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Speaking realistically it is physically impossible for infinity to exist. I believe that infinity is ONLY available on computers! Here is my reason why. On a computer you can have an infinite number (but is it really infinite?). If you were to look at an entire infinite number by scrolling through the computer it would be impossible (your life isn't infinite so you will never see the end of the number). People think that math has infinite numbers, but that is WRONG. On a calculator you will never see the entire number AND if you were to convert the number from the calculator to paper you WOULDN'T write the entire number! Infinity has to have an end, but at the moment it just serves as a "placeholder" for an unknown integer that we are not yet (or maybe never) able to define. So tell me, what do you think is infinite? Our universe has an end, our lives have ends, etc. Everything must end if it has once begun...
It depends what you mean by 'exist'.
We can consider 3 different contexts.
1) Number systems. No, infinity does not exist in number systems.
Any number system includes a set of operations (add, subtract etc) which obey the rules of arithmetic. Infinity (∞) does not obey those rules. For example ∞ - 1 = ∞
2) Topography (spaces). Yes, here infinity does exist. See HERE for a longer explanation
3) In set theory (ie measuring the size of a set of things). Yes, here infinity does exist but there is more than one infinity. See HERE for a longer explanation.
First of all, infinity exists as a concept. We can then apply this concept to real situations. A time interval or a space interval can be divided in infinite intervals. A second is equal to 1000 ms, which can be subdivided and so on...
I disagree with the equality:
infinity - 1 = infinity
1 is too small compared with infinity but is not zero. So, I'd correct this statement to:
infinity - 1 is infinitesimally equal to infinity
Anyway, it's somehow odd to subtract 1 from infinity. What's infinity minus one? It's abstract because these two quantities can't be compared. Bikerman says it's the same because infinity prevails from finite quantities. Although it prevails there's still an infinitesimal difference.
The link provided by Bikerman is interesting. Integer numbers (with infinite possibilities that's right!). Infinity has many different forms, no doubt. You can divide infinity by infinity and several results might happen.
I'm not a mathematician but I think there's something to say about that article. By simplifying things, it doesn't explain what's this thing called dimension and might confuse some ideas.
In my humble opinion, the following equality still holds:
the infinite number of integer numbers = infinite number of odd numbers + infinite number of even numbers
The article says the "dimension" of the three sets is the same. No doubt. Because "dimension" is something used to explain that infinity might be layered in various infinities.
For each integer there are infinite real numbers between it and the following integer number.
| romaop wrote: |
I disagree with the equality:
infinity - 1 = infinity
1 is too small compared with infinity but is not zero. So, I'd correct this statement to:
infinity - 1 is infinitesimally equal to infinity
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infinity - 1 = infinity
It seems to me that you were almost trying to give infinity an end. If something is infinite, it never ends. What you were saying is sort of like saying 1,000,000,000,000 - 1 doesn't equal 1,000,000,000,000. It equals 999,999,999,999, which is extremely close to 1,000,000,000,000. But 1,000,000,000,000 ends. Since it ends, subtracting 1 from it will have an effect. Since infinity never ends, subtracting 1 from it will not have any affect on it because it's going to keep going on forever, even if you subtract 1 from it..
| fpwebs wrote: | | Infinity has to have an end, but at the moment it just serves as a "placeholder" for an unknown integer that we are not yet (or maybe never) able to define. |
Try dividing 1 by 0. That answer will be undefined. The reason why it is undefined is because the result will be so infinitely great that it cannot be defined by numbers. Try raising zero to the zero power; that will also be undefined, which means that the result will be so infinitely great that it cannot be defined by numbers.
The concept of infinity has to exist in numbers. If it didn't, calculus wouldn't be able to exist. In calculus, when you fine the area under a curve, you're using an infinite amount of infinitely small rectangles.
Also, in algebra (or any type of math involving graphing) you may have an asymptote. This is a line in which a graph is approaching constantly, getting infinitely closer, but will never reach.[/b]
| Quote: | | I believe that infinity is ONLY available on computers! |
How so? It would take an infinite time for a computer to calculate, or count to, infinity.
| romaop wrote: | First of all, infinity exists as a concept. We can then apply this concept to real situations. A time interval or a space interval can be divided in infinite intervals. A second is equal to 1000 ms, which can be subdivided and so on...
