Let's take another look at the concept of Infinity, shall we...

Ok, where do we start... 

How about with a small intro...

Now one school of thought in mathematics is of the view / advocates that there are multiple theoretical / conceptual infinities, and thus that some theoretical / conceptual infinities are greater / larger than some other theoretical / conceptual infinities... ( aka ∞ * ∞ > ∞)

This article will discuss a few points associated with this debate.

Point no 1:

Realise that what we are talking here / addressing here, is the theoretical / conceptual infinity (or the mathematical concept of infinity), and not regarding infinity we come across in applied physics or engineering. 

Thus for example, for all practical purposes in optics, we can consider the sun to be a light source at infinity (∞). The reason being : the angle between two light rays emanating from our sun, when measured on earth is negligibly small (literarily = 0). Aka the light rays are parralel, to each other for all practical purposes.

BUT 

Mathematically, in the pure theoretical / conceptual / mathematical sense, the distance to our sun is NOT infinite (∞) and the angle between two light rays is NOT zero (0).

Thus we have to make this Point no 1, clear before proceeding... Thus what we are addressing here, is the pure concept / notion of mathematical infinity (and not necessarily the applied notions of infinity). 

Point no 1, thus clearly defines the scope / the turf of what we are talking here (aka the scope of the conceptual infinity).

Point no 2:

Let's take a look at the very interesting deductions / logical arguments we come across / we are presented with, in the case study of the hypothetical Hilbert's Hotel.

Now this Hilbert's Hotel has infinite (∞) rooms. 

(Note: Immediately those in engineering or project management or quantity surveying may raise their objections, pointing out that this Hotel in Infeasible due to financial, material, temporal and spacial constraints... 

Answer: Kindly refer to Point no 1... and just "enjoy the movie" and don't spoil the fun by veering out of the context...! Thank you... 😉)

Now where were we...? Ah yes, the Hilbert's Hotel...

Case no 1:

Now you can refer to the online contents describing the mathematical notions regarding how for example on how to add a new guest to this Hotel, even if all the rooms are currently filled. 

Thus all one has to do is : ask each guest to move to the adjacent room, thus making room no 1 vacant, to which we can assign the new guest, and as the hotel has conceptually infinite (theoretically no end), number of rooms, there will never arise the theoretical scenario where the current guest at room no [...], will not have a room to move to. Realise  that ∞ + 1 = ∞ (or ∞ + n = ∞). Thus there will always be ∞ more rooms  after every room number [...]. 

Another way to visualize this, is via an open ended linear tube / tunnel with no defineable end (just goes on and on and on... upto infinity... This is the mathematical concept of infinity).

Case no 2:

Now what will happen if "out of nowhere" a Bus arrives with infinite number of guests. (Of course, why not... if we can have a Hotel with infinite rooms, why not a Bus with infinite guests... it whould probably be pretty jammed but... that's none of our business. Or maybe the bus itself is infinitely long...)

No problemo, says the Hotel Manager / front desk ... (they always knew one day this is going to happen, so they were ready...)

Just ask each guest to move to the room number, which is double / twice their room number. Thus room 1 guest moves to room 2, room 2 to room 4, room 3 to room 6 etc. 

Thus now each odd numbered room is vacant. 

And how many odd numbered rooms are there in our Hotel...?

Thus again problem solved...

Case no 3:

Next comes infinite busses, with each having infinite guests...

Again no problem, says our Hotel Manager (whom has a Degree in Mathematics and an MSc in Project Management & Conflict Resolution). So he/she opens the Hotel Room Management / Assignmnent Spreadsheet, and via a simple and yet ellegant serpentine path / diagonal zigzag, manages to assign a room to each of the infinite guests of the infinite Busses. Aka it is possible to draw a serpentine path /  a diagonal zigzag, through the above spread sheet (going only once through each cell), and then via elongating that thread, we can obtain a one to one association with each guest in those infinite Busses, to a room in our Hotel. (For further details, kindly refer one of the many online videos / articles well illustrating this elegant technique).

