The Mysteries Of The "Infinity" (Part 2)

Will this discussion → ∞ ?

Strewth!

Help! I am lost in infinity. If you divide any number by itself you always get 1. Why not, then: ∞/∞ = 1 ? When you say ∞/∞ = n, does that not imply that one man's infinity is bigger than another's?

<does that not imply that one man's infinity is bigger than another's>

No. That's indeterminate. <Splutter>

"we can shatter this line producing infinite parts of any length (and, of course, each part has infinite points)… Thus: "∞/∞ = ∞"

If you divide the line into infinite parts you are not dividing by ∞, you are then dividing by a number which is approaching zero. Therefore ∞/0 = ∞.

Hi, Poison...

You have a "line of infinite length" (of course, it consists of infinite points (i.e. "∞ points")... Then you divide this line infinite (∞) times and, so, you get infinite equal "parts" of this line... Each of these parts can have any finite length (it can be as short or as long as you want) and, of course, it consists, also, of infinite points (i.e. "∞ points")... So, you have: "∞ points" / ∞ = "∞ points" ... Hence: ∞/∞=∞ ...

(The opposite: You have "a part of a line" (which consists of "∞ points")... You multiply this part by ∞ (i.e. you add infinite such parts) and, of course, you get a "line of infinite length" (which, of course, consists of "∞ points")... So, you have: "∞ points" . ∞ = "∞ points" ... Hence: ∞.∞=∞ ... )

Note, though, that the "∞/∞=∞" seems to be valid in the case the "line of infinite length" that I mentioned before... But in the case of the "part of a line" (which has finite length but, also, has "∞ points") I cannot say that the "∞/∞=∞" is valid... As I said (in the initial post) if we divide such a part by "∞" we get an infinite number of "infinite small" parts... If we consider such an "infinite small" part as a "single point" we could say that: "∞ points" / ∞ = "1 point" thus "∞/∞=1" ... But if we consider such an "infinite small" part as a "part" (although an extremely tiny one) we could say that: "∞ points" / ∞ = "∞ points" thus "∞/∞=∞" ... The problem is I don't know how to perceive this "infinite small" part: as a "single point" or as a "part" ???...

(It is the same problem as with the "sets" that I mentioned in my initial post: The infinite subsets (i.e. A₁, A₂, A₃, ..., A_∞) that we can produce from a set (i.e. A) (where A is the union of all those subsets)... In this example I saw that A₁={1,...}, A₂={2,...} e.t.c., so the A₁ seems to have just one element (the "1"), the A₂ seems to have just one element (the "2") e.t.c.... How can I consider all these subsets: as "single numbers" or as "subsets of numbers" ???... It is a similar question as before...)

As a matter of fact, I doubt about my initial perception (see (D) in my initial post) that: "∞/∞=n"...

So I have to reconsider this... Afterall, it seems that in some cases (such as the infinite division of a "part of a line") the "∞/∞" has "no meaning"... I really don't know...

A) "n/0" has "no meaning" and is not allowed (and is not ∞ )

Right, it's not allowed in the number system of the reals, because there is no number WITHIN the set of reals with which we can multiply 0 to get n. As simple as that. If we extend the reals to include the ∞, then we DO get n/0=∞, only that we have to trade off some of the nice properties of the number system of reals (OK, someone might say, "who cares?"). Otherwise we can only approach infinity using limits of sequences.

So there is a question if the "∞.n=∞" and "∞/∞ =n" are valid and equivalent…

If we approach infinity arithmetic in the more relaxed way we are doing throughout this discussion, then indeed these two equations are equivalent and valid, BUT, as long as we are talking about infinities of the same order, or more technically, of the same cardinality. For example, if you divide the infinity of the reals (e.g. the points in a line) by as many time as the number of the integers, then you get infinity again. (If you do the inverse, you get 0.)

Referring to your train of thought involving dividing a line infinite times, it seems that what you do is dividing a higher order of infinity (2nd order maybe) by a countable infinity (i.e. 1st order), that's why you get infinity as a result. Try instead to divide the line by as many times as the total number of the reals, in which case you'll get infinite pieces of finite point long lines! But even if you are an eternal and never-getting-bored being (and there is no Big-Crunch, and...), I doubt you can divide the line so many times as many are the reals!

