On Tuesday, May 23, 2023 at 3:16:57 AM UTC+2, FromTheRafters wrote:
> 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.

Exactly!

Le mardi 23 mai 2023 à 13:02:08 UTC+2, markus...@gmail.com a écrit :
> 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.

That is true

kp

On Tuesday, May 23, 2023 at 4:02:08 AM UTC-7, markus...@gmail.com wrote:
> 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.

The hotel either starts empty or full.

Filling it can involve a scheme, leaving rooms empty, for later.

Otherwise, there must be some effect that makes for Dirichlet or
pigeonhole principle that pushing tenant 1 into room 2 pushes tenant
2 to room 3 and so on.

Otherwise the mere existence of bijections among countable infinite
sets, does _not_ effect all the moves so related, unless you imagine
an infinitely wide hall and infinitely fast communications among
the pair-wise distinct tenants and pair-wise distinct rooms.

Clearly the metaphor of it fails, though it can be approached by
an arbitrarily large bounded number.

In physics there are things like Jordan measure for the line or path integral,
it's got infinitely many parts each infinitesimal to make a unity.

Anyways the notion that there are bijections between countably infinite sets
is due Galileo, i.e. it was known long, long before Hilbert.

Then, the asymptotic analysis does of course give bounds for relations of functions
that have order for ordering theory before number theory then after that counting theory.

Sometimes the asymptotic analysis is called complexity or density.

So, there are models of infinite and transfinite Dirichlet principle or
pigeonhole principle, that indeed have it's full.

On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
> On Tuesday, May 23, 2023 at 4:02:08 AM UTC-7, markus...@gmail.com wrote:
> > 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.
> The hotel either starts empty or full.
>
> Filling it can involve a scheme, leaving rooms empty, for later.
>
> Otherwise, there must be some effect that makes for Dirichlet or
> pigeonhole principle that pushing tenant 1 into room 2 pushes tenant
> 2 to room 3 and so on.
>
> Otherwise the mere existence of bijections among countable infinite
> sets, does _not_ effect all the moves so related, unless you imagine
> an infinitely wide hall and infinitely fast communications among
> the pair-wise distinct tenants and pair-wise distinct rooms.
>
> Clearly the metaphor of it fails, though it can be approached by
> an arbitrarily large bounded number.
>
> In physics there are things like Jordan measure for the line or path integral,
> it's got infinitely many parts each infinitesimal to make a unity.
>
> Anyways the notion that there are bijections between countably infinite sets
> is due Galileo, i.e. it was known long, long before Hilbert.
>
>
> Then, the asymptotic analysis does of course give bounds for relations of functions
> that have order for ordering theory before number theory then after that counting theory.
>
> Sometimes the asymptotic analysis is called complexity or density.
>
>
> So, there are models of infinite and transfinite Dirichlet principle or
> pigeonhole principle, that indeed have it's full.

You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped to 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i.e.to those rooms with numbers greater than 1.

f(room 1) = room 2
f(room 2) = room 3
f(room 3) = room 4
: :

Dan

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

On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
[snip

>
> Then, the asymptotic analysis does of course give bounds for relations of functions
> that have order for ordering theory before number theory then after that counting theory.
>
> Sometimes the asymptotic analysis is called complexity or density.
>
>
> So, there are models of infinite and transfinite Dirichlet principle or
> pigeonhole principle, that indeed have it's full.

Yikes! You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i.e. to those rooms with numbers greater than 1.

f(room 1) = room 2
f(room 2) = room 3
f(room 3) = room 4
: :

Dan

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

On Tuesday, May 23, 2023 at 12:04:12 PM UTC-7, Dan Christensen wrote:
> On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
> [snip
> >
> > Then, the asymptotic analysis does of course give bounds for relations of functions
> > that have order for ordering theory before number theory then after that counting theory.
> >
> > Sometimes the asymptotic analysis is called complexity or density.
> >
> >
> > So, there are models of infinite and transfinite Dirichlet principle or
> > pigeonhole principle, that indeed have it's full.
> Yikes! You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i.e. to those rooms with numbers greater than 1.
>
> f(room 1) = room 2
> f(room 2) = room 3
> f(room 3) = room 4
> :
> :
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

It's an arbitrary (capricious) metaphor that fails.