So far so good... We have no problems to report, no contradictions. Both sides of this debate are in agreement upto this point... But things are going to get heated up / flare up pretty soon / pretty quickly... So here goes...

Case no 4:

Now, next comes a Bus with a problem...

Now... to describe the problem with this Bus, one technique used is the usage of alphabetical letters A & B (aka. via utilising Cantor's Diagonal Rule). Now this is where we would like to present, the first err/ falascy.

Now this Bus has passengers / guests, whom each have a name of infinite length. And their names are made of letters A & B. 

Thus the passenger / guest name list whould be / could be, as follows...

Name List I:

1:

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

...

2:

B

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

...

3:

A

B

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

...

4:

B

B

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

...

5:

A

A

B

A

A

A

A

A

A

A

A

A

A

A

A

A

A

...

 :

upto ∞

Now the first possible logical- mathematical err we would like to point, is around the technique used by Cantor's Diagonal Rule, to define / describe that by inverting each letter in the diagonal, we get a name that whould not be in the original name list.

Thus for example, if the name list of the guests whom arrive at the reception is in the following order...

Name List II:

1:

A

A

A

A

A

B

B

B

A

A

A

A

A

A

B

A

A

...

2:

A

B

A

A

B

A

A

A

A

B

A

A

A

A

A

B

A

...

3:

A

A

A

A

B

A

A

A

A

A

A

B

B

A

B

A

A

...

4:

A

A

B

A

B

B

A

A

B

A

A

A

B

A

A

A

A

...

5:

A

A

B

A

B

A

B

A

A

A

A

A

A

A

B

A

A

...

 :

upto ∞

(Observe that in above list, we have highlighted the diagonal letters in each name...)

Now the argument some present, is that : by flipping the A or B one finds along the diagonal, we could generate a new name that should not appear anywhere in the name list.

But the issue with this logic is that, we have a name list that is infinitely long. Thus this infinitely long list, should have all the permutations / combinations permissible with A & B. 

Another more logical / simple way to look at this, whould be to visualize A & B as binary numbers. Thus if we rearrange Name List I: with 0 & 1 (instead of the A & B), we observe that what we get is the binary number pattern (written in reverse). Thus Name List I (which whould be the names of our infinite guests in Alphabetical Order), should have the equivalence of all binary number combinations possible from zero to infinity.

1:		0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	...
2:		1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	...
3:		0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	...
4:		1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	...
5:		0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	...

:

upto ∞

The Panel of Judges: Your time is up, present your closing remarks.

Mr. Chairman... dear colleauges... As our closing remarks we would like to table two Objections.

Objection no 1: Our Team Proponent whould thus enumerate, that by flipping the character along the diagonal and generating a name with those flipped characters, WHOULD NOT generate a new / unique name, that does not already appear in the infinite list of guests.

Objection no 2: The attempt to represent the infinite set of Real (ℝ) numbers, via the usage of alphabetical characters A & B is thus proven to be error prone / does not justify / highlight, the CORE / CRUX of the point that needs to be highlighted in this part of the infinite numbers argument. Aka. that there are infinite number of Real (ℝ) numbers, between EACH Natural (ℕ) number. (That between 1 and 2 OR between 2 and 3 etc. etc. there whould be an infinite number of Real (ℝ) numbers.)

Thus to conclude, we would like to point out that the Cantor's Diagonal Rule is logically/ mathematically flawed/ incorrect.

Thank you...

Half of the audience  : YEAAAAH!!!!!!

The other half.            : BHOOOOO!!!!!

Mr. Chairman...           : Order ! Order!

Time slot next passes to Team Opponent. They present their Blah Blah Blah Blah Blah... And then some more... in response to our Blah Blah Blah...

Next, it's again our turn...

Point no 3:

Now this whould be a good location to highlight another point regarding Hilbert Hotel. Realise / observe, that Hilbert Hotel is a Hypothetical Construct, used to highlight certain concepts / properties of conceptual Infinity. Thus for example in our Hypothetical Hilbert Hotel, entities / information can hypothetically travel faster than light. Thus the moment one is assigned a room, that guest appears in that room. Thus for example the question: whoudn't it take infinite amount of time, for the infinite guest to reach room number infinity, is not a valid question. Such is not in the scope / not in the context of what mathematicians are trying to address via / in Hilbert Hotel.