So we come to a conclusion that "a part of a straight line" and "a straight line of infinite length" are not "infinities" of the same kind (or "size") (although both have infinite points)… We could consider the "straight line of infinite length" as kind of a "larger infinity" (than the "part of a straight line") because, in this case, the expression "∞/∞ = ∞" is valid…

It is not so. Each part of a straight line has as many points as any other part or even as the whole line itself. It can be easily shown that each point of the part can be mapped one-to-one with each point of the whole.

As I said before, the expression ∞/∞ = ∞ is valid when the numerator is infinite of higher cardinality. Otherwise we cannot tell, unless we know the sequence that limits to this expression: in this case, the result depends on the relative rate at which the infinities in the numerator and denominator are approached.

Hi tkot,

IMHO ∞ is not a number (NAN) and cannot correctly be used in an equation as a number. While it can be shown that the result of an equation, if carried to its ultimate conclusion may result in ∞ such as, for example:

lim _{x -> n} 1 / (x - n) = ∞, this is not the same as using ∞ as a number such as ∞/∞, although symbolically, as I think you pointed out, the symbol ∞ can correctly by used to represent sets and sub sets.

I believe that infinity does represent a situation in where the laws of physics and mathematics break down, such as in equations that deal with QM, M-theory, cosmology, etc.

So you could probably say that ∞ can represent the destination where the singularity (in a BH or at the BB) resides, or simply ∞ = singularity.

Still, it's NAN and cannot be treated as one.

-John

Hi Tasos...

"Referring to your train of thought involving dividing a line infinite times, it seems that what you do is dividing a higher order of infinity (2nd order maybe) by a countable infinity (i.e. 1st order), that's why you get infinity as a result."

By saying "1st order infinity" you mean i.e. the set of the natural or integer numbers???... And by saying "2nd order infinity" you mean i.e. the set of the real numbers???...

"Try instead to divide the line by as many times as the total number of the reals, in which case you'll get infinite pieces of finite point long lines! But ..... I doubt you can divide the line so many times as many are the reals!"

The total number of the elements of the "set of the real numbers" is obviously infinite (∞)... So you just divide the "line of infinite length" by "∞"... I don't get what you are trying to say... And by dividing a "line of infinite length" by "∞" how do you get infinite pieces of finite point long lines???... What "finite point long lines" means???... Is it obvious that each of these line parts (that are produced) have infinite points???... (So I need a clarification)...

I said: "We could consider the "straight line of infinite length" as kind of a "larger infinity" (than the "part of a straight line") because, in this case, the expression "∞/∞ = ∞" is valid…" ... You said:"It is not so. Each part of a straight line has as many points as any other part or even as the whole line itself." ... I agree... Both consists of infinite points... But are they "similar infinities"???... Are they of the same "size"???... And if so, how we get "∞/∞=∞" in the case of the "line of infinite length" but not in the case of the "part of a line"... Pls, read carefuly my posts #5 & #7... Am I wrong in sth???...

Hi guys

There are different orders of infinities. Infinities are classified by the their so called cardinality, which shows how many members are in a particular infinite set. For example the number of the naturals 1,2,3... are classified as infinite of order 1 (or Aleph-1). Another way to cal this infinite is "countable" because you can put all numbers in a row and assign an index number to each of them. Same cardinality has the set of rationals, or the set of even integers and so on.

On the other hand, the cardinality of the set of real numbers is something more than that. If you try to put these numbers in a row and say "this one is the first, this one is the second, ..." you will find that there will always be another one in between any two of them. This set is not countable. It is certainly of higher order, most probably of 2nd order (or Aleph-2) although this has to be proven. (You can go on and imagine even higher order infinities, like e.g. Aleph-100 or Aleph-Aleph-1 or Aleph-Aleph-Aleph-.....infinite times.... -Aleph-1. Or Aleph-Aleph^Aleph^... Oops, I got dizzy...)

When you divide a line, what you do is divide it by, say, 10 or 100 or 1000. You get pieces that can be arranged one after the other, so you have a countable set of pieces. Now if you go on dividing more and more, eventually reaching infinity, then what you do, is to divide the Aleph-2 points of the line by Aleph-1 times. What you get is Aleph-1 pieces of line with still infinite (Aleph-2) points each!

