On 5/24/2023 8:41 AM, PengKuan Em wrote:
> 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.

When you say "after infinity", that implies that infinity is finite.
Whats wrong with this picture?

On 5/24/2023 12:19 PM, Chris M. Thomasson wrote:
> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>> 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.
>
> When you say "after infinity", that implies that infinity is finite.
> Whats wrong with this picture?
>

infinity can hold all, and more, and more, more, forever more... It is
never full.

On 5/24/2023 8:39 AM, PengKuan Em wrote:
> 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.

A finite set that has no end, is infinite. ;^)

On 5/24/2023 8:37 AM, PengKuan Em wrote:
> 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

There is no "after infinity". Infinity goes on forever. There is no last
term.

Do You know, that You have been nominated for the nobel price in math. ?

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

On Wednesday, May 24, 2023 at 12:27:07 PM UTC-7, Valeri Njuhan wrote:
> Do You know, that You have been nominated for the nobel price in math. ?
> On Sunday, May 21, 2023 at 3:33:57 PM UTC+2, 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

There's Wolf, or Abel, but no Nobel, in mathematics. Physics....

The Fields medal is awarded to outstanding mathematicians
who make and promise great contributions, for the under-40 set.

On Wednesday, May 24, 2023 at 10:10:15 AM UTC-7, Dan Christensen wrote:
> 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

Oh, what's class/set distinction?

Fairly enough there's none, ..., or one.

Up into schemas and class/set distinction is "ZF with classes" or
as about "von Neumann-Goedel-Bernays" set theory.

So, there are set theories where you have proper classes,
and set theories where you don't, that it would be a false antecedent
if there isn't, so, material implication would prove all things.

Wrongly....

Anyways if direct implication doesn't suffice, it's improper.

I know you think it's amusing that "material implication" via "simple derivation
rules" seems to establish an implication from a false antecedent, but it's an oxymoron,
a contradiction in terms, a pollution of the truth tables, that many reject..
(And have boilerplate surrounding logic that always rejects it.)

On Wednesday, May 24, 2023 at 8:33:04 AM UTC-7, PengKuan Em wrote:

> 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

Congratulations. You have now stumbled onto the differences between/among the order types
omega
omega*
omega* + omega.

What would be the result if the rooms were arranged with the order type omega + omega*, i.e. if the positive integers came first and then the negative integers?

Suppose the rooms were arranged in the order of the rational numbers?

On Wednesday, May 24, 2023 at 8:37:09 AM UTC-7, PengKuan Em wrote:
> I do not understand for what purpose Hilbert designed his paradox. Just to say that after infinity is a great void?

To demonstrate that the ordinals omega and 1+omega are equal:
1+omega = omega

On Wednesday, May 24, 2023 at 12:18:00 PM UTC-7, Chris M. Thomasson wrote:
> 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...

You are merely exploiting a difference in terminology: Does 'full' mean that each and every room is occupied or does it mean (as you suggest) that it cannot hold any more.

Those are two very different properties. Do not confound them.

On Wednesday, May 24, 2023 at 12:25:18 PM UTC-7, Chris M. Thomasson wrote:
> On 5/24/2023 8:39 AM, PengKuan Em wrote:
> > 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.
> A finite set that has no end, is infinite. ;^)

'A finite set that has no end' is a contradiction. One of the characteristics of infinite sets is that they can be ordered with no end.

On Wednesday, May 24, 2023 at 12:26:19 PM UTC-7, Chris M. Thomasson wrote:
> There is no "after infinity". Infinity goes on forever. There is no last term.

Theological dogma.

Have the courage to question.

There are infinitely many rational numbers between zero and one. Yet two comes after all of them.

On Wednesday, May 24, 2023 at 12:16:25 PM UTC-7, Chris M. Thomasson wrote:
> On 5/23/2023 10:13 PM, FromTheRafters wrote:
> > Chris M. Thomasson was thinking very hard :
> >> 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... ;^)

More important than merely 'how many coins it is holding' is 'what is their ordering?'