I disagree with the equality:
infinity - 1 = infinity |
You can disagree if you like - it is still the case however. Math is not a democracy so I'm afraid disagreeing will make no real difference... | Quote: | 1 is too small compared with infinity but is not zero. So, I'd correct this statement to:
infinity - 1 is infinitesimally equal to infinity |
Well, I'm afraid you would be wrong. | Quote: | | Anyway, it's somehow odd to subtract 1 from infinity. What's infinity minus one? It's abstract because these two quantities can't be compared. Bikerman says it's the same because infinity prevails from finite quantities. Although it prevails there's still an infinitesimal difference. |
That's why I said infinity does not exist in number systems. I did NOT say that 'infinity prevails from finite quantities'. I don't actually know what that means but I certainly did not say it. I said that infinity, in a number system, is impossible. No single number can ever be infinite (x+1, x+2). Whatever number you assign to the concept of infinity it is always possible to add 1, therefore infinity does not exist as a number. Your statement about 'infinitesimals' does not make sense. You can only have an 'infinitesimal' difference between concrete (absolute) values. You cannot have an 'infinitesimal' difference between infinity and infinity - 1, since infinity is not bounded, and cannot therefore be assigned an absolute value. You would also run into the problem of an infinity of infinitesimals. Infinity-1, infinity-0.9, infinity-0.98...
Infinitesimal is a rigorous concept in math. It is a relative term - it only exists relative to another number or set of numbers. You cannot relate infinitesimal to infinity since infinity is not a number.
The mathematically correct way to deal with infinity is to use limits. Thus your example of all the even and odd numbers to infinity could be written as S1 and S2 as follows:
S1n→∞ = 2^0 + 2^1 + 2^2 + . . .+ 2^n
S2n→∞ = (2^1)-1 + (2^2)-1 + . . .+ (2^n)-1
| Afaceinthematrix wrote: | | The concept of infinity has to exist in numbers. If it didn't, calculus wouldn't be able to exist. In calculus, when you fine the area under a curve, you're using an infinite amount of infinitely small rectangles. |
Yes, it exists as a concept but not as an 'entity'.
Think of it this way:
you can construct a simple polynomial - x+1. There is no justification for calling x+1 infinity because it is just one of the many polynomials one could construct for any given value of x (x+1, x+2, x+3...x+n). You cannot make x=infinity since x+1 would be larger. Thus there is no number which can be given the value infinity.
Any computer calculator or whatever would NEVER be able to display an infinite number. Because if some one was to write it down you'd need an infinit amount of paper (to print it) or an computer/calculator with infinite "bits" to "write it down" to display this number would need an infinit long computer screen or you need an infinit amount of time to scroll through it. So its logical you cant display it since it needs a infinit amount of matter (paper,chips etc.) so there's no way to display it.
In philosophy, some people used it like spinoza but this is science.
An example of the number ∞
We mostly use infinit numbers in math: Most well known function when this ocurs is.
F(x) = 1/x
when X becomes ∞
F(∞) = 1/∞
so F becomes infinitly small so the answer will always aproach 0 but never reach it. An `Asymptote`
Same when X becomes really small, 1/∞
You get an infinitely large number again ∞.
Since F(∞) = 1/(1/∞).
(1/1)/(1/∞) = (1/1)*(∞/1)
X/1 = X
So 1* ∞ =∞
In maths you cant devide something by 0 because:
10divided by 5 is 2 because
5times 2 = 10
10 divided by 0 is `x` would mean that
0 times x = 10
So x doesnt excist.
You learn this at school. But when you have a complex analysis you´ll get again (guess what).. ∞
As for ´∞ - 1 = ∞ its true that its ∞.
Fot more info http://en.wikipedia.org/wiki/Infinity#Infinity_symbol
o and who knows what happens when you do ∞/∞. what do you get? Infinite high or low number, R or 1?
Edited little mistake.
| Klaw 2 wrote: | Any computer calculator or whatever would NEVER be able to display an infinite number. Because if some one was to write it down you'd need an infinit amount of paper (to print it) or an computer/calculator with infinite "bits" to "write it down" to display this number would need an infinit long computer screen or you need an infinit amount of time to scroll through it. So its logical you cant display it since it needs a infinit amount of matter (paper,chips etc.) so there's no way to display it.
In philosophy, some people used it like spinoza but this is science.
An example of the number ∞
We mostly use infinit numbers in math: Most well known function when this ocurs is.
F(x) = 1/x
when X becomes ∞
F(∞) = 1/∞
so F becomes infinitly small so the answer will always aproach 0 but never reach it. An `Asymptote` |
We would write this, mathematically, with infinity as a limit (ie x→∞).
| Quote: | 10 divided by 0 is `x` would mean that
0 times x = 12 |
Surely 0 times x = 10? | Quote: | | As for ´∞ - 1 = ∞ its true that its ∞. |
It is indeed (or it would be if infinity were a number, but since infinity is not a number then the illustration only serves to show that the idea is impossible. The equation has no real merit mathematically, it is simply a demonstration of the ridiculous).
| Quote: | | o and who knows what happens when you do ∞/∞. what do you get? Infinite high or low number, R or 1? |
Again the 'sum' (or, I should say 'division') has no real meaning or merit, since you cannot treat infinity as a number and it therefore makes no sense to do arithmetic on it.
