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?
Posted by Stephan (Member # 7549) on :
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?
Posted by Puppy (Member # 6721) on :
That's my question Posted by ClaudiaTherese (Member # 923) on :
Some branches of mathematics admit of different "sizes" of infinity, comparatively speaking.
Edit: I'm thinking of Cantor's diagonal theorem and implications for calculus in particular.
Posted by Stephan (Member # 7549) on :
I don't think there really is an answer then, at least that our primitive minds could grasp.
Posted by vonk (Member # 9027) on :
ok, here's the answer: no. (i don't really know, but if i were to guess, that would be it.)
Posted by Dagonee (Member # 5818) on :
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.
Posted by ClaudiaTherese (Member # 923) on :
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.
Posted by MrSquicky (Member # 1802) on :
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.
[ March 17, 2006, 04:09 PM: Message edited by: MrSquicky ]
Posted by ClaudiaTherese (Member # 923) on :
Philosophy of Logic classes: they're what's for dinner. *grin
Posted by Stephan (Member # 7549) on :
I liked my calculus course so much in college, that I took it twice.
Posted by Dagonee (Member # 5818) on :
To generalize,
Suppose there are r red balls between each pair of green balls.
For any finite r, there are the same number of green and red marbles in the infinite set.
If r is infinite, then the red marbles represent the real numbers between two integers, which is an infinite set in and of itself.
So the set of red balls would be an infinite number of infinities (to use some sloppy language).
Posted by Dagonee (Member # 5818) on :
quote:I liked my calculus course so much in college, that I took it twice.
By that standard, I LOVED differential equations.
Posted by SenojRetep (Member # 8614) on :
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).
Posted by Dagonee (Member # 5818) on :
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.
Posted by HollowEarth (Member # 2586) on :
I'm pretty sure that in this case, the axiom of choice doesn't change anything (since both sets can be indexed by integers.)
Posted by King of Men (Member # 6684) on :
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.
Posted by BaoQingTian (Member # 8775) on :
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.
Posted by Morbo (Member # 5309) on :
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.
Posted by Puppy (Member # 6721) on :
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?
Posted by TomDavidson (Member # 124) on :
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. Posted by Morbo (Member # 5309) on :
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.)
Posted by Mike (Member # 55) on :
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.
Posted by Mike (Member # 55) on :
Not sure this was intentional or not, but Posted by Kwea (Member # 2199) on :
Hey Mike!
I should have know you would turn up in this thread.
BTW, I hate you all, I now have a headache but still don't understand what you are talking about. Posted by A Rat Named Dog (Member # 699) on :
Whoah, I could follow Dagonee's link again and again and again ... it's infinite!
Posted by Dagonee (Member # 5818) on :
It wasn't intentional, but it's darn funny.
Posted by human_2.0 (Member # 6006) on :
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?
Posted by Tante Shvester (Member # 8202) on :
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.
Posted by Mike (Member # 55) on :
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.
Posted by Eldrad (Member # 8578) on :
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.
Posted by King of Men (Member # 6684) on :
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.
Posted by HollowEarth (Member # 2586) on :
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.
Posted by Eldrad (Member # 8578) on :
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.
Posted by A Rat Named Dog (Member # 699) on :
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.
Posted by Eldrad (Member # 8578) on :
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).
Posted by Ron Lambert (Member # 2872) on :
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?
Posted by King of Men (Member # 6684) on :
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.
Posted by Papa Moose (Member # 1992) on :
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?
Posted by King of Men (Member # 6684) on :
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.
Posted by Eldrad (Member # 8578) on :
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.html Posted by HollowEarth (Member # 2586) on :
Check the first post again papa moose.
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.
Posted by King of Men (Member # 6684) on :
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.
Posted by JustAskIndiana (Member # 9268) on :
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.
Posted by Papa Moose (Member # 1992) on :
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.
And welcome to Hatrack.
--Pop
Posted by HollowEarth (Member # 2586) on :
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.
Posted by Will B (Member # 7931) on :
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.
Posted by Avin (Member # 7751) on :
Great thread!
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!
Posted by Avin (Member # 7751) on :
WillB, you seem to be confusing the Alephs and Beths. I've seen people do this before, but here is a more accurate definition of the cardinalities:
Aleph_0 is defined to be the cardinality of the smallest infinite set. You can prove that the set of natural numbers is such a set, therefore its cardinality is Aleph_0 (it is often also pronouned "Aleph-nought").