On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7, Chris M. Thomasson 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...

Congratulations. You have stumbled onto the difference between a 'potentially infinite' process and an 'actually infinite' set. Contrary to popular opinion around here, there is room for both tools.

FredJeffries pretended :
> On Wednesday, May 24, 2023 at 12:16:25 PM UTC-7, Chris M. Thomasson wrote:
>> On 5/23/2023 10:13 PM, FromTheRafters wrote:
>>> Chris M. Thomasson was thinking very hard :
>>>> 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... ;^)
>
> More important than merely 'how many coins it is holding' is 'what is their
> ordering?'

I presume some are on top.

Chris M. Thomasson brought next idea :
> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>> 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.
>
> When you say "after infinity", that implies that infinity is finite. Whats
> wrong with this picture?

Intuition fails. Learning about the surreals you find that a particular
class of numbers have early birthdays. In his construction after
infinitely many of these he goes on to the next infinitely many and
then the next.

On 5/24/2023 2:39 PM, FredJeffries wrote:
> On Wednesday, May 24, 2023 at 12:26:19 PM UTC-7, Chris M. Thomasson wrote:
>> There is no "after infinity". Infinity goes on forever. There is no last term.
>
> Theological dogma.
>
> Have the courage to question.
>
> There are infinitely many rational numbers between zero and one. Yet two comes after all of them.

Touche. I mean if we get "bogged down" with trying to iterate the
infinity between zero and one, we will never be able to actually reach two.

Is there an an infinity after the naturals since there is no largest
natural?

On 5/24/2023 2:52 PM, FromTheRafters wrote:
> Chris M. Thomasson brought next idea :
>> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>>> 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.
>>
>> When you say "after infinity", that implies that infinity is finite.
>> Whats wrong with this picture?
>
> Intuition fails. Learning about the surreals you find that a particular
> class of numbers have early birthdays. In his construction after
> infinitely many of these he goes on to the next infinitely many and then
> the next.

If one tries to iterate all of the rationals between 0 and 1, it will
never be able to actually reach one. If one tries to iterate all of the
rationals between 1 and 2, it will never be able to actually reach two.
On and on.

On 5/24/2023 2:34 PM, FredJeffries wrote:
> On Wednesday, May 24, 2023 at 12:18:00 PM UTC-7, Chris M. Thomasson wrote:
>> 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...
>
> You are merely exploiting a difference in terminology: Does 'full' mean that each and every room is occupied or does it mean (as you suggest) that it cannot hold any more.
>
> Those are two very different properties. Do not confound them.

Strange to me. Full as in each and every room is occupied makes me want
to say this hotel is full and cannot hold any more. However, if the
hotel is truly infinite, then it can hold infinitely more, forever. It
never runs out of space. Think of a fractal that creates infinitely many
rooms, they is always room for another person, it can never run out of
space.

On 5/24/2023 2:42 PM, FredJeffries wrote:
> On Wednesday, May 24, 2023 at 12:16:25 PM UTC-7, Chris M. Thomasson wrote:
>> On 5/23/2023 10:13 PM, FromTheRafters wrote:
>>> Chris M. Thomasson was thinking very hard :
>>>> 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... ;^)
>
> More important than merely 'how many coins it is holding' is 'what is their ordering?'

Say each coin is given an index number. Say this infinite sack is
empty... I drop a coin in. It is given an index number of:

sack_index = 0; // empty state

drop a coin in:
coin = sack_index;
sack_index = sack_index + 1;

So,

sack contains nothing

Drop some coins in:

sack contains coin[0]
sack contains coin[1]
sack contains coin[2]
sack contains coin[3]
....

This sack can hold an infinite amount of coins. It can never get full.

On Wednesday, May 24, 2023 at 5:01:35 PM UTC-4, Ross Finlayson wrote:
> On Wednesday, May 24, 2023 at 10:10:15 AM UTC-7, Dan Christensen wrote:

[snip]

> > > > 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.