Using the 'division by zero' fallacy it is possible to do a number of spurious things in math. One of the most famous is the proof that 1=2.
Thus:
a=b=1 (a and b are equal to 1)
a^2 = ab (multiply both sides by a to give a squared is equal to a times b)
a^2 - b^2 = ab - b^2 (subtract b squared from both sides)
(a+b)(a-b)=b(a-b) (factorise the terms)
a+b=b (divide both sides by a-b) ***alert alert alert***
2=1
(of course you can choose different start values for a and b, in which case you have the general 'proof' that 2x=x)
| fpwebs wrote: | | I believe that infinity is ONLY available on computers! |
That is the only place I'm sure about where infinity is not AVAILABLE!!! Universe has much greater potential to be infinite, however the discussion on this topic is pure philosophy. Infinity is very helpul in physics and math. Without it, work in these fields would be virtually impossible. So I think something so useful has to be something real, at least in our mind:)
| redace wrote: | | fpwebs wrote: | | I believe that infinity is ONLY available on computers! |
That is the only place I'm sure about where infinity is not AVAILABLE!!! Universe has much greater potential to be infinite, however the discussion on this topic is pure philosophy. Infinity is very helpul in physics and math. Without it, work in these fields would be virtually impossible. So I think something so useful has to be something real, at least in our mind:) |
There seems to be some confusion between infinitesimal and infinity. Calculus uses the concept of the infinitesimal rather than infinity. In general terms infinity is bad news in math/physics. There are some occasions in which it is used as a tool but not many. When infinities start popping-up in physics it is nearly always an indication that either the theory is breaking down or that the data is bad.
There is an interesting example of infinite that has to do with a mathematical comfortable way of avoiding a singularity:
Consider the common case of two straight lines in a plane. Lets fix in the plane one of them and the other lets rotate by an angle theta around a fixed point in the plane, such that this point is not over the fixed straight.
Lets messure the angle theta from zero, with zero corresponding to the mobile straight being parallel to the fixed one. Then consider the following function:
c(theta)=1 if straights cross
c(theta)=0 if straights dont cross
in our example we would have c(0)=0, because when straights are parallel they don't cross
and c(theta!=0)=1 because when straights are not parallel, so they have to cross somewhere.
That gives us a very singular function. c(theta) is not well behaved in the point theta=0. it has all kind of discontinuities in that place. So, to avoid that, we make the following definition:
*** Infinite is where two parallel straights cross. ***
then we have to modify the c(theta) function
c(0)=1, because two parallel straights cross at infinity.
In this way we have now a comfortable continuous and well behaved c(theta) function.
| Bikerman wrote: | | redace wrote: | | fpwebs wrote: | | I believe that infinity is ONLY available on computers! |
That is the only place I'm sure about where infinity is not AVAILABLE!!! Universe has much greater potential to be infinite, however the discussion on this topic is pure philosophy. Infinity is very helpul in physics and math. Without it, work in these fields would be virtually impossible. So I think something so useful has to be something real, at least in our mind:) |
There seems to be some confusion between infinitesimal and infinity. Calculus uses the concept of the infinitesimal rather than infinity. In general terms infinity is bad news in math/physics. There are some occasions in which it is used as a tool but not many. When infinities start popping-up in physics it is nearly always an indication that either the theory is breaking down or that the data is bad. |
Yes I understand. It is bad when we need to do renormalization in Quantum Field Theory in order to get some realistic results and so on, but there are many situations in physics when you need to do some approximation and you suppose that some variable goes to infinity. It helps you to get some insight into the problem and then you can solve it more precisely. So I think infinity is not good in some situations, but can be of great help in others. And by the way the terms infinitesimal and infinity are very closely related in mind. When you get familiar with one of them you understand them both.
| redace wrote: | | Yes I understand. It is bad when we need to do renormalization in Quantum Field Theory in order to get some realistic results and so on, but there are many situations in physics when you need to do some approximation and you suppose that some variable goes to infinity. It helps you to get some insight into the problem and then you can solve it more precisely. So I think infinity is not good in some situations, but can be of great help in others. And by the way the terms infinitesimal and infinity are very closely related in mind. When you get familiar with one of them you understand them both. |
I think the point is, though, that we generally use infinity as a limiting case rather than an 'entity' or 'term' in the equation. Otherwise I agree with what you are saying.