On Tuesday, May 23, 2023 at 5:15:19 PM UTC-4, Ross Finlayson wrote:
> On Tuesday, May 23, 2023 at 12:04:12 PM UTC-7, Dan Christensen wrote:
> > On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
> > [snip
> > >
> > > Then, the asymptotic analysis does of course give bounds for relations of functions
> > > that have order for ordering theory before number theory then after that counting theory.
> > >
> > > Sometimes the asymptotic analysis is called complexity or density.
> > >
> > >
> > > So, there are models of infinite and transfinite Dirichlet principle or
> > > pigeonhole principle, that indeed have it's full.
> > Yikes! You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i.e. to those rooms with numbers greater than 1.
> >
> > f(room 1) = room 2
> > f(room 2) = room 3
> > f(room 3) = room 4
> > :
> > :
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

> It's an arbitrary (capricious) metaphor that fails.
Not entirely. Better, of course, is my metaphor of a walk through a finite village. It is much more grounded in physical reality and more intuitive IMHO. I first develop the non-numeric notion of a finite set (from Dedekind).. Then an infinite set is just one that is not finite. See 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

On 5/22/2023 6:15 AM, PengKuan Em wrote:
> 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.

For some reason, the idea of an infinite pool being "full" makes my mind
hurt a little bit. Think of an infinite waterfall pouring into a vessel
that can hold an infinite amount of water. It is never full... Humm...

Chris M. Thomasson used his keyboard to write :
> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>> 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.
>
> For some reason, the idea of an infinite pool being "full" makes my mind hurt
> a little bit. Think of an infinite waterfall pouring into a vessel that can
> hold an infinite amount of water. It is never full... Humm...

The vessel would have to be so large that it takes forever to build it.

Stop wracking your brain trying to make mathematical objects act like
real objects.

On 5/23/2023 4:13 PM, FromTheRafters wrote:
> Chris M. Thomasson used his keyboard to write :
>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>>> 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.
>>
>> For some reason, the idea of an infinite pool being "full" makes my
>> mind hurt a little bit. Think of an infinite waterfall pouring into a
>> vessel that can hold an infinite amount of water. It is never full...
>> Humm...
>
> The vessel would have to be so large that it takes forever to build it.
>
> Stop wracking your brain trying to make mathematical objects act like
> real objects.

I have to go into fictional magic land to create an infinite pool:

https://oeis.org/A179759

The land of creativity. The magic sack that can hold an infinite number
of coins... ;^)

On 5/23/2023 5:59 PM, Chris M. Thomasson wrote:
> On 5/23/2023 4:13 PM, FromTheRafters wrote:
>> Chris M. Thomasson used his keyboard to write :
>>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>>>> 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.
>>>
>>> For some reason, the idea of an infinite pool being "full" makes my
>>> mind hurt a little bit. Think of an infinite waterfall pouring into a
>>> vessel that can hold an infinite amount of water. It is never full...
>>> Humm...
>>
>> The vessel would have to be so large that it takes forever to build it.
>>
>> Stop wracking your brain trying to make mathematical objects act like
>> real objects.
>
> I have to go into fictional magic land to create an infinite pool:
>
> https://oeis.org/A179759

OOPS! Wrong link. Sorry about that for that was meant for Dan Joyce.

The link I meant to post was:

https://youtu.be/vhI3bMOjHfI

Infinity is magical... ;^)

>
> The land of creativity. The magic sack that can hold an infinite number
> of coins... ;^)

On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7, Chris M. Thomasson wrote:
> On 5/22/2023 6:15 AM, PengKuan Em wrote:
> > 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.
>
> For some reason, the idea of an infinite pool being "full" makes my mind
> hurt a little bit. Think of an infinite waterfall pouring into a vessel
> that can hold an infinite amount of water. It is never full... Humm...

How about an infinitely-divisible gas tank Empty <-> Full.

Or, you know, a pool, with water in it, ..., like a kiddie pool, filled from a hose.

Once every half hour everybody gets out of the pool: to check for D.B.s.
(Or loose poops.)

"Related rates" problems of course solve most these sorts systems
in continuous quantities, usually in terms of the most prototypical
continuous and infinitely-divisible quantity: time.

When the odometer rolls over, it goes 999, then 000.

