My argument why Hilbert's Hotel is not a veridical Paradox

[dakiprae]

TL;DR

Hilbert's Hotel often shown as a veridical Paradox. I want to show my argument, why the argument 'every guest n moves into the next room n+1' is not provable true.

Hello there,

I had another similar post, where asking for proof for Hilbert’s Hotel.

After rethinking this topic, I want to show you a new example. It tries to show why that the sentence, every guest moves into the next room, hides the fact, that we don’t understand what will happen in this infinite thought experiment (mathematically and logically).

If you don't know Hilbert's Hotel you can read it on Wikipedia:

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Wikipedia article of Hilberts Hotel captor Analysis:

Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true.

My argument, why I think it is not provable true:
First I want to explain my own example (A), than we go back to Hilbert's Hotel (B):
Guest 1 moves out and knocks on Guest 2’s door. Guest 2 goes out. Guest 1 moves in the Room 2 and Guest 2 knocks on Guest 3’s door. Repeat this process every second. If you repeat this forever, there is always one guest n outside knocking on n+1's door. Here we have potential infinity, but never reach infinity. But after an infinite amount of time, every single guest moved. Finally, every guest n has moved in n + 1 Room. No more guest is outside anymore, because all moved. Something happens, which used to be impossible before.

Back to Hilbert's Hotel (B):
The mathematical or logical argument for Hilbert's Hotel Paradox is: Every guest can move to n + 1 room. So you can make room for any new guest (Peano axioms).

I would say, there is no logical or mathematical proof, that every single guest will move into the next room in this thought experiment. It's not clear what will happen in this infinity scenario. If you go with my argument, Hilbert’s Hotel is an unsolved topic and not a veridical paradox. Saying every guest moves into the next room in Hilbert's Hotel is like saying every guest moves into the next room in my example (A) above. I am not sure, if every guest moves into the next room, because I don't know how this infinite sets interact with each other. If we don't know how this two infinite sets, Guests and Rooms, interacts with each other, than it is an unsolved topic.

 

[PeroK]

If you go with my argument, Hilbert’s Hotel is an unsolved topic ...

That's why your argument has little or no value. Mathematics is not going to retreat 200 years just because one novice doesn't understand it!

 

[etotheipi]

Guest 1 moves out and knocks on Guest 2’s door. Guest 2 goes out. Guest 1 moves in the Room 2 and Guest 2 knocks on Guest 3’s door. Repeat this process every second.

I think you're going to have a problem with this part. Guest 2 has to get out of bed, pack away all of his luggage, ransack the free pens and sewing kits, put his shoes on, answer the door and proceed to go and hassle his hotel neighbour in the space of just 1 second.

 

[dakiprae]

I think you're going to have a problem with this part. Guest 2 has to get out of bed, pack away all of his luggage, ransack the free pens and sewing kits, put his shoes on, answer the door and proceed to go and hassle his hotel neighbour in the space of just 1 second.

Time is relative :D

 

[Mark44]

The mathematical or logical argument for Hilbert's Hotel Paradox is: Every guest can move to n + 1 room. So you can make room for any new guest (Peano axioms).

What you wrote isn't what you actually meant. What you wrote is that every guest moves to room n + 1.

I would say, there is no logical or mathematical proof, that every single guest will move into the next room in this thought experiment. It's not clear what will happen in this infinity scenario.

On the contrary, it's very clear what will happen. At a given time, each guest moves to the next higher room number. Infinity doesn't play a role here.

If we don't know how this two infinite sets, Guests and Rooms, interacts with each other, than it is an unsolved topic.

But we do know how these two infinite sets interact by virtue of a one-to-one mapping between guests and rooms. If Guest_i is in Room_j before the move, with $1 \le i, 1 \le j$, then after the move, Guest_i will be in Room_j+1, and Room₁ will be unoccupied.

 

[Mark44]

Guest 1 moves in the Room 2 and Guest 2 knocks on Guest 3’s door. Repeat this process every second. If you repeat this forever, there is always one guest n outside knocking on n+1's door.

This is why your argument fails. In the Hilbert Hotel scenario, all guests move at the same time, not one after another.

Keep in mind that the idea that all guests can move at the same moment is just as plausible as the idea that such a thing as a hotel with an infinite number of rooms can exist. This is, after all, purely a thought experiment.

 

[PeroK]

This is, after all, purely a thought experiment.

I'd say it's pure mathematics!

 

[dakiprae]

This is why your argument fails. In the Hilbert Hotel scenario, all guests move at the same time, not one after another.

I. In (B) all guests move at the same time.

II. In (A) all guests move after an infinite amount of time.

I is an argument like II, both bridge infinity. But in II. the problem gets visible directly, because one guest should stay out in every case.

 

[Mark44]

I. In (B) all guests move at the same time.

Yes

II. In (A) all guests move after an infinite amount of time.

No, that's not part of the scenario.

I is an argument like II, both bridge infinity.

No. Infinity itself never plays a role in any of the calculations.

But in II. the problem gets visible directly, because one guest should stay out in every case.

Thread closed.
There were 61 posts in the first thread you opened, and any questions you had were amply discussed in that thread.

Since you still don't seem to get the logic behind the Hilbert Hotel, and are still espousing the same ideas that were refuted in the earlier thread, I am closing this thread.