Greg Chaitin*, amongst others, knows of the existence of 'infinities' bigger than either the countable infinity of integer numbers, ∞, or the uncountable infinity of reals, C = 2^∞ = 2 x 2 x 2 ...
His discovery of Ω = ∑'p halts' 2^(-|p|) as an incalculable, incompressible, unpredictable real number is interesting too.
source: http://fr.wikipedia.org/wiki/Image:Omega_among_reals_french.jpg
* http://www.umcs.maine.edu/~chaitin/
Dude, very nice picture. But my French isn't that good could you translate the words??
| Klaw 2 wrote: | | Dude, very nice picture. But my French isn't that good could you translate the words?? |
Probably. 
Parlez-tu pas,
"La Belles Franglaise"?
Je pensed,
qu'il compulsoire,
these days.
Well there are several ways to define it.
One is a practical way as to get an undeterminate scale for your model,
although this might not be the TRUE infinity, it is still a number too large to count.
This is the definition that I think is relevant for physical purposes.
but on a philosophical or mathematical angle...
The interesting thing is that Lebell said that infinity could represent N1 and another as N2
adding each infinity together N1 + N2 = infinity
But then you could go beyond this to N1 + N2 + N3.....etc.
or N1 * N2 * N3....and so on
or even better
N1 ^ N2 ^ N3...on to infinity.
Interestingly even the concept of infinity has scales.
A good book to read on this is "Infinity & the Mind" by Rudy Rucke
1. What is the count of stars in the Sky ?
2. What is the total capacity of water in earth - Provide the figure in Litres.
3. How many have you opened and closed your eyes right from your birth ? Or atleast for the past one week ?
4. How much time your mom would have called your name till now, right from your birth ?
5. What is the total time in seconds that you have wasted in your life ?
6. How many females have you seen in your life time ?
7. How many times have you taken bath ? [may be you can answer this] 
8. What is the total number of times you would have said a "hi" - "hai" or "hey" to your friends in your life time ?
9. What is total sleeping time till today - provide the figure in hours.
10. How many hair do you have now (atleast get me an figure on your head) ?
If you got a figure for any of these, then INFINITY does not exist man.... You are the perfect !!!!
Keep Smiling - Keep Living
| shakib wrote: |
but on a philosophical or mathematical angle...
The interesting thing is that Lebell said that infinity could represent N1 and another as N2
adding each infinity together N1 + N2 = infinity
But then you could go beyond this to N1 + N2 + N3.....etc.
or N1 * N2 * N3....and so on
or even better
N1 ^ N2 ^ N3...on to infinity.
Interestingly even the concept of infinity has scales.
A good book to read on this is "Infinity & the Mind" by Rudy Rucke |
Next Einstein !!!!
| indianinworld wrote: | 1. What is the count of stars in the Sky ?
2. What is the total capacity of water in earth - Provide the figure in Litres.
3. How many have you opened and closed your eyes right from your birth ? Or atleast for the past one week ?
4. How much time your mom would have called your name till now, right from your birth ?
5. What is the total time in seconds that you have wasted in your life ?
6. How many females have you seen in your life time ?
7. How many times have you taken bath ? [may be you can answer this]
8. What is the total number of times you would have said a "hi" - "hai" or "hey" to your friends in your life time ?
9. What is total sleeping time till today - provide the figure in hours.
10. How many hair do you have now (atleast get me an figure on your head) ?
If you got a figure for any of these, then INFINITY does not exist man.... You are the perfect !!!! |
i can be the perfect. ^_^
1. ~1500 visible on a clear night (7 x 10^22 in total)
2. ~1.20 x 10^21 L
3. i didn't count, but i could. a safe estimate would be between 60480~302400 times for the past week. It's not a big deal to multiply that by the number of weeks in my life. Hell, even for a 100 year old, that's only 5200 weeks, so between 0.31 x 10^9 ~ 1.57 x 10^9 for their life.
4. i was raised by wolves.
5. Including this one?
6. Oddly enough, far fewer than i would have liked, yet far more than i can stand. Women, eh?
7. ∝ the answer to 6.
8. Never. i always say "whaaaassssuuuuup?"
9. Not hard to give a ballpark figure for. But, let's be logical. Even if you lived to be a thousand, that's only around nine million hours. If you slept for all of that... it's still not that big a number.
10. Lots! Oh... on my head....
You know, just because it takes a lot of effort to count to a value, or getting the data for that value is difficult (because who counts eye blinks), that doesn't imply infinity. All of the eye blinks over the entire life of every person alive on the world today is probably just going to be on the order of 10^15. That's a lot. It ain't infinity.
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