Let's dwell back again, at the case scenario where infinite number of Busses each with infinite guests arrived at the Hilbert Hotel (Point no 2: Case no 3).

∞ Room Assignment Technique 2:

Observe how in the scenario of Point no 2: Case no 3, mathematicians have resolved this problem, via utilising the "serpentine thread / diagonal zigzag technique". 

Now alternatively, we could also come up with a solution, where we assign rooms for guest in the order of the arrival of the Buses. Thus we utilise technique discussed under Point no 2: Case no 2, where each guest is asked to move to Room number * 2, each time a Bus arrives. And as at Hilbert Hotel, everything happens instantaneously (even the arrival of the Busses), all the infinite guests of all the infinite Busses gets assigned rooms (even before the Hotel Manager has fully pressed the Enter Key). 

But now setting aside the pun, let's go back at the case of the Problem Bus (remember we ran out if time, in our previous round...)

If one recalls we already demonstrated that Cantor's Diagonal Rule has a falascy/ err.

Now to address the actual issue of this Problem Bus a better notion whould probably be, to utilise Real (ℝ) numbers themselves. 

Thus now, this Problem Bus would be presented having infinite number of seats, with each seat having a unique Natural (ℕ) number. But this Bus, whould be of an infinite height. Aka for example in between seat number 1 and 2, there are infinite other guests, into the vertical / z direction. 

Now if one were to ponder into this / look deeper, one may think that this situation is totally equivalent to the case where we had infinite Busses, with infinite guests. Thus we could ask the infinite guests At & Above seat 1 to proceed, followed by guest At & Above seat 2... Thus yet again, even  before the Hotel Manager has fully pressed the Enter Key... problem whould have got resolved... Right...?

Well not so fast, not so fast ... Let's ponder  / investigate a bit deeper / a bit further into the seating arrangement of this Bus.

This Bus, though has Natural (ℕ) numbers as seat numbers... In the vertical direction (z axis) the guests have IDs assigned with / as / in the form of Real (ℝ) numbers. And now, how many Real (ℝ) numbers should there be, between any two adjacent Natural (ℕ) numbers (say between the numbers 1 and 2 for example)?

Answer: Take any two Natural (ℕ) numbers, and you can find infinite number of Real (ℝ) numbers, between those two Natural (ℕ) numbers. 

Now utilising the [∞ Room Assignment Technique 2] discussed above, we can make a 1 to 1 assignment between the infinity between two adjacent Natural (ℕ) numbers, to the infinity of rooms in Hilbert's Hotel / Hilbert Construct. 

(Why the above has been marked in red will be discussed later... when Georg Cantor -from Team Opponent- will take the stage...)

Now repeat the above procedure ∞ number of times, and all guests in our Problem Bus, whould be assigned rooms inside Hilbert's Construct. 

And thus from dath day on, dath Problem Bus was Problemeth No More... (Oh! you think so... Wait till Georg Cantor takes the stage... 😋 🙃 🤣)

And thus we have provetheth :

∞ * ∞  = ∞

Or have we? 🤔

And now, send infinite number of "Problemeth No More Buses", and Hilbert will warmly welcome and accommodate them all... 

For (∞ * ∞)  * ∞ will still / alway be nothing more than = ∞.

And with that, we rest our case Mr. Chairman.

[Epilogue to Team Proponent's Round 2... 

And they lived happily ever after in Hilbert's Hotel / Construct... 

No... That did not happen... 

Though Hilbert Construct did accommodate them all, the bickering between guests was beyond even the scope of Hilbert...]

Time slot next passes again to Team Opponent. And who could be their next speaker...? Who is taking the stage... It's Georg Cantor...!

Oh boy...!

Georg Cantor : Ok, let me try to explain this point, utilising a slightly different approach...

