Is there anything bigger than infinity?

What's the biggest number you can imagine? You probably won't say 1000, neither will you say 1 billion, because no matter how big of a number you can think of, you can always simply add 1, and that number will be bigger. So what about infinity? Infinity is a sort of vague concept, but is there something beyond infinity?

Image Credit: Vectorium via Shutterstock / Edited by Universal-Sci

Image Credit: Vectorium via Shutterstock / Edited by Universal-Sci

Let's start by revealing that, yes, there really is something bigger than infinity. But to explain this, we first need to take a detour to the tail end of the 18th century. 

Back then, there was a lot of confidence in the world of science. People thought that physics was pretty much complete, and there was a lot of optimism within the field of mathematics as the idea prevailed that proof could be found for everything that was 'true.' One prominent mathematician that lived at that time was Georg Cantor.

Cantor was the first person that showed there was something beyond infinity. His explanation was very simple and elegant and, at the same time, completely shattered the world of mathematics.

A hotel with infinite rooms

According to Cantor, two types of infinity exist. This can be best explained by using David Hilbert's paradox of the Grand Hotel (Hilbert was another prominent mathematician of that time). Hilbert's hotel is a thought experiment that illustrates a counterintuitive property of infinite sets.

Gregor Cantor in the early 1900’s - Image Credit: Wikimedia Commons

Gregor Cantor in the early 1900’s - Image Credit: Wikimedia Commons

Let's try the thought experiment:

We will need to appeal to your sense of infinity: imagine a hotel with an infinite amount of rooms. It has numbered rooms: so, room 1,2,3,4,5,6,7... (this goes on to infinity).

The hotel is booked up; every room has a guest. Then suddenly, a new guest appears at the reception. The guest asks: ''do you have any rooms available?'' The receptionist replies, ''yes, of course,'' and continues by moving all guests up a room. (So the guest residing in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on)

This action frees up the first room for the new guest. (Infinite + 1 = infinite)By repeating this procedure, it is possible to make room for any finite number of new guests. 

So what if an infinitely long bus arrives with an infinite amount of new guests and all these guests were to ask for a room? Again the receptionist says: ''that this is no problem at all''. He devises the following plan: 

Everyone has to leave their room, look at their room number, multiply it by 2 and go to the room with that number. (So the guest residing in room 1 moves to room 2, the guest in room 2 moves to 4, the guest in room 3 moves to room 6, and so on.)

You see that all even-numbered rooms get occupied with this plan, freeing up all odd-numbered rooms. Both even-numbered and odd-numbered rooms are infinite, so the vacant rooms can now be occupied by the infinite amount of new guests. In short, there is now room for the bus with an infinite amount of guests.  (Infinite + infinite = infinite)

Making room for the new guests - Image Credit: Jan Beránek - Edited by Universal-Sci - (CC BY-SA 4.0)

Making room for the new guests - Image Credit: Jan Beránek - Edited by Universal-Sci - (CC BY-SA 4.0)

Let's go one crazy step further and imagine an infinite amount of these infinitely long busses with an infinite amount of occupants and that all of these people were to ask for a vacant room. Again the cunning receptionist would be able to forge a plan to solve this problem:

All current guests will need to go to even-numbered rooms as before. Then the new guests from bus 1 go to the rooms with the numbers 3, 9, 27, and so on. (so powers of 3). The new guests from bus 2 move into rooms 5, 25, 125, and so on (so powers of 5). New guests from bus 3 will take rooms 7, 49, 343, and so on (so powers of 7). We now know for certain that there is room for all the new guests. 

In general, any so-called pairing function can be used to solve this problem (assuming that the seats on each bus are numbered as well). This goes a bit beyond the scope of this article though; nonetheless, we can conclude that infinite x infinite = infinite. 

How to go beyond infinity?

So if infinite + 1 = infinite, infinite + infinite = infinite and infinite x infinite = infinite how do we get beyond infinity?

