posted
Can an infinite number be divided into fractions or proportions?
What I mean is, let's say I have an infinite number of marbles. Some of them are red, and some of them are green. Therefore, I have an infinite number of red marbles AND an infinite number of green marbles.
However, if you take a random sample anywhere in my vast field of marbles, you will find that the red marbles outnumber the green marbles two to one.
Or, if you arrange them in an infinite line, that line will go RED-RED-GREEN-RED-RED-GREEN, forever, into infinity.
Is that actually possible, mathematically?
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posted
If you have an infinite number of red and green marbles, how can one outnumber the other? Isn't infinity more of a place holder in mathematics?
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Two infinities are the "same size" if there is a one-to-one correspondence between them. For example, the set of perfect squares and the set of positive integers have a one to one correspondence, even though there are clearly many more positive integers than there are perfect squares for any set 1..n.
This is because every single perfect square has exactly one positive integer which is it's square root.
However, there are many infinities that do not have a one-to-one correspondence with positive integers:
quote:Cantor used construction like this to show that the set of points on a real line constitutes a higher infinity than the set of all natural numbers. One can see this intuitively when we consider the sequence of rational numbers between 0 and 1 (for example, 1/2, 1/3, 1/4, 1/101, 1/1,000,001). There are an infinite number of them, but always a gap between each value. For the real numbers on a line segment from 0 to 1 there is a continuum with no gaps, i.e. a larger infinity that rational numbers.
In other words, he showed that there were degrees of infinity. This fact runs counter to the naive concept of infinity: that there is only one infinity and this infinity is unattainable and not quite real. Cantor keeps this idea in this theory and calls it the Absolute Infinity, but he allows for many intermediate levels between the finite and the Absolute Infinity. We call these stages transfinite numbers, numbers that are infinite, but none the less conceivable.
In your case, I believe that there is a correspondence. There are only a finite number of red balls between any two green balls. I'm not sure if it would be two-to-one or one-to-one correspondence.
Edit: Drat! Beaten by CT.
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Therefore, for every Gn, there is one and only one Rn. Therefore there is a one-to-one correspondence. So there are just as many Gs as Rs in the infinte set.
But, for any n, there are aproximately twice as many Rs as Gs less than n.
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posted
Not really. You can have different orders of infinity but sets of the same order are the same size. So, for example, the infinite set of Real numbers is larger than the set of Integers, because there are an infinite number of real numbers between each pair of integers so the difference between them goes to infinity/infinity = 1. However, if I recall correctly, the set of positive integers is the same size as the set of all integers, because the difference between them goes to 2/infinity or 0.
edit: Of course, the correspondence thing works much better, I think so nevermind.
edit 2: I just realized I forgot the magic words: taking the limit. The differences involved where when you take the limit as it goes to infinity.
posted
Does the fact that the number of red and green marbles is equal have anything to do with the Axiom of Choice? If one rejects the AoC, would the sets still have equal measure? (Not that I know what I'm talking about, I'm just wondering).
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posted
I don't think so, because the red and green marbles are well ordered (at least as Geoff laid out the sequence), so the AC isn't necessary.
Edit: OK, the red and green marbles might be well ordered simply because there is an injectable function from the natural numbers. Which means that, according to this:
quote:If the set A is infinite, then there exists an injection from the natural numbers N to A (see Dedekind infinite).
the AC is required.
I'm officially out of my depth, by the way.
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I'm pretty sure that in this case, the axiom of choice doesn't change anything (since both sets can be indexed by integers.)
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While there are as many green marbles as red marbles, I think it's fair to say that the probability of picking a green marble (choosing at random) can be different from the probability of getting a red one. Certainly this is trivially true if you start with a finite number and just take the limit to infinity. I'm not sure if that was the original question, though.
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posted
There are also infinately countable and infinately uncountable set considerations
To answer your original question, no I don't believe infinite sets can be expressed in traditional ways such as fractions and proportions.
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posted
One way to look at the sets is: Green set = {set of all positive integers evenly divisible by 3} = {x| x>0, x is an integer, x=0 mod 3}
Red set = {set of all positive integers not evenly divisible by 3} = {x| x>0, x is an integer, x=1 mod3 or x=2 mod 3}
Both sets are countably infinite, and can be put into a 1 to 1 correspondance with each other or with the set of all positive integers. So the 2 sets are the same size, by definition of size of infinite sets. Yet intuitively it seems like the red set is twice as big as the green set.