Now, let's consider the Set of Real (ℝ) numbers one whould find, in between any two adjacent Natural (ℕ) numbers. Now if one attempts to make a one to one association, between the Real (ℝ) numbers and the Natural (ℕ) numbers, first of all we will have to identify / locate the first Real (ℝ) number that follows the said Natural (ℕ) number. For example, what is the first Real (ℝ) number that follows the Natural (ℕ) number 1 ?

Is it

1.0000000001    or is it

1.000000000000000001      or is it

1.000000000000000000000000000001

Thus observe that there whould always be a smaller Real (ℝ) number, than any given Real (ℝ) number.

Thus observe that to get the first Real (ℝ) number, that follows a given Natural (ℕ) number, we whould literally have to go all the way to infinity (of zeros).

Now this is why in the Hilbert's Hotel, this particular Bus is deemed as a Problematic Bus. Aka how do we find the guest / how do we define the guest, that follows / is next inline to the guest after seat number ℕ. To find that guest, we whould have to go all the way to infinity (which in-turn is an entity, beyond the comprehensible /  reachable).

Observe that in each of the previous Buses, we could start / begin, assigning Guests to Rooms of the Hilbert Hotel, as the first guests were at finite entities. Thus even in the case where we used a spreadsheet and a serpentine trace route / diagonal zigzag, observe that it was possible to obtain / get a starting point (to get the algorithm going).

But in the case of the Problem Bus, we cannot even obtain a starting point, to start the room allocation process, because the first Real (ℝ) guest him/her/itself, is at infinity. Aka we can't even find  / locate / reach the first Real (ℝ) guest (let alone the others).

Thus without even being able to find / locate / reach the first Real (ℝ) guest, how does one intend to make any association with Natural (ℕ) Rooms?

Observe how in the cases of the non Problamatic Buses  (aka those having infinite number of Natural (ℕ) guests) , we could Defer dealing with the guests at infinity upto / until the end of / tail of the algorithm. But not so with the Problem Bus, having Real (ℝ) guests. Here the first guests themselves whould be at infinity (one needs to reach upto infinity to locate even the first guests). Thus our logic / our algorithms fails / get stuck, at the start of the process itself.

For example, if one were to write a computer program / an algorithm to locate the first guest, that program will enter into an infinite loop, trying to locate the first Real (ℝ) guest (aka requiring infinite number of zeros...)

Observe that none of the previous Buses had any such issues / complications.

Thus this Real (ℝ) number Bus, is again confirmed as a Problematic Bus.

And with that short burst Georg Cantor, closes his case.

Is it our turn again...!?

Oh Boy...! Shall we ask for a recess?

You take the stage... No you take the stage... Why should I? I did it in the last round, you go... I can't, I've got laryngitis. If youv'e got laryngitis, then Iv'e got...

Ok ok shut up you guys... I think I got an idea...

Let me directly address the following to Mr. Georg Cantor of Team Opponent.

Now Mr. Cantor, as you pointed out, we definitely seem to have an issue of making a one to one association, between Real (ℝ) numbers and Natural (ℕ) numbers.

As Mr. Cantor highlighted, the issue / inability of / to locating even the first Real (ℝ) number, whould definitely defeat any attempts of / at making any such associations...

Thus in this regard, let me table a possible / probable solution (and open it for peer review).

Now as it was pointed out, as the issue of locating the first (and all subsequent) Real (ℝ) numbers, gets hampered by the Resursive Nature of reaching a lesser Real (ℝ) number. 

The following section / notation has been deprecated...

Kindly Refer Following article...

https://projectsofacademicinterests.blogspot.com/2025/03/a-new-perspective-on-infinity-via-2d.html

THUS: What this solution proposes, is to box / confine / bind the Recursive Component into the inside of a Black Box. Thus now we have something as the following.

 ℝ : (±)? (ℕ)* ([∞↻])?(.)? (ℕ)* ([∞↻]  (ℕ)* )?

