Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.

Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.

Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.

Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.

In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.

For more detail of this study please read the complete paper here:
« Is Hilbert's Grand Hotel a paradox? »
https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox

On Sunday, May 21, 2023 at 6:33:57 AM UTC-7, PengKuan Em wrote:
> Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.
>
> Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.
>
> Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.
>
> Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.
>
> In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
>
>
> For more detail of this study please read the complete paper here:
> « Is Hilbert's Grand Hotel a paradox? »
> https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox

It's sort of like transfinite Dirichlet, and yes, moving everybody does involve
a sort of infinite quantification.

Being sort of like transfinite Dirichlet, there's Cohen who introduced forcing
to make it so that the Cantorian Continuum Hypothesis was independent of ZFC,
so, similarly, there are models of Hilbert's hotel that are full and models that
are not, then that whether guests can relocate to make space is an extra rule.

So, it's fair of you to describe one where it's not.

On Sunday, May 21, 2023 at 9:37:01 AM UTC-7, Ross Finlayson wrote:
> On Sunday, May 21, 2023 at 6:33:57 AM UTC-7, PengKuan Em wrote:
> > Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.
> >
> > Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.
> >
> > Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.
> >
> > Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.
> >
> > In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
> >
> >
> > For more detail of this study please read the complete paper here:
> > « Is Hilbert's Grand Hotel a paradox? »
> > https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox
> It's sort of like transfinite Dirichlet, and yes, moving everybody does involve
> a sort of infinite quantification.
>
> Being sort of like transfinite Dirichlet, there's Cohen who introduced forcing
> to make it so that the Cantorian Continuum Hypothesis was independent of ZFC,
> so, similarly, there are models of Hilbert's hotel that are full and models that
> are not, then that whether guests can relocate to make space is an extra rule.
>
> So, it's fair of you to describe one where it's not.

Infinite degrees of freedom is not a contradiction.
That is how Uranium vibrates.

On Sunday, May 21, 2023 at 9:33:57 AM UTC-4, PengKuan Em wrote:
> Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests.

It isn't a paradox. It is a "thought experiment" to illustrate the essential property of all infinite sets, namely that there exists a bijection from that set to a proper subset of it.

As an alternative to the mind-blowing scenario of Hilbert's Hotel, you might consider my walk through an ordinary, finite village. With this thought experiment, I develop the essential non-numeric property of a FINITE set. Then an infinite set is just one that is not finite.

https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

Le dimanche 21 mai 2023 à 18:37:01 UTC+2, Ross Finlayson a écrit :
> On Sunday, May 21, 2023 at 6:33:57 AM UTC-7, PengKuan Em wrote:
> > Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.
> >
> > Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.
> >
> > Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.
> >
> > Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.
> >
> > In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
> >
> >
> > For more detail of this study please read the complete paper here:
> > « Is Hilbert's Grand Hotel a paradox? »
> > https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox
> It's sort of like transfinite Dirichlet, and yes, moving everybody does involve
> a sort of infinite quantification.
>
> Being sort of like transfinite Dirichlet, there's Cohen who introduced forcing
> to make it so that the Cantorian Continuum Hypothesis was independent of ZFC,
> so, similarly, there are models of Hilbert's hotel that are full and models that
> are not, then that whether guests can relocate to make space is an extra rule.
>
> So, it's fair of you to describe one where it's not.

Thanks. I do not know that there are models that says the hotel is not full..

KP

Le dimanche 21 mai 2023 à 20:57:25 UTC+2, mitchr...@gmail.com a écrit :
> On Sunday, May 21, 2023 at 9:37:01 AM UTC-7, Ross Finlayson wrote:
> > On Sunday, May 21, 2023 at 6:33:57 AM UTC-7, PengKuan Em wrote:
> > > Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on.. The new guest is called guest G.
> > >
> > > Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.
> > >
> > > Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.
> > >
> > > Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.
> > >
> > > In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
> > >
> > >
> > > For more detail of this study please read the complete paper here:
> > > « Is Hilbert's Grand Hotel a paradox? »
> > > https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox
> > It's sort of like transfinite Dirichlet, and yes, moving everybody does involve
> > a sort of infinite quantification.
> >
> > Being sort of like transfinite Dirichlet, there's Cohen who introduced forcing
> > to make it so that the Cantorian Continuum Hypothesis was independent of ZFC,
> > so, similarly, there are models of Hilbert's hotel that are full and models that
> > are not, then that whether guests can relocate to make space is an extra rule.
> >
> > So, it's fair of you to describe one where it's not.
> Infinite degrees of freedom is not a contradiction.
> That is how Uranium vibrates.

?

