1=0.999... infinities and box of chocolates..Phliosophy of Math...

[Pete]

how can you just shift one passenger to another room when they are all occupied infinitely?

Everyone moves at the same time.
Is there any guest that can't move? No.
So what's the problem?

On what logical grounds can you say infinity some how implies an open ended situation when you have already claimed all infinite rooms are full?

one possible slant:

The manager says to the incoming guests, "You can't be looking for a room, as you already have one...after all are you not a part of the infinite guests to begin with? So what room do you already have?the rational being explored is:
Infinity may be "never ending" but it is always a complete series [ with in a given scenario ].
In this case we have an HOTEL with infinite rooms and infinite guests. [ a "complete" yet "never ending" scenario ]
or;
Since when has an infinite number of guests not included all possible guests?

You seem to imply that 'infinite' means 'everything'. That's not what it means.
For example, there are infinite even integers, right? But that does not mean that all integers are even.
So, we can have infinite guests in the hotel, and still guests outside.

this also suggests a contradiction or more a hypocrisy.

We can have an infinite number of rooms but a less than infinite number of guest....
"We just move them along one..." implies to me that whilst the rooms are infinite the guests aren't.

No, there are infinite guests in the hotel.
If the guests aren't infinite, then they must be finite. So how many are there?

(R1,G1) + (R2,G2) + (R3,G3) ....
whereby the rooms and guest are directly associated infinitely.
All possible rooms and guests are accounted for. [in this single HOTEL]

Correct. But the whole point of Hilbert's Hotel is that we can manipulate that association.
Look at this direct association:
(R2,G1) + (R3,G2) + (R4,G3) ....

All previous guests are accounted for, but R1 is now a spare room.

 

[Undefined]

Hi Pete, QQ.

Everyone moves at the same time.
Is there any guest that can't move? No.
So what's the problem?

You seem to imply that 'infinite' means 'everything'. That's not what it means.
For example, there are infinite even integers, right? But that does not mean that all integers are even.
So, we can have infinite guests in the hotel, and still guests outside.


No, there are infinite guests in the hotel.
If the guests aren't infinite, then they must be finite. So how many are there?


Correct. But the whole point of Hilbert's Hotel is that we can manipulate that association.
Look at this direct association:
(R2,G1) + (R3,G2) + (R4,G3) ....

All previous guests are accounted for, but R1 is now a spare room.

Hmmm. About that Room 'Number' aspect:

What if the rooms are NOT numbered, but are just built to infinity and the builder forgets to nail on the room numbers? What then? What becomes of all those abstract 'associations' between present/potential guests and room numbers?

Just a 'silly question' to see what your 'take' would be in that 'anonymous infinity' case, guys!

 

[Pete]

Hmmm. About that Room 'Number' aspect:
What if the rooms are NOT numbered, but are just built to infinity and the builder forgets to nail on the room numbers?

Then all is lost, unless there is some other method of mapping besides room numbers (such as location in 3D coordinates.)

 

[Quantum Quack]

Everyone moves at the same time.
Is there any guest that can't move? No.
So what's the problem?

You seem to imply that 'infinite' means 'everything'. That's not what it means.
For example, there are infinite even integers, right? But that does not mean that all integers are even.
So, we can have infinite guests in the hotel, and still guests outside.


No, there are infinite guests in the hotel.
If the guests aren't infinite, then they must be finite. So how many are there?


Correct. But the whole point of Hilbert's Hotel is that we can manipulate that association.
Look at this direct association:
(R2,G1) + (R3,G2) + (R4,G3) ....

All previous guests are accounted for, but R1 is now a spare room.

Your going to have to offer your definition of infinity Pete..

You seem to imply that 'infinite' means 'everything'. That's not what it means.

it certainly means all integers odd and even. [not just one or the other but both]

 

[Quantum Quack]

so another hotel opens up in competition next door and an infinite number of guests move out and into the 2nd hotel, how many guests remain in our infinite room and guest Hilbert Hotel?

infinity + infinity = infinity is the proposition...yes?
well
infinity - infinity = infinity ...should also be valid then.

and Hilbert's hotel is out of business or does it still have an infinite occupancy... which?

Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.

this wiki explanation fails to account for the fact that all rooms are already occupied infinitely.

famous splurge "up and onward to infinity and beyond" [chuckle]

it is a bit like adding a digit to the decimal series of pi and justifying it by saying that well it goes on infinitely so we can..

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128481121745028410270193852110555964462294895493038196
4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...

 

[Motor Daddy]

Everyone moves at the same time.
Is there any guest that can't move? No.
So what's the problem?