What the above notation describes is a Real (ℝ) number: with optional (±) followed by zero or more Natural (ℕ) numbers : (ℕ)*, followed by an optional decimal point : (.)? , followed by zero or more Natural (ℕ) numbers : (ℕ)*, 

Next we have the boxing / confining of the Recursively to Infinity [∞↻] , followed by zero or more Natural (ℕ) numbers : (ℕ)*

Thus in above notation, all / everything to the left of [∞↻], aka (±)? (ℕ)* (.)? (ℕ)*  is in the / belongs to the Finite Domain / Realm, where as all / everything to the right of [∞↻], aka  (ℕ)* , belongs to the Infinite Domain / Realm.

Thus observe that in above, the Infinite Domain / Realm has been marked as optional aka ([∞↻] (ℕ)* )? Meaning that one has the option (optional) to invoke, the Recursively to Infinity, if and only if one so desires / only if needed.

Now in most all physical / practical / engineering world applications, the invocation of the ([∞↻] (ℕ)*), factor whould not be required.

But, in cases of for example theoretical mathematics, one may find the above handy / applicable (for example in deriving / analysing Set Associativity between Real (ℝ) numbers and Natural (ℕ) numbers).

Thus, to solidify the point further, one need to realise that everything to the right of [∞↻], aka (N)* belongs to the Infinite Domain / Realm. Thus such is OUTSIDE of the Finite Realm and thus cannot be reached physically. 

But though (N)* is physically  unreachable, there is nothing preventing one from conceptualising such. For the activity of conceptualising is not bounded by any confines / constrains (spacial nor temporal).

The following are thus a few examples of Real (ℝ) numbers... 

ℝ : (±)? (ℕ)* (.)? (ℕ)* ([∞↻] (ℕ)* )?

       +      10    . 00000  [∞↻]   1

       +      10    . 00000  [∞↻]   2

       +      10    . 00000  [∞↻]   55555

       +      10    . 00000  [∞↻]   7001002

Thus now, with above notation, the resursively reaching the smallest element, is confined into the Black Box of [∞↻]. Thus we have delagated / confined / boxed / abstracted,  this resursive / infinite loop into a Black Box. 

Now the definition of Black Box, in the confines of Science is : an entity who's internal functioning is either not known to the outside world or we do not care to know its internal functioning. (One can compare this to the Concept of White Box which is...).

For example for most all end users of say an engined vehicle or most electrical / electronic appliances, whould care the least to know how such actually functions on the inside / what goes on the inside (as long as it does what is supposed to do). Thus such could be considered / viewed as Black Boxes from the perspective of the end users.

Likeso our [∞↻], as well is a Black Box, whose sole purpose is to keep churning onto eternity (so that you and me don't have to), until it reaches absolute infinity.

And once we have delegated that  resursive task (of reaching infinity) onto an automata, we can at ease conceptualise the last digit(s) of the Real (ℝ) number(s), we are interested in / working on. For... conceptually the last digits should start from 0 and increase in increments of 1...

10. 0000 [∞↻] 0

10. 0000 [∞↻] 1

10. 0000 [∞↻] 2

...

10. 0000 [∞↻] 55555

etc.

Now via the utilisation of the above Black Box, whould it be possible for us, to try to come up with a one to one association between Real (ℝ) numbers  & Natural (ℕ) numbers...?

Let's go back into that Problem Bus (which is still parked outside Hitbert's Hotel, because of the conundrum of not being able to find the first guest  / give a unique ID to the first guest et al.)

Thus what we shall do, is provide each passenger of that Bus, with a Black Box of [∞↻], and a Serial Number / ID, we Conceptually define (ex.1, 2, .... , 55555, .... , 1000001, .... ∞).

Now with such, whould it be possible, to assign all / each of the infinite Real (ℝ) number guest / passenger (between each Natural (ℕ) number seat), a unique ID from 1 to ∞ ?

Thus for example, whould it be possible to arrive at a 1 to 1 association between the Set of Natural (ℕ) numbers and the Set of Real (ℝ) numbers (say between 10 - 11 for example...)

