Hilbert’s Paradox of the Grand Hotel


Today I read about Hilbert’s paradox of the Grand Hotel (shame, only today) and I was struck by the illogical human thinking.
That’s right!
Paradoxes are nothing but illogical thinking.

Hilbert’s paradox goes like that:

Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.[Read the rest in Wikipedia]

When for some stupid reason we create infinity, we create it from objects with same properties. Right?
We can have an infinite number of apples, or an infinite number of shoes, or an infinite number of shoe pairs.
What do we have in Hilbert’s case?
We have an infinite number of hotel rooms, all of which are occupied by guests.
What we actually have is an infinite number of room/guest pairs.
No, the right way to say it is that we have an infinite number of occupied rooms.
(If the mathematicians knew the importance of word choice, they would know how difficult it would be to create a paradox.)

Hilbert's paradox
Hilbert’s Grand Hotel
Now we have to accommodate one guest in that hotel.
It appears that if we choose to think of the occupied room as of room + guest, we can move the guests with one room up and vacate room №1 for the new guest.
“Hold on! – one would say. – All the rooms are occupied! How do you move the guests with one room up.”
Well, apparently that choice have a good explanation; since the rooms are infinite, there is no last room which will block the guests’ movement with one room up. So it is possible according to that way of thinking, although it contradicts the idea that all rooms are occupied.
Mentioning “all”, don’t you think that this word is inappropriate when it comes to infinity, because “all” implies a number, but infinity is not a real number.

“Not a real number” is the first step for creating a paradox.
How many unicorns can we accommodate in fully occupied unreal number?
The answer could be any real and unreal number and it would be the right answer because you can adjust the unreal parameters in any convenient for you way.

But isn’t there any way to make sense of this and put it in a logical explanation?
Can we refute the illogical statement that infinite occupied rooms can accommodate one and even infinite number of guests?
Sure, there is nothing easier than refuting a paradox.

The logic for a limited set of numbers tells us that we cannot create a pair in a row of room/guest pairs by adding only a guest in that row.
Simply said, we cannot accommodate a guest when all the rooms are occupied.
As Hilbert tells us that’s not the case with the infinite number of occupied rooms.
Then let’s change Hilbert’s choice of words!
Here is how our change would look:

  1. empty room = bucket of water
  2. guest = 1 kg of cement
  3. occupied room = concrete

Let’s accommodate the cement in the bucket of water in the same way we accommodate the guests in the empty rooms.
Now we have an infinite number of concrete buckets.
What happened here?
We paired a with b and the result is c
To make it even easier, you can use an infinite number of 10L buckets filled with 10L water, and try to add 10L water in the infinite bucket-water pairs.

It doesn’t work that way, you say?
Oh, I see, you prefer unicorns.

So, where the paradox comes from?
It comes from the concrete, of course.
Hilbert wasn’t clever enough to see the real property of “occupied room”.
He saw it as 1+1, not as 1

Now you’d tell me that the logic says that you can do pairing and mapping and blah-blah, and all the blah seems to be logical, but if there is a paradox, one or all of your blahs are nonsense.
To avoid paradoxes, we should create common sense rules like:

  1. do not assign wrong properties
  2. do not misplace properties
  3. do not misuse properties
  4. do not try to think but think


Here is a TED-Ed video lesson about the Hilbert’s Grand Hotel

Подобни Писаници:

  • Няма намерени подобни писаници!
Споделете, ако ви е харесало

Оставете Мнение

2 Comment threads
0 Thread replies
Most reacted comment
Hottest comment thread
1 Comment authors
Denislav Sisoev Recent comment authors
  Запиши се за отговори  
най-нови най-стари най-гласувани
Уведоми ме за
Denislav Sisoev
Denislav Sisoev

The paradox breaks in its initial definition. It says that Infinite amount of rooms are totally Full. Infinite suggests a set of numbers without an end. Full suggests a set of numbers WITH an end. So those two words cannot actually exist together in one statement as they contradict each other and making a statement that includes both of them is like saying “The sky is on the ground in this sunlit night”. Either the number of rooms is not infinite and they do have an end, or they are not all full 🙂 I don’t understand how that “paradox”… Прочети нататък »

Denislav Sisoev
Denislav Sisoev

But then again looking at the definition of the word paradox i think that there is nothing wrong with this one as well 🙂

a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true.
“the uncertainty principle leads to all sorts of paradoxes, like the particles being in two places at once”

Is it seemingly absurd and contradictory? Yes it is. Then it must be a paradox 🙂