The problem is with the coordination of the timing of the big switch. Light only travels at 299,792,458 m/s. Information of the time for all guests to switch to a new room will not make it to the last guest due to the finite size of each guest current room. It is simply impossible for the big switch to occur simultaneously. Furthermore, if all rooms are at an intersection of grid lines of the x,y persuasion, then the hypotenuse of the right triangles are not correct given the use of incorrect SR, ie, the length is contracted in the x direction but not the y or z direction. So your numbers don't add up! All guests can not switch rooms simultaneously therefore there is chaos in the hotel due to the unknown continuous quantity that infinity represents! Infinity is an unknown, not a known.

 

[Pete]

You're going to have to offer your definition of infinity Pete.
it certainly means all integers odd and even. [not just one or the other but both]

Are you saying that there are not infinity even numbers?

so another hotel opens up in competition next door and an infinite number of guests move out and into the 2nd hotel, how many guests remain in our infinite room and guest Hilbert Hotel?

You haven't given enough information to tell.
If all guests move into the hotel next door, then there are none left.
If only those from even numbered rooms move, for example, then there are infinity left.

 

[hansda]

Hence the question in all such exercises offered so far to 'sector/halve' MD's real disc:

"How does one 'share' a central point that is 'dimensionless'; or a middle line that has 'no dimension other than length' ?"

Can you or anyone else answer that for me? I would be very interested to see how you and others would do this 'sharing' process in reality. Thanks.

As per wiki a point is dimensionless and a line has no dimension other than length. So by sharing of a point or a line, do not add any dimension in reality.

 

[Billy T]

so another hotel opens up in competition next door and an infinite number of guests move out and into the 2nd hotel, how many guests remain in our infinite room and guest Hilbert Hotel?

infinity + infinity = infinity is the proposition...yes?
well
infinity - infinity = infinity ...should also be valid then.

and Hilbert's hotel is out of business or does it still have an infinite occupancy ...

Yes QQ, your two equations are correct, but in some special cases, infinity - infinity = 0 or some other finite number.

No, Hilbert Hotel is still marginally functing with an infinite number of guests, due to Hilbert's policy of "No charge, if in a room number is a prime number with more than five digits*," but more than 99% of the former occupants have moved into the new Billy T infinite hotel, even though rooms cost 5% more than at non-prime rooms at Hilbert, mainly because:
(1) Rooms are more modern with free internet in each room;
(2) New arrivals are instantly given their room number, as they step off the bus, even if infinite capacity buses are full and an infinite number of buses arrive at the same time;
(3) The elevator delay to your floor is only 25% of that at the Hilbert hotel. . More details on all of this at:
Here:http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3132475&viewfull=1#post3132475
and
(4)The Hilbert hotel is so old that its mortgage is fully paid off, yet with such a current low fractional occupation rate, it may go bankrupt too as the Monimonika infinite Hotel recently did. (It was not filling all its rooms even with infinite arrivals due to poor room assignment algorithm, and was still paying the mortgage on the empty rooms.)

Monimonika, (owner of the now bankrupt infinite hotel with his name) had asked how the Billy T infinite hotel algorithm could immediately give room numbers with no "empty room gaps" via so simple calculation say for new passenger called (1,37) - Passenger 37 who got off bus 1 and to all the other infinite passengers from all the simultaneously arriving infinite number of infinite capacity buses. I told him and the basic information each new arrival gets.

A personal Cartesian coordinates point (b , p), unique for Passenger number, P#, on Bus number, B#, has been specified.

Welcome (1,37) to the Billy T infinite hotel, more modern ( & faster elevators), than the old Hilbert Hotel. The empty room numbers are: 2{( S'-1)^2 + O'}. Please go directly to your room: 2{36^2 + 1} = 2,594. That plus your credit card number, extending it as a decimal fraction, will be your internet pass word and will be used for billing.
Towels and sheets are changed every week, and as Mr. Hilton always requested: Please make sure the shower curtain is INSIDE the tub.....

The hotel agents, not the passengers, know the values of S' and O' for each arriving passenger, and you can too by reading one more short paragraph at the above link. The quote above is the first two paragraphs.

*It takes so long to prove it is a prime with more than five digits that most move out and are billed. (Hilbert was a mathematician and loved prime number and wanted others to also.)

 

[Quantum Quack]

Are you saying that there are not infinity even numbers?

Not true... of course both odd and even numbers are infinitely represented... Why would you think otherwise?

 

[Quantum Quack]

Just to clarify my take on the use of infinity in the Hilbert hotels scenario.

Instead of Hilbert we call this hotel Pi
Pi Hotel is a 5 star Hotel with an infinite number of floors.
Each floor hold one apartment or room.

The managers name is "Three-zy" which rhymes with sleazy.

and in the lobby we have a "bell boy" called "Decimo Pointius" who is even more sleazy than the manager...and he sleeps on the couch between servicing the guests. [especially the female guests]


So,
we have this infinite floored hotel and each room is given a number:
the number of the first floor is the first ten digits of Pi. [with a floor "witness" designate of incremental integers.]
Edit: the witness numbers are typically used in data bases as the Key or record ID and also provide a count of the number of floors

floor one #1-1415926535
the subsequent floors are given number of subsequent 10 digits of Pi.
floor two #2- 8979323846
floor three #3- 2643383279
and so on infinitely.