(ℕ) :         1                2                 ...   ➛ ∞

(ℝ) : 10.[∞↻]1  10.[∞↻]2  10.[∞↻]...

That is the question...

If possible, then we can extend it to make a 1 to 1 association between all Real (ℝ) numbers and Natural (ℕ) numbers (via the spreadsheet / serpentine string path / diagonal zigzag solution...)

(Yes, it is true that following the decimal point of a Real (ℝ) number, there could be infinite number of digits, but the set of Natural (ℕ) numbers itself is infinite / can extend all the way to infinity... Thus / if so...)

- Mr. Cantor...?

Mr. Cantor : I whould have to take a deeper dive / further look into that...

- Perfect...

Also another solution would be to come up with a 2D Hilbert Hotel. Thus each guest in our problem bus, having seat numbers between (ℝ) 0 and 1, would be assigned to the infinite rooms starting at X axis, Natural (ℕ) number 0 and extending towards theY axis infinity.

And guests, having seat numbers between (ℝ) 1 and 2, would be assigned to the infinite rooms starting at X axis, Natural (ℕ) number 1 and extending towards theY axis infinity. etc...

Thus in this scenario it would again be possible to assign all the guests of the problem bus, into the ∞ * ∞ rooms of our 2D Hilbert Hotel.

Thus via this 2D Hilbert Hotel, we can visualise that ∞ * ∞  > ∞ (as L * L > L for all non zero L).

Thus we would like to suggest, point out that all infinites are uncountable. For if any quantity is countable, that negates the notion of infinity (Thus regarding this point we disagree with Cantor that some infinites are countable.)

And we agree with Cantor, that some infinites are larger than some other infinites. (Thus the infinites of real numbers is larger than the infinite of natural numbers BUT both infinites are in the domain/ realm of abstract / / undefinable ∞ realm).

Addendum 1...

Observe that one technique to conceptualise what absolute / theoretical  ∞ is: is to visualize a tube / tunnel with no definable end. Aka a tube that goes on and on and on... Thus one end of this tube is unreachable, is beyond the Confines of the Finite Realms...

And this is why we can add as many guests as we want, into Hilbert's Hotel, because one end of the Hotel is unbounded...

Now there are two distinct points of thoughts regarding absolute infinity.

One school of thought is that all absolute infinities are equal.

The other school of thought is of the view that some infinities are greater than others ...

You whould have noticed, that the author of this text, prefers the notion / view that advocates that all theoretical infinities are undefinable/ unreachable. (Aka the infinite tube with no definable end at one end).

Of course, some sequences whould reach further / farther / faster into the infinity than others... But the infinity they all reach (after infinite steps / infinite time), is the abstract / / undefinable ∞.

Next, consider the following:

Spacecraft 1 traveling at 100, 000 m/s

Spacecraft 2 traveling at 0.01 c

Spacecraft 3 traveling at 0.1 c

HyperSpace 1 reaching 1000 c

HyperSpace 2 reaching 1,000,000 c

(where c is velocity of light in vacuum)

Question1: After an ∞ timespan, how far has each of the above travelled?

Question2: After an ∞ timespan, which one has travelled farthest?

Pause for a moment, and reflect at the intricacies / complications of the answers you just gave...

Thus does this demonstrates that ∞ * ∞  > ∞ ?

And yet mathematically ∞ * ∞  = ∞ 

... 🤔

... much much much later ... 🤔

... are we there yet...?

And such are the beauties , misteries & intricacies of the Absolute / Conceptual ∞. A Mysterious and yet Charming Entity we can only dream of and never be able to reach via the Finite Realms...

Addendum 2...

If / when we attempt to define infinite sets (or sets that goes all the way to infinity), via the such of Venn diagrams, one limitations one comes across is the bounded nature of a Venn circle.

Thus instead if we use open ended extruded circles... (extruded circles which are open at one end...)

The above section / notation has been deprecated...

Kindly Refer Following article...

https://projectsofacademicinterests.blogspot.com/2025/03/a-new-perspective-on-infinity-via-2d.html

Ly De Sandaru.