Aleph_(n+1) is defined to be the cardinality of the smallest infinite set that is greater than but not equipotent to Aleph_(n). So for instance, Aleph_1 is the next cardinality higher than Aleph_0.
Note that the subscripts (0,n,n+1) for the Alephs are not cardinal numbers themselves; they are ordinal numbers. Hence, there is an Aleph_0, Aleph_1, Aleph_2, .... and infinitely so, then Aleph_(omega) [omega is the smallest infinite ordinal number, whereas Aleph_0 is the smallest infinite cardinal number], Aleph(omega+1), Aleph(omega+2)... and infinitely so, then Aleph_(2omega) ....and on and on with infinite hierarchies of infinities.
Now, Beth_0 is defined to be the same as Aleph_0.
Beth_(n+1) is defined to be the cardinality of the power set of Beth_(n). Thus for instance Beth_1 is the power set of Beth_0, which we know is equipotent to the Continuum (Real Numbers), therefore Beth_1 is the cardinality of the real numbers.
Now, presumably the Aleph hierarchy and the Beth hierarchy overlap somehow. The problem is, within ZFC set theory you cannot determine how they overlap. One hypothesis that has been suggested is the Beth_1 = Aleph_1 (in other words, that Aleph_1 is the cardinality of the reals). This is called the Continuum Hypothesis. Your post, Will B, seemed to assume that it is true. However, it is important to note that there is some mathematical work done assuming that the Continuum Hypothesis is false. It can be shown that the Continuum Hypothesis is independent (that is, can neither be proved nor disproved) from ZFC set theory. The same applies to the more generalized version of the Continuum Hypothesis (known as GFC) which states that Aleph_n = Beth_n for all n.
Posted by human_2.0 (Member # 6006) on :
Where did all the marbles come from? I would like an infinite amount of Legos. Now, I wonder what the percentage of red bricks there would be...
Posted by Eldrad (Member # 8578) on :
quote:Originally posted by Eldrad: 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).
If you would have read my previous posts, you'd see that I already spoke of what you just pointed out. I'm not misreading anything.
Posted by Will B (Member # 7931) on :
I pulled the name "beth" out of the air (or else the Hebrew alphabet), not knowing it was already in use . . . ok, I'll use Aleph-prime. Unless that one's taken too, in which case I'll call it zignabrump. I'm pretty sure that has no previous definition!
Posted by BaoQingTian (Member # 8775) on :
Avin- I loved the 0% prime numbers example. If I remember right in my 4000 lvl math proofs class (I forgot the course name), we proved that the probability of picking an integer from the set of all real numbers was in fact 0% as well. I may be wrong though, it has been a little while.
Posted by Avin (Member # 7751) on :
That's correct, but for an almost different reason. In this case, the cardinality of the two sets is different, so it's much more straightforward.
Posted by The Rabbit (Member # 671) on :
This has been pretty thoroughly addressed in set theory. There are different sizes of infinity but its not intuitively obvious whether one infinite set is bigger than another infinite set.
For example the number of even integers is the same as the number of integers (both odd and even) which is the same as the number of rational numbers. The number of irrational numbers is however bigger than the number of rational numbers. What's more, the number of irrational numbers between any two rational numbers no matter how close, is greater than the number of rational numbers.
Posted by Dagonee (Member # 5818) on :
I remember when we did the proof that there were "more" irrationals than rationals. It was very confusing* - up until them, the only irrationals I ever thought about were pi, e, phi, and roots - mostly square roots, but I knew there were others.
Of course, had I thought about it, I would have realized that just the irrational roots would be a "bigger" infinite set, nevermind all the other irrationals one can make.
There are a lot of infiinite, non-repeating series of the digits 0-9.
It's just one of those blindingly obvious things one doesn't think about until confronted with it.
*By confusing I meant "upsetting to my mind's view of the number system," not hard to understand.
Posted by rivka (Member # 4859) on :
quote:Originally posted by Will B: aleph, beth, gimel, etc.?
Oh, look! A part of this thread that doesn't make my head hurt! Posted by BaoQingTian (Member # 8775) on :
Hehe, Rabbit said "intuitively obvious." If you ever want to teach higher math, engineering, or sciences at a university, there's a secret requirement that you must use that phrase in your lectures. Or did you have to swear an oath to use it when you got your doctorate? Posted by HollowEarth (Member # 2586) on :
Ha. My quantum professor like to tell us that "you'll show this on your homework, but don't worry its easy." HA.