So at the end of construction the Pi hotel is made up of floors /apartments numbered by Pi. [with a "witness" integer]
Which is managed by a guy called "Three-zy" which rhymes with sleazy.
All guests are given an account number that corresponds exactly to the apartment number.

Now a bus arrives at Pi with a sh*t load of possible guests.. [ Yeah ...the women have secretly heard about the bell boy called "Decimo Pointius]


How ever Three-zy [which rhymes with sleazy] can only say,

"Sorry, but all rooms of Pi are taken - I have no more floors, apartments or witness numbers to allocate"

 

[Quantum Quack]

Yes QQ, your two equations are correct, but in some special cases, infinity - infinity = 0 or some other finite number.

No, Hilbert Hotel is still marginally functing with an infinite number of guests, due to Hilbert's policy of "No charge, if in a room number is a prime number with more than five digits*," but more than 99% of the former occupants have moved into the new Billy T infinite hotel, even though rooms cost 5% more than at non-prime rooms at Hilbert, mainly because:
(1) Rooms are more modern with free internet in each room;
(2) New arrivals are instantly given their room number, as they step off the bus, even if infinite capacity buses are full and an infinite number of buses arrive at the same time;
(3) The elevator delay to your floor is only 25% of that at the Hilbert hotel. . More details on all of this at:
Here:http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3132475&viewfull=1#post3132475
and
(4)The Hilbert hotel is so old that its mortgage is fully paid off, yet with such a current low fractional occupation rate, it may go bankrupt too as the Monimonika infinite Hotel recently did. (It was not filling all its rooms even with infinite arrivals due to poor room assignment algorithm, and was still paying the mortgage on the empty rooms.)

Of course 50% of infinity is still infinity.
But is it the same qualified infinity as first started with?
In Hilbert's hotel it simply states with the qualifications that the 1] Hilbert Hotel has 2] infinite rooms and 3] infinite guests. One must assume that it is only qualified by the words rooms and guests and not anything else.
So one can only presume that all possible rooms are fully occupied in Hilbert's Hotel.

 

[Billy T]

... So one can only presume that all possible rooms are fully occupied in Hilbert's Hotel.

In the context of the puzzle that must be presumed or there is no puzzle if for example All Modulo 3 = 0 rooms are empty as then there would be an infinity of even and an infinity of odd numbered rooms without occupants so then there would be an infinity of rooms available to accommodate an infinity of new guests, even with each guest allowed to choose either even or odd numbered room.

 

[Quantum Quack]

In the context of the puzzle that must be presumed or there is no puzzle if for example All Modulo 3 = 0 rooms are empty as then there would be an infinity of even and an infinity of odd numbered rooms without occupants so then there would be an infinity of rooms available to accommodate an infinity of new guests, even with each guest allowed to choose either even or odd numbered room.

I am sorry Billy T but I do not understand what you are saying with the above...

 

[Undefined]

Hi handsa.

As per wiki a point is dimensionless and a line has no dimension other than length. So by sharing of a point or a line, do not add any dimension in reality.

Didn't I just tell you that a point was dimensionless and line had only length dimension? ...as per my previous post, now bolded...

Hence the question in all such exercises offered so far to 'sector/halve' MD's real disc:

"How does one 'share' a central point that is 'dimensionless'; or a middle line that has 'no dimension other than length' ?"

Can you or anyone else answer that for me? I would be very interested to see how you and others would do this 'sharing' process in reality. Thanks.

Hence my original question to Tach when he suggested 'sharing' a point, ie:

How does one (mathematically) 'share' a (mathematically) DIMENSIONLESS CENTRAL POINT of REAL solid DISC 'divided' into 3 REAL sectors; or how does one 'share' a CENTER LINE of a 'halved' REAL solid DISC when that center line has only length dimension and (mathematically) no 'sideways' dimension extending into either of the REAL 'halves'?

In short, how does one 'share' nothing at all (mathematical point); or something not extending into the halving results at all (mathematical line)? That is the REAL question.

 

[hansda]

Hi handsa.



Didn't I just tell you that a point was dimensionless and line had only length dimension? ...as per my previous post, now bolded...


Hence my original question to Tach when he suggested 'sharing' a point, ie:

How does one (mathematically) 'share' a (mathematically) DIMENSIONLESS CENTRAL POINT of REAL solid DISC 'divided' into 3 REAL sectors; or how does one 'share' a CENTER LINE of a 'halved' REAL solid DISC when that center line has only length dimension and (mathematically) no 'sideways' dimension extending into either of the REAL 'halves'?

