the eye.

Perhaps we first encountered it in when we were very young, arguing over who had more toys to the point where we exclaimed that we had ‘infinity’ of them. However, along with the obvious physical impossibility of having infinite objects, the thing about infinity is that it isn’t really a number – it’s an idea. Infinity simply doesn’t follow the same rules as ‘normal’ numbers and mathematics breaks down if we try and perform basic operations on it, as we will see in a bit. But the truly bizarre nature of infinity is best explained through its uniquely intriguing paradoxes (logical arguments leading to self-contradictory statements) so let’s make our way back to Ancient Greece to view one of the first ones ever created.

**Zeno’s Dichotomy Paradox**

The earliest paradoxes on infinity originated from the Greek philosopher Zeno of Elea who thought of a number of paradoxes in his time. One of the most famous among these was the dichotomy paradox meaning ‘cutting in half’, and involves Achilles completing a race.

Zeno begins by stating that for Achilles to run a 100m race he must first run half of it. But once he has run the 50m he must then run half of the remaining distance. And thus, since after completing every ‘step’ there will always be another half of the race left to run, Achilles has the seemingly impossible task of completing an infinite number of steps in a finite amount of time (which is formally known as a supertask). Hence, Zeno concludes that motion is impossible.

But surely this is nonsense? We observe motion on a daily basis without any such problem so how could Zeno arrive upon such a conclusion? The flaw here arises from infinite series. Zeno essentially assumes that an infinite number of terms must add to an infinite amount, but upon inspection it is clear to see that after a point the terms become so negligibly small that the corresponding difference they make to the sum of values is also negligible. And so, as common sense also tells us, 50 + 25+ 12.5 + … simply converges to 100. Nonetheless, despite being resolved this paradox is paramount in explaining just how easy it is to reach surprising conclusions when dealing with infinity and the next example, conceptualised by German mathematician David Hilbert, portrays this much more evidently.

**Hilbert’s Infinite Hotel**

Let’s start by considering that there is a manager in charge of a hotel with an infinite number of rooms. One night, when the hotel is completely full, a bus arrives with a new guest looking for a spare room. Now the manager doesn’t want to turn away the guest but the hotel is completely full so, after some deliberation, he comes up with an innovative plan: he tells the guest in Room 1 to move to Room 2, the guest in Room 2 to Room 3 and so on. Since there are infinitely many rooms, this process can repeat forever as there will always be a ‘next room’ to move into. And voilà – Room 1 is now free for the new guest to move into.

What does this mean? Well this actually proves a rather peculiar result: that ∞ + 1 is also infinity. In fact, even if there were a bus with, say, 100 new guests, the manager could simply repeat the process but this time move every guest to Room n+100. Therefore, bizarrely enough, ∞ + any finite number is still infinity.

Now let’s say the next day an infinitely long bus arrives with an infinite number of passengers, each looking for a room in the hotel. Initially the manager is confused but then he comes up with a new plan: he now moves every guest to the room that is double their current room number i.e. Room n to Room 2n. And again, since there are infinitely many rooms to move into, this strategy successfully leaves every even-numbered room occupied and every odd-numbered room empty for the infinite new guests to move into. And with that, we have proved a new result – that there are, as weird as it seems, as many even numbers as there are natural numbers.

But what happens if there are infinitely many buses each with infinitely many guests? It must surely be impossible to accommodate all these new arrivals, right? Yet there is actually still an ingenious way to do so, and it involves the fact that there are infinitely many prime numbers.

The manager begins by moving each existing guest to the room number which is 2 (the first prime number) raised to the power of their current room number. So the guest in Room 1 moves to Room 2 (2^1); the guest in Room 5 moves to Room 32 (2^5) and so on. Then, he assigns a prime number to each bus, giving the first bus 3 and each subsequent bus the next largest prime number.

And now comes the incredible part – every passenger can now be moved to the room that is the prime number of their bus raised to the power of their seat number. So someone in seat number 5 in the first bus would go to Room 243 (3^5), whilst someone in seat number 2 in the second bus would move to Room 25 (5^2).

Somehow, despite being completely full, the hotel has still managed to make room for an infinite number of buses with an infinite number of passengers, meaning that ∞ x ∞ = ∞. And what’s more is that there are actually now empty rooms – since every occupied room has a number which is the product of only one specific prime multiplied by itself a given number of times, all the rooms which comprise of more than one prime (e.g. 6, 20) are vacant!

You may wish to pause for a minute – the past examples have shown some rather unusual results, undeniably surprising at first. If Hilbert’s Grand Hotel Paradox proves anything it genuinely is the fact that infinity is indeed perplexing. But there is still one weird phenomenon left to prove (as far as this article is concerned) – that there are, in fact, different types of infinity.

**Countability**

So far, we have actually only been discussing one type of infinity – countable infinity. This, in essence, refers to a set of numbers which, if there were an infinite amount of time, could be exhaustively listed down. In other words, every number has a one to one correspondence with the set of natural numbers (1, 2, 3 etc.) and so every countably infinite set is equal, which is why the hotel manager never had any problem making arrangements for his new guests.

The question now emerges – how can an infinity be bigger than this? And the answer came from German mathematician Georg Cantor, one of the leading figures in the topic of infinity, in his famous Diagonal Argument.

Cantor stated that if you attempted to compose a list of all the real numbers between 0 and 1, you would always fail to do so. This is because if you take the nth digit from each number and change it (e.g. add 1) you can create a new number every single time, and you would know that it is unique since every existing number differs by at least one digit (its nth digit) from this new number.

What’s more is that since this can be done with the numbers in any order, the diagonal argument itself can be repeated an infinite amount of times. The list simply cannot be completed. And so, Cantor not only proved the existence of a new, uncountable infinity but also the fact that there are more real numbers between 0 and 1 than there are natural numbers.

Take a moment to reflect on this – it’s a lot to take in and requires some focus to truly grasp. No matter how much you tried, even with an infinite amount of time, you simply could not list every rational number. And perhaps reason agrees – attempting to write the smallest positive real number is, in itself, a task equivalent in nature to listing every natural number. Because what would it be? 0.1? 0.001? 0.0000000000001? There will always be another 0 that could be included in the infinite string of 0s required, just like there will always be a real number missing from an infinite set of real numbers.

In the end, there is far too much to cover about infinity than can be contained in this article, or to be completely honest, in any article. Maybe the last 1500 words made very little sense; after all, infinity is one concept which we may never truly be able to fathom in all its complexity. But perhaps the very fact that it lies beyond complete comprehension is what makes it one of the most alluring and spellbinding topics in all of mathematics.

References

- Wesley C. Salmon – A Contemporary Look at Zeno’s Paradoxes – Space, Time and Motion: A Philosophical Introduction
- Colm Kelleher – What is Zeno’s Dichotomy Paradox? https://www.youtube.com/watch?v=EfqVnj-sgcc
- Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/spacetime-supertasks/
- Hilbert’s Infinite Hotel Paradox https://medium.com/i-math/hilberts-infinite-hotel-paradox-ca388533f05
- Understanding Hilbert’s Grand Hotel Paradox http://mathandmultimedia.com/2014/05/26/grand-hotel-paradox/
- Jeff Dekofsky – The Infinite Hotel Paradox https://www.youtube.com/watch?v=Uj3_KqkI9Zo
- Math Insight https://mathinsight.org/definition/countably_infinite
- Pursuit of Wonder – The Mystery of Infinity https://www.youtube.com/watch?v=tgSJGGGd4no