Also, the sets satisfy this condition
quote: However, if you take a random sample anywhere in my vast field of marbles, you will find that the red marbles outnumber the green marbles two to one.
The probabilities are different from from the size considerations.
Take a more extreme example, primes. In the larger numbers, primes become very sparse or rare, yet there are an infinite number of primes as well.
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Exactly! Although technically there are "as many" prime numbers as non-prime numbers, since both sets are infinite, the chance of a randomly-chosen number being prime is relatively small, compared to the alternative.
If you could find the fraction that the ratio of primes to non-primes approached as you approached infinity, couldn't you state that fraction as being the proportion of numbers that are prime?
Similarly, couldn't I say that of the infinite number of marbles that I am imagining, one-third of them are green?
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posted
Isaac Asimov blew my mind with this in one of his books on math, years ago. If there are an infinite number of whole numbers -- and there are -- then since there are an infinite number of fractions between each whole number, the infinite number of fractions is in fact a full power larger than the infinite number of whole numbers. Posts: 37449 | Registered: May 1999
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quote:Originally posted by Puppy: Exactly! Although technically there are "as many" prime numbers as non-prime numbers, since both sets are infinite, the chance of a randomly-chosen number being prime is relatively small, compared to the alternative.
If you could find the fraction that the ratio of primes to non-primes approached as you approached infinity, couldn't you state that fraction as being the proportion of numbers that are prime?
Similarly, couldn't I say that of the infinite number of marbles that I am imagining, one-third of them are green?
As far as primes go, there is such a ratio or function:if P(N) is the probability that N is prime, for some large N, then P(N)~ 1 / ln(N) , with the approximation getting better as N-> infinity. In fact, P(N)->0 as N-> infinity. In spite of this, there are still as many primes as positive integers!
quote: In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers.
Roughly speaking, the prime number theorem states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime.
In other words, the prime numbers "thin out" as one looks at larger and larger numbers, and the prime number theorem gives a precise description of exactly how much they thin out.
In answer to the second question, I would say ...Yes? I guess. I'll have to think about that.Edit:I don't know. One difference is the relative scarcity of primes asymtotically approaches zero as N->infinity, while the scarcity of numbers evenly divisible by 3 is constant.
Anyway, sizing of infinite sets can be counter-intuitive, so don't get too wrapped up in using regular language to describe them, Geoff.
And now I'm close to exhausting my knowledge of analytic number theory, needed to prove the prime number theorem (calculus applied to plain-vanilla number theory is called analytic number theory.)
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quote:Originally posted by TomDavidson: Isaac Asimov blew my mind with this in one of his books on math, years ago. If there are an infinite number of whole numbers -- and there are -- then since there are an infinite number of fractions between each whole number, the infinite number of fractions is in fact a full power larger than the infinite number of whole numbers.
And yet the set of rational numbers (whole number fractions) is still a countable set, so you can't really say that the set of rational numbers is larger than the set of natural numbers. The set of real numbers, on the other hand, is larger than the rationals or the naturals.
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posted
I'm confused. If you have an infinite number of marbles, where is there room for anything else in the Universe if there is nothing but marbles? Wouldn't the spaces between the marbles even have to be full of marbles?
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If the universe is infinite, then it should have no trouble holding an infinite amount of marbles, plus an infinite amount of other stuff, too. It can even have an infinite amount of space between the marbles, I suppose.
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Depends on how big the universe is. Or maybe how big the marbles are.
Or not: if you have a finite universe and your marbles have to be at least a certain size, then you couldn't have an infinite number of marbles in the universe. It seems that by arranging things so you can have an infinite number of marbles, you are also arranging the possibility of however much free space you want.
Edit: Tante beat me to it.
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posted
To answer your question, Puppy, look at it like this: 3 out of every 10 marbles are green, while the rest are red. 30% of an infinite number is still infinite, yielding you the fraction of green marbles as infinity/infinity, which is not admissible in mathematics. The answer, then, is no, you can't divide an infinite number into fractions.
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I don't see why not. Infinity / infinity most certainly is permissible in mathematics; it's defined as a limit. In this case, the limit, as n goes to infinity, of 2n/3n, which is perfectly well behaved. Now, not all such limits are defined, but fractions are generally pretty forgiving.
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KoM, thats not dividing an infinite number into parts. Your just saying that the limit of an expression with respect to a variable it doesn't depend on is just that expression (lim(x->inf)y=y).