In short, how does one 'share' nothing at all (mathematical point); or something not extending into the halving results at all (mathematical line)? That is the REAL question.

A conclusion can be made that:" Finite dimensions of any entity, are made of infinite dimensionless constituent entities."

 

[Undefined]

Hi handsa.

Hi handsa.

Didn't I just tell you that a point was dimensionless and line had only length dimension? ...as per my previous post,...

Hence my original question to Tach when he suggested 'sharing' a point, ie:

How does one (mathematically) 'share' a (mathematically) DIMENSIONLESS CENTRAL POINT of REAL solid DISC 'divided' into 3 REAL sectors; or how does one 'share' a CENTER LINE of a 'halved' REAL solid DISC when that center line has only length dimension and (mathematically) no 'sideways' dimension extending into either of the REAL 'halves'?

In short, how does one 'share' nothing at all (mathematical point); or something not extending into the halving results at all (mathematical line)? That is the REAL question.

A conclusion can be made that:" Finite dimensions of any entity, are made of infinite dimensionless constituent entities."

handsa, your 'conclusion' there basically implies that an infinity of nothing ( zero ) can somehow become something ( >zero ). Ie, like Universe/Maths 'ex-nihilo', hey?


See the problem.....UNLESS there IS some 'infinitesimal' penultimate 'state change' boundary condition/zone 'entity/state' in physics and maths to reflect that change from nothing to something (and back again)?


Hence my continuing observation about there being in fact a non-zero 'smallest step' (ie, infinitesimal) being a necessary to make the maths/physics logics/modeling work consistently.

In physics, there must be an 'infinitesimal of effectiveness' between infinity and finite process entity/step.

And in Maths there must be an 'infinitesimal number' between zero and the next non-zero 'real number' (according to axiomatic Number Line Points 'understanding' so far).


In short, as I have previously suggested, the "0" and the "1" are actually BOUNDARY CONDITION states symbols, indicating a change from one state to another. In other words, the "0" is the infinitesimal between nothing and something; and the "1" is the singularity between the something of one kind/value becoming another something of another kind/value etc etc.


Only after we bring all these further insights consistently into the mathematical axiomatic set/treatments will the logic/modeling 'gap' between maths and reality be bridged, and we can proceed to COMPLETE both the mathematical modeling and the physical theory of reality.

 

[Quantum Quack]

....And in Maths there must be an 'infinitesimal number' between zero and the next non-zero 'real number' (according to axiomatic Number Line Points 'understanding' so far).


In short, as I have previously suggested, the "0" and the "1" are actually BOUNDARY CONDITION states symbols, indicating a change from one state to another. In other words, the "0" is the infinitesimal between nothing and something; and the "1" is the singularity between the something of one kind/value becoming another something of another kind/value etc etc.


Only after we bring all these further insights consistently into the mathematical axiomatic set/treatments will the logic/modeling 'gap' between maths and reality be bridged, and we can proceed to COMPLETE both the mathematical modeling and the physical theory of reality.

@underfined
Mathematics and physics recognize without question that 1/infinity =/= 0. The mere fact that they impose deliberately a limit on infinity so that the 1=0.999.. can be defined is proof that they are well aware of what you are contending about the axiomatic use of infinitesimals etc.
In fact a whole system of numbers called the hyperreals was developed to help in "accommodating" this "key" issue. [ I believe ]

Mathematics has provided the world with enormous benefit even with such a "paradox" embedded in the out puts.
So what you are contending is not new, in fact it goes back quite a few thousand years. [as you already know]

So Undefined can I ask?

What is the solution you propose to this paradox of 1/infinity = ?
What would you like to see happen?
btw an infinitesimal is an abstraction and doesn't actually exist as anything other than a [ 0 + (1/infinity) ] "wide" dimensional boundary. [as far as I can tell]

 

[Tach]

@underfined
Mathematics and physics recognize without question that 1/infinity =/= 0.

Contrary to your claim, mainstream mathematicians and physicists KNOW that $$\frac{1}{\infty}=0$$.
Since you claim the opposite , I will ask you to provide proof to your claim.

The mere fact that they impose deliberately a limit on infinity so that the 1=0.999.. can be defined is proof that they are well aware of what you are contending about the axiomatic use of infinitesimals etc.

Show a reference or admit that you are just making things up.

In fact a whole system of numbers called the hyperreals was developed to help in "accommodating" this "key" issue. [ I believe ]

Rubbish.

 

[Quantum Quack]

Contrary to your claim, mainstream mathematicians and physicists KNOW that $$\frac{1}{\infty}=0$$.
Since you claim the opposite , I will ask you to provide proof to your claim.



Show a reference or admit that you are just making things up.




Rubbish.

to answer you question:
Why did mathematics develop the use of limits when dealing with infinities?