There are an infinite number of points between 0 and 1.
There are an infinite amount of points between 0 and 2.
The distance between 0 and 2 is twice that of 0 and 1.
Therefore the infinity between 0 and 2 is twice as large as the infinity between 0 and 1.
Is there anything wrong with this?
My prof. said that it is flawed, and that there are an equal ammt. of points between 0-1 and 0-2, however his explanation was lost on me. His claim, though, seems to fly in the face of logic
Also, he said that the only comparison that you can do is on infinite orders of magnitude. ie, the number of points between 0 and infinity is infinitly bigger then the number of points between 0 and any number n.
OH MY GOSH I had this EXACT idea a little while ago, I can’t belive someone else did too. I think you’re right man, and did you ever think of applying it to time? If you accept this as true, then there is an infinite number of infinitely small pieces of time in one second or in one decade. makes you really see…i don’t know…something. I’ve also thought about why we, as people, can’t really comprehend infinity, and I think it’s because we’re so surrounded by it. I mean, if you want someone to give you a really good description of what “cold” is, you don’t ask someone who’s been cold all their life. You find someone who’s been hot all their life and then expose them to cold and see how they describe it. I think we’re the same way. Since there is always an infinite amount of time going by and infinitely small and large spaces all around us at all times, we’re too immersed in infinity to be able to describe it. We’re too “cold”, so to speak. Hope I didn’t digress too much there, I’m just so excited that someone had this same idea. If someone would explain why they think this idea is flawed, I’d really appreciate it, because I really don’t see any flaws at all
Your professor is correct when he says the number of real-valued points between 0 and 2 is the same as the number of real-valued points between 0 and 1. If he said that the number of real-valued points between 0 and infinity is infinitely bigger than the number of real-valued points between 0 and 1 then he was wrong, they are sets of the same size.
All of this is based on the work of Georg Cantor, in case you are interested.
Cantor proved that the set of rational points between 0 and 1 is in 1-to-1 correspondence with the set of all rational points in n-dimensional space, for arbitrarily large n.
Well, you can map any point in 0-1 to any point in 0-2, with the simple formula x = 2y, where x is a point in 0-2, and y is a point in 0-1. Every point in 0-2 has a correspondence in 0-1, and vice versa, with no duplicates. There is an exact 1-1 correspondence. Therefore, we can say that there are the same number of points in both segments.
Muad’Dib: you are trying to apply your common sense intuitive notion of comparison of magnitudes to infinite sets : unfortunately common sense breaks down in such cases. You have to abandon your intuition and agree that two sets have the same cardinality (informally, are the same “size”) if their elements can be put in one-to-one correspondence. Think of a line as an (infinite) set of points. Another example is two concentric circles: you can draw radii through both circles, making each point on the inner circle correspond to one point on the outer cirlcle. Therefore both circles have the same number of points.
youreblurry - I believe current scientific thinking says that time may be “granular” rather than continuous, with the smallest unit being the “Planck Time” (this has been discussed quite a bit on this forum). Or at least, our laws of physics can’t tell us what happens at time intervals shorter than the Planck time. Hopefully Chronos and ultrafilter will be along to explain better than I can - though you might want to search the archives since these subjects have been debated quite a lot here.
Could it just be that your common sense is just wrong? It makes a whole lot more sense mathematically to claim the two infinities the same, than to claim that one is twice as large as the other. Why should two lines being made up of the same number of points mean they’re the same length?
Infinity is a theoretical value, not set by numbers. By any number, you may draw the same amount of digits, and, therefore one number, by its value, may be as accurate as another. By saying infinite, you may give any accurate value of any number, large as it may be (therefore you are quantifying it), but infinity is still bigger.
To describe different types of infinity is more difficult though. One, is that any number may be evaluated to infinite accuracy, which means that you will never know the true value of a number, unless it is exact ,such as 0.3333333333 ad infinitum, or 1.0000…000 (which is a rare case in science, both of which are fractions). By your 0-1-2 case, all three (0-1, 1-2, 0-2) series are equally “infinite”. There are even others which are not so easy to describe, such as “pi”, who, by definition, have no end.
The other is in terms of powers of infinity, such as infinity^3 which is infinitely larger than infinity^2, because it is infinity^2 times infinity. Was the original infinity^2 just as large as the infinity that you are now multiplying it by? All we know, is that mathematically, infinity^8, is greater than infinite^7. This is basically the childish arguement that infinity + 1 will always beat infinity, except extended. Infinity times infinity, is , logically, infinitly larger than anything you can imagine. This is all really theory, and nothing you can imagine is as big or small as you think it really is.
Infinity is NOT an exact number, and therefore you cannot say that one infinity is bigger than another infinity, except in terms of other infinities. That is like saying that you went out to the edge of infinity, and then you went out even further, which is false. The truer statement of this is that there is no end to infinity, only that that one infinity is bigger than another.
Woops! My last post was in reference to Flymaster not DarrenS.
Grrr… I still don’t like that! It just seems terribly irrational, and like I said before, the irrational has no place in Mathmatics. I see what you are trying to say with you examples, but the conclusions still leave me feeling quesy. I had a geometry teacher tell me the same thing before years ago, and I thought that it just seemed wrong then too.