To answer this, let's go back to Hilbert's hotel once more. A strange bus has just arrived: the decimal bus. This enormous bus has occupants with very peculiar surnames consisting of decimals. A decimal is a number between 0 and 1 like 0.384882... (this continues to infinity) or between 1 and 2 like 1.489383... (this continues to infinity), and so on. The bus contains all the decimal numbers are represented...  

The bus driver askes the receptionist if the hotel has room for his passengers. To the bus driver's surprise, the receptionist replies: ''no, I'm sorry, that's not going to work.'' 

Angerly, the bus driver says: ''but you had room for the infinite amount of busses with an infinite amount of passengers, why don't you have room to accommodate my passengers?'' The receptionist replies by stating that there is simply no room. The annoyed bus driver doesn't believe the receptionist, so the receptionist suggests playing a game to determine who is right. 

The bus driver is allowed to submit a list defining which room each bus occupant would get. The receptionist, on his turn, is allowed to point out someone who isn't on the list but does have a seat on the bus. (So a passenger that the bus driver has missed).

Some infinities are bigger than others.

Let's start by playing the game using a hotel with a finite number of rooms to better illustrate the game's merits. Let's take a hotel with 4 rooms and a bus with 5 seats. In this case, the bus driver will never be able to get a room for all his passengers. Five people in four rooms do not fit, so with every imaginable list the bus driver submits, the receptionist can say that there is a missing passenger. 

But if the hotel had had 5 rooms, it would have been possible to provide a list where each bus passenger gets a room. However, the receptionist wants to prove that the infinite decimal bus is bigger than the infinite Hilbert Hotel, so what can he do?

This is where Cantor comes in. Cantor claims that in this scenario, the receptionist will always win. There is an algorithm that the receptionist can apply so that he always wins.

Let's take a look at a section of a list that the bus driver could have made:

Image Credit: Universal-Sci - (CC BY-SA 4.0)

On this list, room 1 will accommodate passenger 0.81077415… (this continues to infinity), room 2 will accommodate passenger 0.32148673… etc. So now the receptionist has to point out a bus passenger who is not on the list. 

He looks at the surname of the listed guest for the first room and notices that the first digit to the right of the decimal point is 8 (0.81077415…). He then adds 1 to this number, making it 9 (0.9...); from this point on, we know that any number that starts with 0.9... does not occupy room 1.

He continues to the listed occupant for the second room and looks at the second digit to the right of the decimal point. Which, in this case, is a 2 (0.32148673…). Again, he adds 1 to this number, making it 3 (0.33...). We know now is that any number that starts with 0.93... does not occupy room 1 or 2. 

If the receptionist continues doing this (room 3: 0.530.., room 4: 0.7697..., room 5: 0.83680..., room 6 0.879344, etc.) We now know that any number that starts with 0.930704... does not occupy rooms 1,2,3,4,5, or 6. The receptionist can go on like this forever and create a number that does not exist on the bus driver's list, which means the receptionist wins. 

Whatever list the bus driver prepares, the receptionist can always follow this recipe and come up with an occupant that is not on the list. 

We can conclude that the infinity of the decimal bus is bigger than the infinity of rooms in Hilbert's hotel. In other words, real numbers (Numbers that can represent a distance along a line. They can be positive, negative, or zero) are more numerous than natural numbers (counting numbers 1,2,3... like we encountered in Hilbert's hotel)

For many people, it takes some time to let this sink in; it is something you have to learn to appreciate. As a matter of fact, a lot of mathematicians at the time of Cantor did not like his ideas on infinity. It was described as utter nonsense and laughable. Cantor even suffered some personal attacks, being called a charlatan and a 'corruptor of youth.' Some theologians at the time even took it as an attack on God. 

Nonetheless, at a later point in time, the royal society rewarded Cantor with its Sylvester Medal, the highest honor it can bestow for work in mathematics. In addition, David Hilbert himself defended Cantar's views on infinity as they really spoke to him. So much so that he went on in saying, "No one shall expel us from the paradise that Cantor has created."

Nowadays, we remember Cantor as the man who discovered different types of infinity.

Sources and further reading on the subject of infinity, numbers, and math:

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