Posted by Althai (Member # 9275) on :
quote:Originally posted by Puppy: 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.
To be really clear mathematically as to what you mean by a "random sample", you need to define what you mean by random - you need to define a procedure for choosing a number at random. Traditionally, what you do is define a "probability measure", which assigns to every subset a probability in a way which is compatible with the laws of probability (in this context, this means things like the chance of picking an even number + the chance of picking prime number - the chance of picking 2 = the chance of picking a number which is either even or prime.) But here's the rub - there is no "invariant probability measure" on the natural numbers. In other words, there is no way of randomly choosing natural numbers that treats all natural numbers equally. So given a countably infinite collection of marbles, you can't use probability as a way of saying that some fraction of them are red, green, or otherwise, in a way that treats all marbles on equal footing. Of course, you can put them in an order, and say that in the given order, you alternate two red marbles and one green marble. Then, if you choose any marble in the order, the fraction of red marbles before that marble in the order will be about 2/3. But doing so puts different marbles on different footing (because they have different positions in the order.) So without specifying an order to the marbles, or some other way of treating some marbles differently than others, there's no way of talking about the fraction of marbles which is red as opposed to green. The exception to this would be if there were only finitely many red marbles, in which case it would make sense to say that the fraction of red marbles is 0, or if there were only finitely many green marbles, in which case it would make sense to say that the fraction of red marbles is 1.
quote: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?
As mentioned before, this makes perfect sense, mathematically. However, as someone already pointed out, when you talk about the fraction of red marbles in this way, it's not really well-behaved, in the following way: In this order, it seems perfectly clear that there are two red marbles for every green one. However, you can rearrange the marbles in line, and get two green marbles for every red one, or one thousand green marbles, or even infinitely many. (You could rearrange them so that there was one green, then one red, then two greens, then one red, then three greens, then one red, etc. Then as you went further and further along in the list, the ratio of green marbles to red marbles would eventually exceed any fixed number.)
So, while there are ways you can make sense of ratios of infinities as mentioned before, it really doesn't make sense to speak of the fraction of red marbles in an infinite collection of red and green marbles, when the red and green marbles have the same cardinality. However, there are definitely different sizes of infinity, as first discovered by Cantor, and as explained in various links already given.
If this doesn't make sense, I encourage you to think about it. There is a lot about these ideas which is very counter-intuitive and bewildering. As an example of just how non-intuitive infinity can be, I offer the following though experiment, which I stole from somewhere:
Imagine that you have a funnel which contains infinitely many ping-pong balls, and the balls are numbered 1, 2, 3, 4, 5, etc. The funnel is hanging over a barrel, so that the balls will pour into the barrel. The ping-pong balls are arranged so that the lowest numbered balls are near the bottom of the funnel, and at the beginning of our experiment, they start pouring out, in order, at an increasing rate. During the first 30 seconds of our experiment (which will only last one minute), 10 of the balls pour out, numbered 1-10. During the next 15 seconds, another 10 pour out (numbered 11-20). Then during the next 7.5 seconds, another 10 pour out (21-30). It goes on like this, with each set of 10 balls pouring out in half of the time it took the previous set, so that after 1 minute has elapsed, all of the balls have poured out of the funnel and into the barrel.
Yes, we have arranged it so that in infinite number of events have happened in a finite period of time. We can do this, because the events occur faster and faster as we go, and 1/2 minute + 1/4 minute + 1/8 minute + 1/16 minute + ... = 1 whole minute. (Just picture the second hand moving - as it moves around the clock face, it first goes half way, and then half of the remaining distance, and half of the remaining distance, and half of the remaining distance again, and so on.)
While this is going on, a tiny gremlin is chucking balls out of the barrel. During the first 1/2 minute, he chucks out ball #1. During the next 1/4 minute, he chucks out ball #2. During the next 1/8 minute, he chucks out ball #3. And so on. So, during the first 1/2 minute, 10 balls enter the barrel, and 1 ball leaves, so there are exactly 9 balls in the barrel at the 30 second mark. During the next 1/4 minute, the same thing happens, so 45 seconds into the experiment, there are 18 balls in the barrel. After 52.5 seconds, 27 balls are in the barrel. In fact, there will be 9 billion balls in the barrel exactly 1/2^1,000,000,000 second before the end of the experiment. In fact, given any large number, at some point there will be at least that many ping-pong balls in the bucket. So here is the question: after the experiment ends, when the whole minute has elapsed, how many balls are in the barrel?