On Tuesday, May 23, 2023 at 2:32:26 PM UTC-7, Dan Christensen wrote:
> On Tuesday, May 23, 2023 at 5:15:19 PM UTC-4, Ross Finlayson wrote:
> > On Tuesday, May 23, 2023 at 12:04:12 PM UTC-7, Dan Christensen wrote:
> > > On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
> > > [snip
> > > >
> > > > Then, the asymptotic analysis does of course give bounds for relations of functions
> > > > that have order for ordering theory before number theory then after that counting theory.
> > > >
> > > > Sometimes the asymptotic analysis is called complexity or density.
> > > >
> > > >
> > > > So, there are models of infinite and transfinite Dirichlet principle or
> > > > pigeonhole principle, that indeed have it's full.
> > > Yikes! You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i.e.. to those rooms with numbers greater than 1.
> > >
> > > f(room 1) = room 2
> > > f(room 2) = room 3
> > > f(room 3) = room 4
> > > :
> > > :
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com
>
> > It's an arbitrary (capricious) metaphor that fails.
> Not entirely. Better, of course, is my metaphor of a walk through a finite village. It is much more grounded in physical reality and more intuitive IMHO. I first develop the non-numeric notion of a finite set (from Dedekind). Then an infinite set is just one that is not finite. See 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

Well-founded or non-well-founded?

Chris M. Thomasson was thinking very hard :
> On 5/23/2023 4:13 PM, FromTheRafters wrote:
>> Chris M. Thomasson used his keyboard to write :
>>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>>>> 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.
>>>
>>> For some reason, the idea of an infinite pool being "full" makes my mind
>>> hurt a little bit. Think of an infinite waterfall pouring into a vessel
>>> that can hold an infinite amount of water. It is never full... Humm...
>>
>> The vessel would have to be so large that it takes forever to build it.
>>
>> Stop wracking your brain trying to make mathematical objects act like real
>> objects.
>
> I have to go into fictional magic land to create an infinite pool:
>
> https://oeis.org/A179759
>
> The land of creativity. The magic sack that can hold an infinite number of
> coins... ;^)

But alas, it can only hold a countably infinite number of coins.

Ross Finlayson laid this down on his screen :
> On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7, Chris M. Thomasson wrote:
>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>>> 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.
>>
>> For some reason, the idea of an infinite pool being "full" makes my mind
>> hurt a little bit. Think of an infinite waterfall pouring into a vessel
>> that can hold an infinite amount of water. It is never full... Humm...
>
> How about an infinitely-divisible gas tank Empty <-> Full.
>
> Or, you know, a pool, with water in it, ..., like a kiddie pool, filled from
> a hose.
>
> Once every half hour everybody gets out of the pool: to check for D.B.s.
> (Or loose poops.)
>
> "Related rates" problems of course solve most these sorts systems
> in continuous quantities, usually in terms of the most prototypical
> continuous and infinitely-divisible quantity: time.
>
> When the odometer rolls over, it goes 999, then 000.

But when 999 rolls over as a group it becomes 666, no rotational
symmetry. Lets pretend that is group theory.

Le mardi 23 mai 2023 à 18:05:44 UTC+2, Ross Finlayson a écrit :
> On Tuesday, May 23, 2023 at 4:02:08 AM UTC-7, markus...@gmail.com wrote:

> > 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.
> The hotel either starts empty or full.
>
> Filling it can involve a scheme, leaving rooms empty, for later.
>
> Otherwise, there must be some effect that makes for Dirichlet or
> pigeonhole principle that pushing tenant 1 into room 2 pushes tenant
> 2 to room 3 and so on.
>
> Otherwise the mere existence of bijections among countable infinite
> sets, does _not_ effect all the moves so related, unless you imagine
> an infinitely wide hall and infinitely fast communications among
> the pair-wise distinct tenants and pair-wise distinct rooms.
>
> Clearly the metaphor of it fails, though it can be approached by
> an arbitrarily large bounded number.
>
> In physics there are things like Jordan measure for the line or path integral,
> it's got infinitely many parts each infinitesimal to make a unity.
>
> Anyways the notion that there are bijections between countably infinite sets
> is due Galileo, i.e. it was known long, long before Hilbert.
>
>
> Then, the asymptotic analysis does of course give bounds for relations of functions
> that have order for ordering theory before number theory then after that counting theory.
>
> Sometimes the asymptotic analysis is called complexity or density.
>
>
> So, there are models of infinite and transfinite Dirichlet principle or
> pigeonhole principle, that indeed have it's full.

Just as Hilbert suggests, I suppose that all room are occupied and no pigeonhole.