That there maybe a 2 to 3 ratio of red to green in any finite subset of the total set, is fine, but it doesn't effect the count of each color--they're both still infinite.
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As HollowEarth pointed out, that isn't a limit. The limit of an expression is determined as a variable approaches a given value. Sure, there are limits as, say, n goes to infinity, but in the cases of a fraction where infinity/infinity occurs, it is considered to be improper; that's the idea behind L'Hôpital's rule, for example (one of the conditions is infinity/infinity, the other 0/0). That rule exists for a reason, and the case of infinity/infinity is one of them.
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So if I have an infinite number if thingies, and some of those thingies have one trait, while others of those thingies have another trait, then statistically, it is impossible for those traits to occur in different proportions? They are always exactly 50/50 because both are infinite?
That doesn't strike me as applying very well to the example of prime numbers described above.
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quote:Originally posted by A Rat Named Dog: So if I have an infinite number if thingies, and some of those thingies have one trait, while others of those thingies have another trait, then statistically, it is impossible for those traits to occur in different proportions? They are always exactly 50/50 because both are infinite?
That doesn't strike me as applying very well to the example of prime numbers described above.
Not quite. Going back to what King of Men said, say 2/3 of the marbles are red; then 2n/3n marbles are going to be red for every possible value of n. The thing is, when you take the limit, you're supposed to simplify the fraction as much as possible, so the n's cancel out, leaving you with only the proportion of red marbles (2/3), not the actual number of them (which would be infinite if n went off to infinity).
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posted
The illustration of the infinite line of multi-colored marbles is an example of fractional infinity. A simpler illustration would be an infinite dotted line, with alternativing spaces and dots or hyphens.
For example, suppose you start with an infinite line, and then you stipulate that every other inch, a one-inch section of the line is erased, making alternate lines and spaces. And this goes on to infinity.
Then a couple of even better questions would be: Who created the dotted line? How was infinity reached, so that it could be said the dotted line was infinite?
Are we getting a little Zenish, here?
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quote:Originally posted by HollowEarth: That there maybe a 2 to 3 ratio of red to green in any finite subset of the total set, is fine, but it doesn't effect the count of each color--they're both still infinite.
Well, I realise that, but that's not the question I was answering. As I understood Geoff's question, it was 'can I have an infinite amount of red and green marbles such that the probability of drawing a green marble is not 50%?' And the answer is yes, because such a probability could only be defined in terms of a limit. And incidentally, infinity over infinity is definitely not an improper limit; that kind of thing is what limits were invented to deal with. You might as well say that addition doesn't apply to whole numbers.
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quote:That there maybe a 2 to 3 ratio of red to green in any finite subset of the total set, is fine. . . .
Ah, but nobody has stated this (and you probably mean a 2 to 1 ratio).
Here's one to chew on. Since we're agreeing that one of every three marbles is green and two of every three marbles is red, we'll equate them to positive integers n from one to infinity, where n being divisible by three implies red, and not divisible by three implies green. Thus 1,2,4,5,7,8 are green and 3,6,9 are red, etc.
I'm going to reorder them now, as follows: 1,3,2,6,4,9,5,12,7,15,8,18,10,21,11,24,13,27,.... Since the list is infinite, it will include all the positive integers. But in my sequence, every other marble is red. So, what are the chances that a random marble chosen from my list is green?
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posted
One-half, but you've relaxed the original criterion of any random sample having a two-to-one ratio, so it's no longer the same limit being taken.
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quote:Originally posted by King of Men: And incidentally, infinity over infinity is definitely not an improper limit; that kind of thing is what limits were invented to deal with. You might as well say that addition doesn't apply to whole numbers.
Your analogy doesn't hold water, though. I'm not sure where you're getting your source, but infinity/infinity is an improper limit; as I pointed out, that's half the purpose behind L'Hôpital's rule, so that you can actually evaluate a limit that results in infinity/infinity. This helps to explain what I was getting at here: http://mathworld.wolfram.com/Indeterminate.htmlPosts: 143 | Registered: Sep 2005
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quote:However, if you take a random sample anywhere in my vast field of marbles, you will find that the red marbles outnumber the green marbles two to one.
Although I did have the ratio wrong.