Of course, you cannot divide the line by Aleph-2, given we agree what line division means in our context. IF you did, then you could have separate points - as I said in my previous post - but I cannot see how this division can be done. Even if this was technically possible (how?), still it wouldn't suffice even Aleph-1 time to perform, even if you could make Aleph-1 divisions every second!

As for the number of the points in a finite length of line as compared to the whole line of infinite length, they are the same. The trick used when talking about cardinalities, is to map one-to-one each member of one set to each member of the other, and if such a map exists, then you establish that the sets have the same cardinality. In our case, consider the line length between the points 0 (0 not included) and 1 and the line between 1 and ∞. You can easily device map(x)=1/x to prove that each point in the one part has one and only one counterpart in the other.

Infinity is a thrilling topic, although very difficult and I cannot touch it any deeper. But what I understand, is that infinity doesn't mean "just everything". There are certainly things that are not included, even in an infinite set of things. And there are things you cannot do even with infinite power or infinite time. I wish Douglas Adams were right when he claimed that in an infinite and eternal universe you will sometime and someplanet eventually find screwdrivers growing from the trees...

Oops, I discovered a mistake in my previous post. I should have started the Aleph classification with index zero. That is, a countable infinity has in fact cardinality Aleph-0, the set of real numbers Aleph-1 and so on.

Thanks, Tasos... Nice analysis... It helped a lot...

According to your analysis a "line of infinite length" and a "part of a line" are sets (of points) of the same cardinality (probably Aleph 2)... So, in my procedure I was dividing both of them by an "Aleph 1" infinity (the "∞" of the natural numbers)... As such a division of a "line of infinite length" gives infinite number of "line parts" (each of them having any finite length and infinite points), thus "∞/∞=∞", the same must happen for the case of the "part of a line"... e.g. such a division of a "part of a line" must give infinite number of "line parts" (each of them having infinite points)... Thus the result of such a division must be "parts" (although extremely small) and not "points"... So, we should consider again that "∞/∞=∞"... Am I right???...

Oops, now I saw your correction... (Assume the same in my previous post)...

Anyway, I wait for your reply...

In the context we are talking about it is ∞/∞=∞ indeed, but to be more accurate we should have written instead: Aleph1/Aleph0=Aleph1. Things are much clearer now. The symbol ∞ denotes generally the infinity which is relative to the context at hand (mostly the Aleph0). But when it is used in the expression ∞/∞, we lose information about what exactly is the nominator and what is the denominator. They are not the same entities in our example!

I totally agree... I still need your opinion in my message #12...

Let's put my message #12 in the correct terms:

The number of points of a "line of infinite length" is Aleph-1... I divided it by Aleph-0 (the "∞" of the natural numbers)... The result is [Aleph-1]/[Aleph-0]=[Aleph-1]... (this is the more precise expression of the "∞/∞=∞" that I said before...)

The number of points of a "line part" is, also, Aleph-1... I divided it by Aleph-0 (the "∞" of the natural numbers)... The result is [Aleph-1]/[Aleph-0]=[Aleph-1]... (this is, also, the more precise expression of the "∞/∞=∞"...)

Am I right???...

So, it seems that all these infinite "extremely tiny line parts" that are produced by such a division of the "line part" (by the "∞" of the natural naumbers) gives Aleph-1... which means that we must concider these "extremely tiny line parts" as "parts" (with infinite points) and not as "single points"...

Do you agree???...

So, it seems that all these infinite "extremely tiny line parts" that are produced by such a division of the "line part" (by the "∞" of the natural naumbers) gives Aleph-1... which means that we must concider these "extremely tiny line parts" as "parts" (with infinite points) and not as "single points"...

Exactly! In order to get single points, you need to divide the line by Aleph-1, which is not possible as I understand. Dividing a line, as we mean it, is separate it in smaller consecutive regions, something like cutting a string by the scissors. This method can only provide us with at most Aleph-0 cuttings.

That's nice... So, let's finish it...

