Uncountable infinity and Hilbert's paradox of the Grand Hotel

Sometime around the end of the XIX century, a german mathematician, Georg Cantor, created a new look on the infinity. Instead of taking into consideration individual numbers, Cantor thought of making certain number collections, which he called sets. Set's power is the quantity of numbers a given set consists of. And so {1, 2, 3} set's power is 3 and that one's {23, 53, 67, 100} equals 4. Things get more interesting if we ponder a set with an uncountable quantity of elements.

Cantor has introduced a new symbol for infinity – alef zero, and said that it describes the power of the natural numbers set. {1, 2, 3, 4, 5, …}. Any set with an infinite number of elements which we are able to count one by one and have counted each element eventually has power. {1, 4, 9, 16, 25, …}, {2, 3, 5, 7, 11, …}, {666, 1666, 2666, 3666, …} - all these sets' powers equal .

Alef zero

I'm going to present a bigger infinity than and I will use a story that David Hilbert used to show his students in the university. It's a story about a hotel with an infinite number of rooms. That hotel is sometimes called 'Hilbert's Grand Hotel'.

In Hilbert's hotel, there is an infinite number of rooms, which are numbered 1, 2, 3, 4,... One day, a traveller comes to the reception and finds out that the hotel is already full. He asks if it was possible to find one more room for him. Receptionist responds – of course! The staff shifts guests from rooms n to rooms n+1. The first room turns out to be unoccupied by anyone. Perfect!

The next day, we have a much more complicated task to deal with. A coach with an infinite number of passengers arrives to the hotel. Could we somehow manage to find rooms for all those people (in a full hotel, worth to add) ? Receptionist says 'easy peasy!' and orders the staff to move guests from rooms n to rooms 2n. By doing so, every room with an odd number is free for our new guests.

The third day, we are really in trouble. A greater number of coaches arrives to the hotel. An infinite number, to be precise. Each with an infinite number of passengers. Yet again, the smart receptionist comes up with a solution. First of all, he needs an infinite number of rooms. He uses the exact same trick and moves each guest from room n to room 2n. Every room with an odd number is free now.At this moment, he only needs to count the passengers to allocate everyone to their rooms. Each passenger receives an indication in the form of m/n, where m is the coach and n is the seat. Ultimately, we will have counted everyone!

To put all that in numbers, it turns out that :

● 1+ =

● + =

● * =

That's something we had expected from the infinity. Let's stop right here though. If we treated each person marked m/n as a fraction , the table from the previous slide, if extended the right way, would include every possible fraction. In other words, a set of all fractions and a set of all natural numbers share the same power :That's quite unintuitive because it would seem that there are more fractions than natural numbers (between two natural numbers there is an infinite quantity of fractions).Cantor showed us that our intuition,in this case, is wrong. NOTE : [please, see the third slide if something is unclear] :)

mn

Here comes the main attraction. A proof that there is a greater set's power than . Coming back to Hilbert's hotel, this time the hotel is completely empty when an infinite number of people arrives. There aren't any coaches outside but every person has a t-shirt which depicts a decimal expansion of a number between 0 and 1. There are no 2 exact same t-shirts, every decimal expansion is included and shown on those t-shirts.

On this occasion, the receptionist cannot find a room for everyone. There will always be a decimal expansion of a number between 0 and 1 which will not be included in the infinite list of people trying to get into the hotel. I'm about to show you an explanation why.

Imagine that the first man has a shirt with a number 0,83967295..., the second man has 0,02515629... The receptionist starts giving away the free rooms.

Room 1 0,83967295...

Room 2 0,02515629...

Room 3 0,23593593...

Room 4 0,63636124...

Room 5 0,25252525...

Room... 0,...

Our point is to find a decimal expansion that is not on the list.

We create a special number which on the first place after the comma has the same digit as the first digit after the comma from the room 1 number. On the second place after the comma it has the same digit as the second digit after the comma from the room 2 number. We continue in such a way.

0,83967295...

0,02515629...

0,23593593...

0,63636124...

0,25252525...

Our number is : 0,82532...

We've almost finished making our special number, now we only have to do the very last thing. To every digit after the comma of that number, we need to add for example 1.

Our number is 0,82532...

we add 1 to every digit, so : 8+1 = 9 , 2+1= 3, 5+1= 6, …

We get our final number : 0,93643...

This artificially made decimal expansion is the exception we were looking for. It can't be on the receptionist's list. It's not in room number 1 because its first digit differs from the first digit from room number 1. Neither is it in room number 2 because its second digit differs from the second digit from room number 2. As we continue, we find out that our final number can't be in any room n because its nth digit will always differ from nth digit of the decimal expansion from room n. Uhh... that was quite intense. But here we are !

For every possible list, we are always able to create a number with the diagonal method which will not be on the list. Even though Hilbert's hotel has an infinite number of rooms, it can't accommodate an infinite amount of people. There will always be some people waiting outside. The hotel just isn't big enough.

The discovery that there is an infinity bigger than the infinity of natural numbers was crucial for the modern mathematics. It can easily be explained: some infinities are countable and some are not. We call them uncountable infinities. Uncountable infinities are bigger than the countable ones and may have various sizes.

Uncountable infinity which is the easiest to acknowledge is named c – it is the number of people who came to Hilbert's hotel with all the decimal expansions between 0 and 1.

We encounter another strange result here. We know that between 0 and 1 there are c points but we also know that there are fractions on the entire number line. If we have proven that c is bigger than , then there has to be more points between 0 and 1 on the number line than points representing fractions on the entire number line. Cantor yet again brought us to the world which conflicts with our intuition. The infinite amount of fractions is really nothing compared to the quantity of irrational numbers, which flood the number line. The abyss between c and is so huge that if we were to choose a point (shooting in the dark) on the number line, we'd have a 0% chance to hit a fraction. There just aren't enough fractions if we look at the uncountable infinity of irrational numbers.

Though Cantor's ideas were difficult to comprehend at first, the history accepted his arguments ; and c are the basics of modern mathematics. David Hilbert once said, 'No one shall expel us from the Paradise that Cantor has created.'

Bibliography : „Przygody Alexa w Krainie Liczb” Alex Bellos

'Wikipedia, the free Encyclopedia'

'The Infinite Hotel Paradox - Jeff Dekofsky' on Youtube

A quick look on the infinity by Sebastian Pełka

Thank you for your attention.