## Leonhard Euler

The Mystery of Infinities

Emir Alikalfa

In the mysterious realm of mathematics some concepts, such as infinity, catch people's attention and make people wonder about itself. Georg Cantor was just one of many mathematicians who were intrigued about the term “infinity”. But, what actually is infinity?

The word “infinity” conjures endless possibilities and unbounded expanses, dragging people deeper into mathematics. Infinity represents endless possibilities and quantities which exceed any finite measurement. Mathematicians have given the notion “∞” for this term, a symbol that challenges our understanding of numbers and quantity. But, If a number can be larger than another, can an infinity be larger than another infinity or what is infinity plus 1?

In the early 20th century a German mathematician, Georg Ferdinand Ludwig Philipp Cantor, came up with ideas and theorems that revolutionized the understanding of infinity at that time and even today. He introduced the idea that there can be different sizes of infinities and his works introduced us to the “hierarchy of infinite cardinalities”.

Cantor used the concept of sets to prove his point about infinities. He is mostly famous for his results in comparing two sets of natural numbers and the real numbers between 0 and 1. One might think that they are both infinitely large and that they have the same size since infinity symbolizes endless quantities. However, Cantor proved that the set of real numbers has a comparatively larger infinity than the set of natural numbers. So, we know that If two sets have the same amount of pairs, they will be equal in size just like the numbers between 0-1 and 0-2. You may think that isn’t possible but if you try to match each

number between 0 and 1 to its double in 0 and 2, you will see that they match perfectly and since they have the same amount of pairs. So we accept that these two boundaries have infinitely many numbers and that they are infinite.

And even though there are infinite numbers between both 0 to 1 and the natural numbers, when we try to match the numbers, we simply fail to do so. If we try to match every natural number with the real numbers between 0 and 1 when we finally match the final natural number we are still able to produce more real numbers.

Cantor explained this event using his “diagonal argument”. The diagonal argument assumes that you have an infinitely long number of real numbers, and each one has infinitely many decimal places. Cantor suggests that you can create brand new numbers by taking the first decimal from the first element, the second decimal from the second element, and so on. You can continue this process infinitely many times since you have infinite numbers and infinite decimals so every time you do this process, you can come up with a new real number between 0 and 1. But this isn’t the case with natural numbers since you can’t produce new natural numbers out of nowhere so when you try to match the sets and you have matched the last element from the natural numbers, you can always produce more real numbers so that the sets will never match with each other. So this argument solidifies that the infinity of real numbers is larger than the infinity of the natural ones.

This theory and the new concept of infinities has also created a famous problem about infinity. Hilbert’s Hotel Paradox is a problem that has been introduced by German mathematician David Hillbert in 1924. In this problem you are assumed to be a hotel manager who is responsible for assigning new rooms to people who come into the hotel. The problem has 3 questions.

a) One night, when the infinite hotel which has infinitely many rooms with infinitely many guests is fully booked up. When another guest comes to the hotel, the night manager arranges a room for him. How does the night manager provide an available room for this new guest If all the rooms are full?

b) But what If, one train of countably infinitely many guests come to the hotel when all the rooms are booked? The manager still manages to provide room for the new guests. How does he manage to do so?

c) Finally, one night, infinitely many trains which have countably infinitely many guests inside pull up to the hotel. The manager once again manages to provide these guests with room. How?

If you manage to solve them, you can send your answers to us.

If you are also interested in the topic, I highly recommend you to watch the video “The Infinite Hotel Paradox” from the channel “TED-Ed” on Youtube. It is also the first link in the bibliography.

Bibliography

https://www.youtube.com/watch?v=Uj3_KqkI9Zo

https://www.cantorsparadise.com/why-some-infinities-are-larger-than-others-fc26863b872f

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

https://www.youtube.com/watch?v=A-QoutHCu4o

https://www.google.com/search?q=diagonal+argument&sca_esv=590386837&tbm=isch&source=lnms&sa=X&ved=2ahUKEwi64oSjj42DAxVAR_EDHWfcAAsQ_AUoAXoECAEQAw&biw=1440&bih=783&dpr=2#imgrc=4wYJO9Y1T8pywM