Le lundi 22 mai 2023 à 02:29:20 UTC+2, Dan Christensen a écrit :
> On Sunday, May 21, 2023 at 9:33:57 AM UTC-4, PengKuan Em wrote:
> > Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests.
> It isn't a paradox. It is a "thought experiment" to illustrate the essential property of all infinite sets, namely that there exists a bijection from that set to a proper subset of it.
>
> As an alternative to the mind-blowing scenario of Hilbert's Hotel, you might consider my walk through an ordinary, finite village. With this thought experiment, I develop the essential non-numeric property of a FINITE set. Then an infinite set is just one that is not finite.
>
> https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Thanks. It is true that an infinite set is just one that is not finite.

KP

On Sunday, May 21, 2023 at 3:33:57 PM UTC+2, PengKuan Em wrote:

> Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on.

Nope.

Wikipedia: "We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room."

Moreover: There is no second "case".

On Sunday, May 21, 2023 at 6:33:57 AM UTC-7, PengKuan Em wrote:
> Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.
>
> Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.
>
> Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.
>
> Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.
>
> In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
>
>
> For more detail of this study please read the complete paper here:
> « Is Hilbert's Grand Hotel a paradox? »
> https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox

Congratulations!

You have stumbled onto the fact that ordinal addition is not necessarily commutative -- that
1 + omega =/= omega + 1

On Sunday, 21 May 2023 at 15:33:57 UTC+2, PengKuan Em wrote:
<snip>
> In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
>
> For more detail of this study please read the complete paper here:
> « Is Hilbert's Grand Hotel a paradox? »
> <https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox>

Yes, indeed, but you have copied it from here:
<https://seprogrammo.blogspot.com/2014/02/hilberts-impossible-hotel.html>

Julio

On Monday, May 22, 2023 at 5:02:37 PM UTC+2, Julio Di Egidio wrote:
> On Sunday, 21 May 2023 at 15:33:57 UTC+2, PengKuan Em wrote:
> >
> > For more detail of this study please read the complete paper here:
> > « Is Hilbert's Grand Hotel a paradox? »
> > <https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox>
> >
> Yes, indeed, but you have copied it from here:
> <https://seprogrammo.blogspot.com/2014/02/hilberts-impossible-hotel.html>

If you say so.

In any case, it's crank stuff too.

"[...] we can prove that, if the hotel is full, accommodating new guests is in fact impossible:

Indeed, to say that the hotel is fully occupied is to say that, for all n in N, room n is occupied. Thus, a fortiori, room 1 is occupied and, for all n in N, if room n is occupied, room n+1 is also occupied. Which in turn is equivalent to saying that there is no n in N such that room n+1 is available. Hence, no more guests can be accommodated. QED."

An incredibe dumb "argument". (Holy shit!)

Hint: What you are "proving" by induction, is what you already KNOW: "Indeed, to say that the hotel is fully occupied is to say that, for all n in N, room n is occupied."

Hint: If we have: "room 1 is occupied and, for all n in N, if room n is occupied, room n+1 is also occupied", then by induction we get: for all n in N, room n is occupied.

An incredible "result", we knew all along.

"Which in turn is equivalent to saying that there is no n in N such that room n+1 is available."

Errr... well, If you say so. In this case, we can just state: For all n in N, room n is occupied "means" that there is no n in IN such that room an is available.

These claims just clarify the relation between "is is occupied" and "is available".

So your "proof" reduces to: "The hotel is fully occupied. Hence, no more guests can be accommodated. QED."

WOW! What a powerful argument!

We could even reformulate it slightly: "If Hilbert's hotel is fully occupied. no more guests can be accommodated - because I say so."

Very convincing!

On Monday, 22 May 2023 at 19:07:30 UTC+2, Fritz Feldhase wrote:
> On Monday, May 22, 2023 at 5:02:37 PM UTC+2, Julio Di Egidio wrote:
> > On Sunday, 21 May 2023 at 15:33:57 UTC+2, PengKuan Em wrote:
> > >
> > > For more detail of this study please read the complete paper here:
> > > « Is Hilbert's Grand Hotel a paradox? »
> > > <https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox>
> > >
> > Yes, indeed, but you have copied it from here:
> > <https://seprogrammo.blogspot.com/2014/02/hilberts-impossible-hotel.html>
>
> If you say so. In any case, it's crank stuff too.
<snip>

Sure, keep sucking.

> In this case, we can just state: For all n in N, room n is occupied

You miss the point: that is (i.e. represents) the definition of "full".
From that we can derive, in particular, that there is NO room n such
that room n+1 is available, which is directly counter the way Hilbert's
explains it out. In a comment I also offer a characterization that even
the less logico-mathematically proficient should be able to read.

But, indeed, it is all very simple: yet apparently you don't get it, you and
your "arguments" always eventually just by authority.