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*obligatory space before hint*
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Hint: what balls are in the barrel?
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*obligatory space before solution*
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Solution: there are no balls in the barrel. Why? Because during the course of the experiment, every ball drops out of the funnel and into the barrel, and is then later chucked out by the gremlin. Take, for example, ball #287,563. During the 28,756th time interval (that is to say, during the 2^28,755th 1/2^28,756th of a minute), this particular ball will drop into the bucket. Then, during the 287,563rd time interval, the gremlin will pluck it out of the barrel and toss it over his shoulder. So any given ball is no longer in the barrel at the end of the experiment, and therefore, although at every point in the experiment the number of balls in the barrel is increasing, there are, in fact, no balls remaining at the end.
David
Posted by Bokonon (Member # 480) on :
So my question is, math geeks:
Which set is larger, the set of infinite sets, or the set of non-infinite sets?
-Bok
Posted by Althai (Member # 9275) on :
quote:Originally posted by Bokonon: So my question is, math geeks:
Which set is larger, the set of infinite sets, or the set of non-infinite sets?
-Bok
Technically speaking, neither of those are sets at all. Naive set theory (as opposed to axiomatic set theory) was ended by Russel's Paradox. If we allow all possible collections to be considered as sets, then we get get such objects as the "set of all sets", which obviously contains itself, since it is a set. We could then consider the "the set of all sets which contain themselves" and the "set of all non-self-containing sets." Russel's paradox is the question "is the set of all non-self-containing sets self containing"? This is a question with no answer, and led to the formulation of axiomatic set theory, which restricts what we consider to be a set.
Formally speaking, the objects you mentioned are "classes", and not sets. So we have a class of infinite sets, and a class of finite sets. Classes may be sets - one could speak of the class of natural numbers, which is a set - but not all classes are sets. However, classes, unlike sets, do not have cardinalities. I do not know if there is an equivalent sense of "size" for a class, but if a class is a "proper class" (is not a set), then it is bigger than any set (according to the standard mathemeticians give for two collections having the same size, which is whether there exists a 1-1 correspondance.) Since the two "sets" you gave are in fact proper classes, there are a lot of members of both, but I don't think there is a way in which you could say there is more of one than the other.
David
Posted by Mike (Member # 55) on :
Excellent. Nice explanations, Althai. And welcome to Hatrack! Posted by Bokonon (Member # 480) on :
Oh sure, come up with a nifty question, and you just change semantics on me!
-Bok
Posted by SenojRetep (Member # 8614) on :
quote:Originally posted by Althai: While this is going on, a tiny gremlin is chucking balls out of the barrel. During the first 1/2 minute, he chucks out ball #1. During the next 1/4 minute, he chucks out ball #2. During the next 1/8 minute, he chucks out ball #3. And so on. So, during the first 1/2 minute, 10 balls enter the barrel, and 1 ball leaves, so there are exactly 9 balls in the barrel at the 30 second mark. During the next 1/4 minute, the same thing happens, so 45 seconds into the experiment, there are 18 balls in the barrel. After 52.5 seconds, 27 balls are in the barrel. In fact, there will be 9 billion balls in the barrel exactly 1/2^1,000,000,000 second before the end of the experiment. In fact, given any large number, at some point there will be at least that many ping-pong balls in the bucket. So here is the question: after the experiment ends, when the whole minute has elapsed, how many balls are in the barrel?
But, if we state instead that in the first 1/2 minute the gremlin throws out ball #1 and in the next 1/4 minute he throws out ball #11 and in the next 1/8 minute he throws out ball #21, ad infinitum, <edit> note, this preserves the "10 in, 1 out" per time period </edit> do we end up with an infinite number of balls in the barrel?
Posted by Althai (Member # 9275) on :
Yes, then we do. It's clear that when the experiment is complete, we have every ball numbered something not ending in 1. We could also end up with any desired finite number of balls in the barrel, say by having the gremlin throw out the lowest numbered ball that's not #12, #37, or #55.
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