I have added this for moving infinitely many guests:
We can imagine that we are at the point 0 of the number line, we have infinitely many negative numbers on our left and infinitely many positive numbers on our right. Let each number be a room and the hotelkeeper on our infinite left call us all to move right by one room. In fact, the number 0 is infinity for the hotelkeeper and we see that on the right of his infinity (the number 0) the rooms are all occupied. So, nobody can move even he has infinitely many rooms. And then, the guest G is out of room.

The key of our analysis is: “is there a person out of room? ” The answer is definitely Yes for the two cases of room shifting and the case of the number line which is equivalent to a general call for the infinitely many guests to shift room. Because there is a person out of room, Hilbert's Grand Hotel cannot accommodate additional guest.

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

Le mardi 23 mai 2023 à 21:04:12 UTC+2, Dan Christensen a écrit :
> On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
> [snip
> >
> > Then, the asymptotic analysis does of course give bounds for relations of functions
> > that have order for ordering theory before number theory then after that counting theory.
> >
> > Sometimes the asymptotic analysis is called complexity or density.
> >
> >
> > So, there are models of infinite and transfinite Dirichlet principle or
> > pigeonhole principle, that indeed have it's full.
> Yikes! You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i.e. to those rooms with numbers greater than 1.
>
> f(room 1) = room 2
> f(room 2) = room 3
> f(room 3) = room 4
> :
> :
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

I do not understand for what purpose Hilbert designed his paradox. Just to say that after infinity is a great void?

Your woman barber solve the barber paradox.
KP

On Tuesday, May 23, 2023 at 10:18:31 PM UTC-7, FromTheRafters wrote:
> Ross Finlayson laid this down on his screen :
> > On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7, Chris M. Thomasson wrote:
> >> On 5/22/2023 6:15 AM, PengKuan Em wrote:
> >>> 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.
> >>
> >> For some reason, the idea of an infinite pool being "full" makes my mind
> >> hurt a little bit. Think of an infinite waterfall pouring into a vessel
> >> that can hold an infinite amount of water. It is never full... Humm...
> >
> > How about an infinitely-divisible gas tank Empty <-> Full.
> >
> > Or, you know, a pool, with water in it, ..., like a kiddie pool, filled from
> > a hose.
> >
> > Once every half hour everybody gets out of the pool: to check for D.B.s..
> > (Or loose poops.)
> >
> > "Related rates" problems of course solve most these sorts systems
> > in continuous quantities, usually in terms of the most prototypical
> > continuous and infinitely-divisible quantity: time.
> >
> > When the odometer rolls over, it goes 999, then 000.
> But when 999 rolls over as a group it becomes 666, no rotational
> symmetry. Lets pretend that is group theory.

I think you refer to the p-adic integers, another interesting sub-field
about the non-Archimedean or number fields approximated by bounded
fragments, after SImon Stevin or so.

Le mardi 23 mai 2023 à 23:32:26 UTC+2, Dan Christensen a écrit :
> On Tuesday, May 23, 2023 at 5:15:19 PM UTC-4, Ross Finlayson wrote:
> Not entirely. Better, of course, is my metaphor of a walk through a finite village. It is much more grounded in physical reality and more intuitive IMHO. I first develop the non-numeric notion of a finite set (from Dedekind). Then an infinite set is just one that is not finite. See 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
Yes, an infinite set is just a finite set that has no end.
KP

Le mercredi 24 mai 2023 à 00:17:02 UTC+2, Chris M. Thomasson a écrit :
> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>
> For some reason, the idea of an infinite pool being "full" makes my mind
> hurt a little bit. Think of an infinite waterfall pouring into a vessel
> that can hold an infinite amount of water. It is never full... Humm...

Yes, this is what I said above, Hilbert's paradox means that after infinity there is a huge void that can contain all things.

KP

Le mercredi 24 mai 2023 à 01:15:39 UTC+2, FromTheRafters a écrit :
> Chris M. Thomasson used his keyboard to write :
> The vessel would have to be so large that it takes forever to build it.
>
> Stop wracking your brain trying to make mathematical objects act like
> real objects.
Hilbert thinks that infinity is the vessel but he is wrong. Infinity can be full.