HTH, someone, somewhere.

Julio

Julio Di Egidio used his keyboard to write :
> On Monday, 22 May 2023 at 19:07:30 UTC+2, Fritz Feldhase wrote:
>> On Monday, May 22, 2023 at 5:02:37 PM UTC+2, Julio Di Egidio wrote:
>>> On Sunday, 21 May 2023 at 15:33:57 UTC+2, PengKuan Em wrote:
>>>>
>>>> For more detail of this study please read the complete paper here:
>>>> « Is Hilbert's Grand Hotel a paradox? »
>>>> <https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox>
>>>>
>>> Yes, indeed, but you have copied it from here:
>>> <https://seprogrammo.blogspot.com/2014/02/hilberts-impossible-hotel.html>
>>
>> If you say so. In any case, it's crank stuff too.
> <snip>
>
> Sure, keep sucking.
>
>> In this case, we can just state: For all n in N, room n is occupied
>
> You miss the point: that is (i.e. represents) the definition of "full".

WM likes to use the word 'complete' for this. As I have pointed out to
him, complete has other meanings in math. If 'complete' means 'has a
last element' like a teacup which 'completes' the set, then it is a
finite set. Here 'full' has no meaning in infinite sets, at least not
that I am aware of. Hilbert's hotel cannot be 'filled'.

On Monday, 22 May 2023 at 20:16:29 UTC+2, FromTheRafters wrote:
> Julio Di Egidio used his keyboard to write :
> > On Monday, 22 May 2023 at 19:07:30 UTC+2, Fritz Feldhase wrote:

> > > In this case, we can just state: For all n in N, room n is occupied
> >
> > You miss the point: that is (i.e. represents) the definition of "full".
>
> WM likes to use the word 'complete' for this. [and other bullshit]

You snip all the substance and reinject the usual nonsense.

What's *your* real name?

*Spammer Alert*

Julio

On Monday, May 22, 2023 at 8:03:58 PM UTC+2, Julio Di Egidio wrote:
> On Monday, 22 May 2023 at 19:07:30 UTC+2, Fritz Feldhase wrote:

> > In this case, we can just state: For all n in N, room n is occupied
> >
> You miss the point: that is (i.e. represents) the definition of "full".
> From that we can derive, in particular, that there is NO room n such
> that room n+1 is available, which is directly counter the way Hilbert's
> explains it out. In a comment I also offer a characterization that even
> the less logico-mathematically proficient should be able to read.
>
> But, indeed, it is all very simple: yet apparently you don't get it, you and
> your "arguments" always eventually just by authority.

Well, what can I say?

"Cranks characteristically dismiss all evidence or arguments which contradict their own unconventional beliefs, making any rational debate a futile task and rendering them impervious to facts, evidence, and rational inference.."

Le lundi 22 mai 2023 à 15:31:23 UTC+2, Fritz Feldhase a écrit :
> On Sunday, May 21, 2023 at 3:33:57 PM UTC+2, PengKuan Em wrote:
>
> > Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on.
> Nope.
>
> Wikipedia: "We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room."
>

We cannot reach infinity and the infinitely far room, because we begin with 1,2,3... When we are in finite numbers, we always are in finite numbers,

There is an argument that the hotelkeeper can magically inform the infinitely many guests all at once for room shifting. This is contrary to the fundamental property of infinity which is: Infinity cannot be achieved. The word infinite is a Latin word in which “in-” means “contrary to”, “-finite” means end. So, infinite means “no end”. For mathematics reaching infinity means reaching “no end”, that is, never ending. So, the hotelkeeper cannot reach the last guest and make everybody to shift room. Because all guests know that he cannot shift all the rooms, they think: “He is a liar!” and nobody will leave his room. So, the guest G will not have a room.

> Moreover: There is no second "case".

I have noticed that second "case" confuses people, so I have changed it and show it below:

b) All the guests are identical
Let us consider the case where all the guests are identical and indistinguishable from the guest G such that when the guest G gets into a room and the previous occupant gets out, we still see guest G before the room. Just imagine that rooms are boxes and the guests are Ping-Pong balls. Then this guest G knocks the next room and will do this forever. This is illustrated by the letter G before the rows of rooms, see Figure 2. In consequence, the additional guest G is always out of room the same way as Hilbert's Grand Hotel does not accommodate him.

Le lundi 22 mai 2023 à 15:48:36 UTC+2, FredJeffries a écrit :
> On Sunday, May 21, 2023 at 6:33:57 AM UTC-7, PengKuan Em wrote:
> Congratulations!
>
> You have stumbled onto the fact that ordinal addition is not necessarily commutative -- that
> 1 + omega =/= omega + 1

Maybe.