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No, Eldrad, you are misreading that site. It is saying that indeterminate limits can fall into seven kinds, one of which is infinity over infinity; it is not saying that all limits of that form are indeterminate. Let me again mention the example of limit, n to infinity, 2n/3n. Now clearly this is, indeed, infinity over infinity, but it is perfectly well defined.
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Puppy: The answer is YES you can have proportions and it all lies in how you think of infinity. What so many people get wrong is that when infinity comes up, they just think of an infinitely big number. But the proper way to think of infinity is to take the limit of a finite number as it keeps growing and never stops. So in your example, yes the marbles can be in a ratio even if it is infinitely big because the mathematical way to think of it is to have a finite number of marbles, and then you keep adding the marbles in a certain ratio on and on and on....so at any point, the ratio will be the same.
Another famous example is the extent of the universe. People wondered that if everything has a gravitational attraction then why doesn't everything just collapse into one point. Newton reasoned that if the universe was infinitely big, then every attraction would cancel out. Because he was so famous, he was also famously wrong; he thought of infinity in the faulty way--instead you need to think of a finite amount of particles, which WILL collapse to one point and then add a particle until there are an "infinite" amount of particles. At any point in the adding of particles, there will always be a central point to collapse on so even in an "infinite" universe the stars would still come together centrally.
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Yeah, King and Hollow -- I missed that in the first post. I'm now trying to determine how one can guarantee that a random sample can have a certain ratio of green to red marbles, and I don't think it can. To use a simple example, choose one marble. What color is it?
I think you'd have to say that any infinite subset of this infinite field of marbles has that ratio, and then we're back to the question of whether or not the ratio can be achieved in the first place. And I suspicion that to guaranteee the ratio (if it's possible), we'd have to be looking at uncountably infinite subsets of an uncountably infinite set even to have a chance (and marbles are notoriously countable).
Indiana, I'm not sure what you're trying to say, but I have a pretty decent understanding of infinity (at least in n-dimensional Euclidean space), and what you're saying doesn't make sense. That seems like exactly the wrong way to think of an infinite universe. Maybe you can try explaining it again in different words, 'cause I'm not getting it.
posted
Eldrad, look at the limit he's doing. The n's cancel, so its just the limit of 2/3 with respect to n, which is just 2/3.
I think I'm gonna have to agree with papa moose, (and the first part of Dag's second post), in that no you can't.
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posted
Aleph-0 (pronounced "aleph-null") infinity is the number of integers.
Aleph-1 is the number of reals; Cantor's argument, above, shows it's more. Aleph-1 is also the size of the power set of integers. The power set is the set of all subsets: {0}, {0 1}, {0 2}, {1 2}, {0 1 2}...
Aleph-2 is the size of the power set of reals, or the size of the power set of the power set of integers.
And so it can go on indefinitely.
Now if we could bundle them all together and call them {Aleph}, could we then show that there are different sequences: aleph, beth, gimel, etc.? Don't know.
But that wouldn't have any effect on the marble example, of course. There are infinitely many reds and infinitely many greens, and there are also twice as many reds as greens. Aleph-null, since it's equivalent to the integers.
Posts: 1877 | Registered: Apr 2005
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I think your original question has been answered Geoff (FYI, the answer that you can have a non-50% proportion is correct as long as you take a limit).
Note that cardinalities of sets are almost a completely different issue. For instance, consider the probability of randomly picking a number out of every whole number that is divisible by 10. Obviously, the probability of doing so is precisely 10%. Yet the cardinality of numbers divisible by 10 is exactly the same as cardinality of numbers not divisible by 10. This is becaues cardinality touches on a slightly different issue when you're dealing with infinite sets (for finite sets, cardinality actually overlaps), and the probability you're looking for is more closely related to ordinal numbers than cardinal numbers.
Now we can actually get a very interesting result when you consider the probability of picking an extremely unlikely number. For instance, in the prime number example, according to the Prime Number Theorem which Morbo linked to, the probability of picking a random prime number from the set of all counting numbers is equivalent to evaluating the limit at infinity of 1/ln(n), which is 0 because the natural logarithm tends to infinity. So the probability of picking a prime number is actually 0% - not approximately 0% : that is not rounded down, it is actually exactly 0%. And yet the possibility certainly exists that if we pick a random number, that number might happen to be prime! This is a neat demonstration of the fact that to a theoretical mathematician, 0% does not necessarily imply "impossible" nor does 100% imply "always going to happen" - when you're dealing with infinitely likely or unlikely events, that is!
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