The infinite Hotel Paradox - a question.

[koti]

Although I have a question, I've decided to post this in the Brain Teasers & Puzzles section.

 

The infinite hotel paradox goes like this:

We have a hotel with an inifinite amount of rooms filled with an infinite amount of guests. A new guest comes in to check in therefore we move each guest to the next room and are able to free room nr 1 to be occupied by the new guest - nothing changes, we still have an infinite number of room and an infinite number of guests. Same thing if a guest checks out...we move all the guests this time to the previous room and we end up with the same result - infinite number of guests and rooms. Nice classic paradox.

My question is...it occurs to me that any operation on the guests moving from room to room will take an infinite amount of time therefore the result of any operation will never occur. Doesn't this render the paradox/thught experimnet invalid ?

Here's a video explaining the infinite hotel paradox for reference in case I wasn't clear above:

 

To be clear on my question...I have no issue with the math, I'm having difficulties in understanding the validity of such a thought experiment if it contains time infinities.

Any thoughts?

 

 

 

 

 

 

[Sriman Dutta]

The paradox is invalid.

[proximity1]

"We have a hotel with an inifinite amount of rooms filled with an infinite amount of guests."

 

I think the problem is in the first use of the term "filled"--note that, grammatically, "filled" refers to the hotel itself, rather than the collection rooms-- which seems does not and could not possibly ever apply at this hotel. _A_ room is "filled" but the hotel, being infinite, by that fact, is not "filled." So there must always be "unfilled" rooms with guests ready to occupy them, right?

 

I like the "problem" because it points up something incongruous about the concept of infinity--a sort of nice abstraction, but one which breaks down in practical examples like this one where one tries to "add" to it. There does not seem to be any meaningful sense in which "infinity" can be "added to" as it's not a finite and, by the way, not a "quantity." The number series, being thought infinite, is an abstraction. We "add to it" <i>only</i> by mental operations. No one actually adds another digit on to the infinite series as a fact.

 

Since the hotel is infinite in capacity, it is never <i>at</i> "capacity." There is always a vacant room, no matter how many people check in. How else could it be the "infinite hotel"?

[koti]

"We have a hotel with an inifinite amount of rooms filled with an infinite amount of guests."

 

I think the problem is in the first use of the term "filled"--note that, grammatically, "filled" refers to the hotel itself, rather than the collection rooms-- which seems does not and could not possibly ever apply at this hotel. _A_ room is "filled" but the hotel, being infinite, by that fact, is not "filled." So there must always be "unfilled" rooms with guests ready to occupy them, right?

 

I like the "problem" because it points up something incongruous about the concept of infinity--a sort of nice abstraction, but one which breaks down in practical examples like this one where one tries to "add" to it. There does not seem to be any meaningful sense in which "infinity" can be "added to" as it's not a finite and, by the way, not a "quantity." The number series, being thought infinite, is an abstraction. We "add to it" <i>only</i> by mental operations. No one actually adds another digit on to the infinite series as a fact.

 

Since the hotel is infinite in capacity, it is never <i>at</i> "capacity." There is always a vacant room, no matter how many people check in. How else could it be the "infinite hotel"?

 

This is another issue that you raised. I agree that it's difficult if not impossible to implement the concept of infinity to a hotel. There is always an infinite number of vacant rooms btw.

My issue is with time though. Come to think of it, I'm not sure now if it takes a finite or infinite amount of time to perform an operation of checking a guest in/out.

[John Cuthber]

"My question is...it occurs to me that any operation on the guests moving from room to room will take an infinite amount of time therefore the result of any operation will never occur. ".

Why would it take an infinite time?
Each person just walks to the next room along.

Why would that take more than a minute or so?

 

[Strange]

My question is...it occurs to me that any operation on the guests moving from room to room will take an infinite amount of time therefore the result of any operation will never occur. Doesn't this render the paradox/thught experimnet invalid ?

 

 

Well, all the guests could move at the same time so if it only takes 5 minutes for one guest to move from his room to the next, then it will take 5 minutes for all the guests to move. [Edit: Beaten to it by John!]

 

But more importantly, this is an analogy to illustrate some of the mathematical properties of infinities. Your objection is like saying "but there isn't enough wood to build infinite rooms".

"We have a hotel with an inifinite amount of rooms filled with an infinite amount of guests."

 

I think the problem is in the first use of the term "filled"--note that, grammatically, "filled" refers to the hotel itself, rather than the collection rooms-- which seems does not and could not possibly ever apply at this hotel. _A_ room is "filled" but the hotel, being infinite, by that fact, is not "filled." So there must always be "unfilled" rooms with guests ready to occupy them, right?

 

I like the "problem" because it points up something incongruous about the concept of infinity--a sort of nice abstraction, but one which breaks down in practical examples like this one where one tries to "add" to it. There does not seem to be any meaningful sense in which "infinity" can be "added to" as it's not a finite and, by the way, not a "quantity." The number series, being thought infinite, is an abstraction. We "add to it" <i>only</i> by mental operations. No one actually adds another digit on to the infinite series as a fact.

 

Since the hotel is infinite in capacity, it is never <i>at</i> "capacity." There is always a vacant room, no matter how many people check in. How else could it be the "infinite hotel"?

 

I don't think so. Consider the natural numbers (1, 2, 3 ...) to be the rooms. They are all filled (i.e. there is a value in each "position"). But if we add 1 to every number (at the same time ) then we end up with an empty slot followed by 2, 3, 4 ... We can then insert 1 into the empty slot and get back to the original sequence of "filled" slots.

 

Note, you can free an infinite number of rooms by doubling all the numbers (ask the guests to move to the room number that is double their initial room number): then all the odd numbered rooms are empty and you can accept an infinite number of busses with an infinite number of passengers in each.

Edited December 21, 2016 by Strange

[koti]

Yep, youre right John & Strange. My mistake. For some reason I figured that the check in process will take an infinite amount of time.

[imatfaal]

Yep, youre right John & Strange. My mistake. For some reason I figured that the check in process will take an infinite amount of time.

 

I have been to 5 room hotels which are slower than that!

 

Read up on Hilbert's original thought experiment - that and his list of questions are great reading even for a layman like myself.

 

Then for a sobering afterthought read up about David Hilbert himself and the tragedy of the Mathematics department at Gottingen - the loss of one of the greatest flourishings of talent we may witness for many years.

[Delta1212]

 

I have been to 5 room hotels which are slower than that!

 

Read up on Hilbert's original thought experiment - that and his list of questions are great reading even for a layman like myself.

 

Then for a sobering afterthought read up about David Hilbert himself and the tragedy of the Mathematics department at Gottingen - the loss of one of the greatest flourishings of talent we may witness for many years.

*Google "David Hilbert"*

 

"German mathematician"

 

"Died: 1943"

 

Welp, that answers all the questions I'm of a mood to have answered right now.