Le lundi 22 mai 2023 à 17:02:37 UTC+2, Julio Di Egidio a écrit :
> On Sunday, 21 May 2023 at 15:33:57 UTC+2, PengKuan Em wrote:
> Yes, indeed, but you have copied it from here:
> <https://seprogrammo.blogspot.com/2014/02/hilberts-impossible-hotel.html>
>
> Julio

I did not know your work. I have added your contribution to my article.

« Is Hilbert's Grand Hotel a paradox? »
https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox

On 5/22/2023 2:36 PM, Fritz Feldhase wrote:

> "Cranks characteristically dismiss all evidence
> or arguments which contradict their own
> unconventional beliefs, making any rational debate
> a futile task and rendering them impervious to
> facts, evidence, and rational inference."

Making rational points and
_waiting for a rational response_
is futile with certain posters.
Agreed.

The interaction becomes much more pleasant if
one just drops that last part,
waiting for a rational response.
Expect nothing, and you'll never be disappointed.

My own tactic is to _not be controlled_ by
the nature of certain responses I get.
On occasion, you (FF) have questioned my sanity
for _not_ responding with anger to severe
provocation. And I can't really say you don't
have reason to wonder.

But, if I'm crazy,
I think it's one of the good kinds.

It happens that Julio Di Egidio formulated :
> On Monday, 22 May 2023 at 20:16:29 UTC+2, FromTheRafters wrote:
>> Julio Di Egidio used his keyboard to write :
>>> On Monday, 22 May 2023 at 19:07:30 UTC+2, Fritz Feldhase wrote:
>
>>>> In this case, we can just state: For all n in N, room n is occupied
>>>
>>> You miss the point: that is (i.e. represents) the definition of "full".
>>
>> WM likes to use the word 'complete' for this. [and other bullshit]
>
> You snip all the substance and reinject the usual nonsense.
>
> What's *your* real name?
>
> *Spammer Alert*
>
> Julio

I snipped your blather, and kept the essential.

Jim Burns laid this down on his screen :
> On 5/22/2023 2:36 PM, Fritz Feldhase wrote:
>
>> "Cranks characteristically dismiss all evidence
>> or arguments which contradict their own
>> unconventional beliefs, making any rational debate
>> a futile task and rendering them impervious to
>> facts, evidence, and rational inference."
>
> Making rational points and
> _waiting for a rational response_
> is futile with certain posters.
> Agreed.
>
> The interaction becomes much more pleasant if
> one just drops that last part,
> waiting for a rational response.
> Expect nothing, and you'll never be disappointed.
>
> My own tactic is to _not be controlled_ by
> the nature of certain responses I get.
> On occasion, you (FF) have questioned my sanity
> for _not_ responding with anger to severe
> provocation. And I can't really say you don't
> have reason to wonder.
>
> But, if I'm crazy,
> I think it's one of the good kinds.

I agree, and I don't think name calling, which Fritz advocated, is a
very good thing either.

On Monday, May 22, 2023 at 9:42:13 PM UTC+2, Jim Burns wrote:

> On occasion, you (FF) have questioned my sanity
> for _not_ responding with anger to severe
> provocation.

Not really, I just tried to provocate you.

> And I can't really say you don't
> have reason to wonder.

Well, it's your decission.

> But, if I'm crazy,
> I think it's one of the good kinds.

Agree. ( It could be worse. :-P )

On Monday, May 22, 2023 at 10:04:02 PM UTC+2, FromTheRafters wrote:
>
> I don't think name calling, which Fritz advocated, is a very good thing either.

You might be right.

But, you see, it's just a certain communication strategy (sort of).

Fritz Feldhase explained on 5/22/2023 :
> On Monday, May 22, 2023 at 10:04:02 PM UTC+2, FromTheRafters wrote:
>>
>> I don't think name calling, which Fritz advocated, is a very good thing
>> either.
>
> You might be right.
>
> But, you see, it's just a certain communication strategy (sort of).

Yeah, like fighting fire with a flamethrower.

söndag 21 maj 2023 kl. 15:33:57 UTC+2 skrev PengKuan Em:
> Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert's Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.
>
> Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.
>
> Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.
>
> Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.
>
> In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert's Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert's Grand Hotel cannot accommodate additional guest in its infinitely many rooms.
>
>
> For more detail of this study please read the complete paper here:
> « Is Hilbert's Grand Hotel a paradox? »
> https://www.academia.edu/102116805/Is_Hilberts_Grand_Hotel_a_paradox
It's not a paradox. You just need to move everyone at the same time. Whether a such thing can exist in reality is an entirely different question, since it's mainly a thought experiment to prove that you can add an element to N and still have the same cardinality of the set.