> Oh, what's class/set distinction?
>

DC Proof has no built-in axioms for class theory. Just the ordinary set theory implicitly used most math textbooks.
> Fairly enough there's none, ..., or one.
>
> Up into schemas and class/set distinction is "ZF with classes" or
> as about "von Neumann-Goedel-Bernays" set theory.
>
> So, there are set theories where you have proper classes,

Not used in most math textbooks.

> and set theories where you don't, that it would be a false antecedent
> if there isn't, so, material implication would prove all things.
>

If you introduce additional axioms for, say, class theory, and thereby obtain an inconsistencies, you could blame those additional axioms.

> Wrongly....
>
> Anyways if direct implication doesn't suffice, it's improper.
>

Nothing wrong with proof by contradiction. Or vacuous truth.

> I know you think it's amusing that "material implication" via "simple derivation
> rules" seems to establish an implication from a false antecedent, but it's an oxymoron,

Wrong. Your system seems incapable of dealing with contradictions. Ordinary (classical) logic has no problems with them.

Again, that simplest example:

1 A & ~A
Premise

2 ~[A & ~A]
Conclusion, 1

> a contradiction in terms, a pollution of the truth tables, that many reject.

Very few mathematician reject material implication. You are very much outside the mainstream on this score, Ross. Deal with it.

Dan

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

On 5/24/2023 9:00 PM, Chris M. Thomasson wrote:
> On 5/24/2023 2:52 PM, FromTheRafters wrote:
>> Chris M. Thomasson brought next idea :
>>> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>>>> 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.
>>>
>>> When you say "after infinity", that implies that infinity is finite.
>>> Whats wrong with this picture?
>>
>> Intuition fails. Learning about the surreals you find that a
>> particular class of numbers have early birthdays. In his construction
>> after infinitely many of these he goes on to the next infinitely many
>> and then the next.
>
> If one tries to iterate all of the rationals between 0 and 1, it will
> never be able to actually reach one. If one tries to iterate all of the
> rationals between 1 and 2, it will never be able to actually reach two.
> On and on.

I can think of a so-called hyper process that creates an individual
process for each interval:

0 to 1
1 to 2
2 to 3
on and on...

On 5/21/2023 6:33 AM, 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

Here is a hotel that grows infinitely large, it always has room. Here is
breaking down the process into steps, for a little while:

https://youtu.be/AJlgxwdtHp4

A DLA experiment.

Chris M. Thomasson presented the following explanation :
> On 5/24/2023 2:34 PM, FredJeffries wrote:
>> On Wednesday, May 24, 2023 at 12:18:00 PM UTC-7, Chris M. Thomasson wrote:
>>> 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...
>>
>> You are merely exploiting a difference in terminology: Does 'full' mean
>> that each and every room is occupied or does it mean (as you suggest) that
>> it cannot hold any more.
>>
>> Those are two very different properties. Do not confound them.
>
> Strange to me. Full as in each and every room is occupied makes me want to
> say this hotel is full and cannot hold any more. However, if the hotel is
> truly infinite, then it can hold infinitely more, forever. It never runs out
> of space. Think of a fractal that creates infinitely many rooms, they is
> always room for another person, it can never run out of space.

It is not really about being full or not, it is about the size of the
set of rooms and the size of the set of occupants and a bijection
between the two even when one set is a proper subset of the other.

FredJeffries <fredjeffries@gmail.com> writes:

> You have stumbled onto the difference between a 'potentially infinite'
> process and an 'actually infinite' set. Contrary to popular opinion
> around here, there is room for both tools.

Who here does not accept the validity of potentially infinite processes?

There is one poster who talks about "potentially infinite sets" but they
have crazy theorems (well, conjectures actually since no proofs have
been offered) like WMath's sets E and P with E in P and P \ {E} = P.

Do you think there is also room for potentially infinite /sets/ here?
If so, can you give an example of a theorem about them (with proof) that
shows how they are different to the usual infinite sets?

--
Ben.