KP

On Tuesday, May 23, 2023 at 9:06:58 PM UTC-4, Ross Finlayson wrote:
> On Tuesday, May 23, 2023 at 2:32:26 PM UTC-7, Dan Christensen wrote:
> > On Tuesday, May 23, 2023 at 5:15:19 PM UTC-4, Ross Finlayson wrote:
> > > On Tuesday, May 23, 2023 at 12:04:12 PM UTC-7, Dan Christensen wrote:
> > > > On Tuesday, May 23, 2023 at 12:05:44 PM UTC-4, Ross Finlayson wrote:
> > > > [snip
> > > > >
> > > > > Then, the asymptotic analysis does of course give bounds for relations of functions
> > > > > that have order for ordering theory before number theory then after that counting theory.
> > > > >
> > > > > Sometimes the asymptotic analysis is called complexity or density..
> > > > >
> > > > >
> > > > > So, there are models of infinite and transfinite Dirichlet principle or
> > > > > pigeonhole principle, that indeed have it's full.
> > > > Yikes! You are overthinking this, as many have done. Hilbert's Hotel is just a humorous illustration of the fact that an infinite set X can, by Dedekind's definition, be mapped 1-to-1 to a proper subset of itself. Nothing more. In this scenario, X is an infinite set of fictitious "rooms" numbered 1, 2, 3, .... There is the obvious function f such that f(room n) = f(room n+1), thus mapping the set of all these rooms to a proper subset, i..e. to those rooms with numbers greater than 1.
> > > >
> > > > f(room 1) = room 2
> > > > f(room 2) = room 3
> > > > f(room 3) = room 4
> > > > :
> > > > :
> > > > Dan
> > > >
> > > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > > Visit my Math Blog at http://www.dcproof.wordpress.com
> >
> > > It's an arbitrary (capricious) metaphor that fails.
> > Not entirely. Better, of course, is my metaphor of a walk through a finite village. It is much more grounded in physical reality and more intuitive IMHO. I first develop the non-numeric notion of a finite set (from Dedekind). Then an infinite set is just one that is not finite. See https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/

> Well-founded or non-well-founded?

It works for any sets. I explicitly neither assume nor rule out the possibility of set self-membership in ordinary set theory as formalized in DC Proof. Users can, of course, introduce additional axioms if it is required. I routinely introduce Peano's Axioms in proofs, for example.

Dan

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

On Wednesday, 24 May 2023 at 17:45:26 UTC+2, PengKuan Em wrote:

> Hilbert thinks that infinity is the vessel but he is wrong. Infinity can be full.

That's simply upside down, *full* is in the premise, not in the conclusion,
hence correct arguments would e.g. be:

- "Full implies full"! IOW, if it's full, it's full! Or,
- or, specifically contra the "Hilbertian fallacies":
assume full, derive a contradiction, then you have
shown it was *not* and could *not* be "full"... you fool!

Then of course, I say more in my post: the positive part,
what we *can* say logically and not counter-factually.
(Link up-thread.)

Indeed, thanks for the mention, although I was more
mocking the lack of any reference to previous discussions
and how the whole thing has become just spammers, trolls,
and the genuinely insane, flooding this and every channel...

Julio

On 5/23/2023 10:13 PM, FromTheRafters wrote:
> Chris M. Thomasson was thinking very hard :
>> On 5/23/2023 4:13 PM, FromTheRafters wrote:
>>> Chris M. Thomasson used his keyboard to write :
>>>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>>>>> 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.
>>>>
>>>> For some reason, the idea of an infinite pool being "full" makes my
>>>> mind hurt a little bit. Think of an infinite waterfall pouring into
>>>> a vessel that can hold an infinite amount of water. It is never
>>>> full... Humm...
>>>
>>> The vessel would have to be so large that it takes forever to build it.
>>>
>>> Stop wracking your brain trying to make mathematical objects act like
>>> real objects.
>>
>> I have to go into fictional magic land to create an infinite pool:
>>
>> https://oeis.org/A179759
>>
>> The land of creativity. The magic sack that can hold an infinite
>> number of coins... ;^)
>
> But alas, it can only hold a countably infinite number of coins.

The sack knows how many coins it is holding... Yes, it can count... ;^)

On 5/24/2023 8:45 AM, PengKuan Em wrote:
> Le mercredi 24 mai 2023 à 01:15:39 UTC+2, FromTheRafters a écrit :
>> Chris M. Thomasson used his keyboard to write :
>> The vessel would have to be so large that it takes forever to build it.
>>
>> Stop wracking your brain trying to make mathematical objects act like
>> real objects.
> Hilbert thinks that infinity is the vessel but he is wrong. Infinity can be full.

Saying its full implies that it cannot hold any more. Therefore, we are
back at finite. Contradiction? An infinite pool can never be full...