Assume that we have a "line part" (or a "line of infinite length"...it doesn't really matter as they both are Aleph-1)... Then if we could divide it by Aleph-1 (e.g. the "infinity" of the real numbers... suppose that we could do it with a magical way...) then the result would be 1 (I mean just one point) or n (I mean a "line part" which consists of "n points", where n is any finite number)???... What do you think???... I would dare to say that it doesn't matter... Afterall, what is the meaning of a "line part" which consists of some (finite) points???... (I, really, cannot perceive such a "part")... So, we could assume that [Aleph-1]/[Aleph-1]=n ... (This is a general or indefinite result)... Do you agree with this???...

It is not possible to have a compact line part consisted of finite points. You can always find new points in between each pair of those, and this story goes on to infinity! If one could divide a line to Aleph-1 parts, I had in mind that the result would be some scattered individual points, not line lengths. Of course, by cutting the line the way we are talking about, i.e. choosing a point on the line which separates it in two parts, the left and the right, and then cut again and again, what we get is a countable set of line parts, therefore, their number will be no more than Aleph-0. If someone knows a method by which we can divide a line in Aleph-1 parts, is welcome to put it forward.

Of course, Aleph-1/Aleph-1 is not necessarily a number n as you said. It can also be Aleph-0. If we cut a line on every e.g. integer point then we will have: (points of whole line)/(points in each part) = (number of parts) or A1/A1=A0.

Ah, yes... I agree with all of these...

What about Aleph0/Aleph0???... The result is "n" (e.g. any finite number)???...

[ By the way, after your analysis, I realized that the other example that I gave in my initial post (the division of the set of the natural numbers (A) by the "∞" of the natural numbers) actually will give as a result individual numbers (and not subsets)... With the procedure (that I presented) we produce subsets A₁, A₂, A₃, ..., A_∞ that are A₁={1,...}, A₂={2,...} e.t.c., The A₁ seems to have just one element (the "1") as the second element should be in an "infinite distance" from the "1", so, actually this subset is A₁={1} (and in the same way we have A₂={2}, A₃={3} e.t.c.)... So, we have that: "infinite elements"/∞ = "1 element" ... Thus, in this example, we have "∞/∞=1" (with the meaning of Aleph0/Aleph0=1)... But I suppose that we could produce the infinite subsets in another way also... I.e. in my initial post I could produce the subsets as follow:

A₁={1,2,11,12,...}, A₂={3,4,13,14,...}, ….. , A₅={9,10,19,20,...} (thus producing 5 subsets)

Then if I follow the procedure of the "spacing out" of the subsets (producing more and more subsets), as in my initial post, then I get:

A₁={1,2,...}, A₂={3,4,...}, ….. or, actually: A₁={1,2}, A₂={3,4}, …..

Which gives "infinite elements"/∞ = "2 elements", hence "∞/∞=2" … With the same way, by making the appropriate subsets, I could get: "infinite elements"/∞ = "n elements", hence "∞/∞=n" … I think that's a good way to perceive that Aleph0/Aleph0=n (where n is any finite number)… Don't you think???… ]

Hence, we have the followings:

Aleph0/Aleph0=n

Aleph1/Aleph1=n or Aleph0

Aleph1/Aleph0= Aleph1

(where n is any finite number)

So, as a conclusion, we can say (in a more relaxed way) that the expression "∞/∞" may gives as result "n" (as it may be "n" in the case of Aleph1/Aleph1 or it is "n" in the case of Aleph0/Aleph0) or "∞" (as it may be "Aleph0" in the case of Aleph1/Aleph1 or it is "Aleph1" in the case of Aleph1/Aleph0)... Don't you agree???...

Aleph-x/Aleph-y =

• Aleph-x, if x>y
• 0 , if x<y
• indeterminate value <= Aleph-(x-1), if x=y

Hence what I said before:

Aleph0/Aleph0=n

Aleph1/Aleph1=n or Aleph0

Aleph1/Aleph0= Aleph1

(where n is any finite number)

is correct... Isn't it???...

Yes.

You can also check some few references here:

http://www.earlham.edu/~peters/writing/infapp.htm
http://www.earlham.edu/~peters/writing/infinity.htm
http://mathworld.wolfram.com/CountablyInfinite.html
http://mathworld.wolfram.com/Aleph-0.html
http://mathworld.wolfram.com/Aleph-1.html

Thanks, Tasos...