Is it that I must just accept it, or that the proofs and ideas behind it are too complicated for the layman?
I’m heading for the bed, but I have a little time to toss out a few relevant things for you to think about.
First of all, what is a set? Actually, I supppose that’s a somewhat philosophical question more than anything else, but I’m trying to keep it simple here. A set is nothing more than a “collection of objects”, right?
What is a line segment? Again, I’m sure you have a pretty clear concept of what a line segment is–it’s…well, it’s just a finite piece of a line, right?
Notice that a line segment is much more than just a set, however. Certainly, we can think of a line segment as being a set (just the set of points in the line segment), but we can think of line segments in many more ways. Take two points on a line segment, a and b. We can ask, “Is a farther to the left of b, or farther to the right? Or maybe a and b are actually the same point, so neither is left or right of the other.” We can also ask, “How far is it from a to b?”
The point I’m hoping that you see is that a line segment has much, much more structure than a simple old set. A set is…well, just a set (of objects). A line segment is also a set, but a set that has been put into a very special order.
How would you describe the size of a line segment? Well, a very natural way, as you’ve already mentioned, would be to describe the size of line segment as its length.
Now, how would you describe the size of a set? It obviously isn’t length (what the hell is the length of a set?). But, again, there’s a very natural way to describe the size of a set–describe it by the number of elements it has.
It should be obvious that 1. and 2. are quite different ideas; it’s important not to confuse the two. But, as odd as it may seem, it’s entirely possible for two different line segments to be the same size (as in sense 2.) and yet not be the same size (as in sense 1.).
When we talk about the sizes of various infinities, we’re talking about 1. (how many elements are in this set), not 2. (not the length of some line segment that we may be able to construct out of our set, which is something entirely different). The interval [0,2] is twice as big as the interval [0,1] (i.e., has twice the length), but, in fact, the intervals [0,2] and [0,1] have the same number of points.
Anyway, I’m running out of steam, and I haven’t even began to really talk about the sizes of infinite sets (cardinalities). In fact, you are right about one thing–there are different sizes of infinities, just not in the examples you’ve given here. Somebody else can take care of that (or read the thread DarrenS provided); I just wanted to point out the inconsistency of the “size of a set” versus the “size of a line segment”.
It’s just that if you don’t accept it, you have to accept that two sets could be in 1-to-1 correspondance but not have an equal number of members.
In other words, you have to accept the following:
For every thing that is an A there is exactly one thing that is a B, and for every thing that is a B, there is exactly one thing that is an A, but there are somehow still more Bs than As!
So choose. Either you accept the above statement, or you accept that there is no connection between a length and the amount of points that reside within it. Specifically, all lengths have the same amount of points within them.
Oh, and just having noticed your last reply to DarrenS, about the idea he gave as seeming to be irrational, here’s a question to think about, which may help:
Think of the idea that I mentioned–the size of a set as being the number of elements in the set.
It’s easy enough to compare two finite sets. What about infinite sets? Can you think of a way of comparing the number of elements in one set with the number of elements in another set (when both sets are infinite)? What could it mean for one infinite set to have more (or less) elements than other infinite set. (Possible helpful hint: Think about how you compare the sizes of finite sets. Can you extend this idea so that you can compare infinite sets?). And to get anything out of this, don’t think of it as just a bullshit question, actually try thinking of some creative (and consistent) way(s) of doing this. Admittedly, it’s not an easy question, but I, and I’m sure others, will be here to guide you along.
Thinking like this will help you see the light.
What I’m trying to get at is this:
Neither. It’s that you must attempt to understand where these concepts came from (they don’t just pop up out of the blue;)). What was Cantor thinking when he came up with his ideas? What motivation did he have for his thoughts, and for doing things the way he did?
I have something to add which is not intended to make this more difficult; it’s intended to clarify what we’re talking about. That is, there is more than one concept of infinity in mathematics. The most robust, and the one we’re working with here, is the infinity that shows up in set theory. However, there’s also an infinity defined in calculus. At least, we can speak of limits at infinity, which are well-defined even if infinity itself is not. One such case of an infinite limit which is so subtle that it’s easy to overlook is the decimal representation of a number, such as moe.ron’s 0.3333…, or 3.14159…
I just think it’s important that we don’t confuse our infinities. But so far, it’s been pretty consistent. Infinity is the “number” of elements of a set which is larger than any finite set.
One thing that Cabbage just said made me think of something. If we say that A is a proper subset of B, we mean that B contains every element that A contains, and there are some elements of B that are not elements of A. With finite sets, if A is a proper subset of B, then B necessarily has more elements than A, and it makes sense to say that B is bigger. With infinite sets, well, I’m not sure what we do with this concept, but it’s not true anymore.
I wanted to mention something here for Muad’Dib’s benefit (hopefully, anyway) about this.
Back to where I left off, about trying to think of a way of comparing the sizes of infinite sets. I wanted to mention that the above quote is a perfectly valid attempt at trying to do this. (Remember, we’re approaching this in the spirit that it’s never been done before (or I am anyway, I don’t know about anybody else, actually, but I’m trying to give Muad’Dib a picture of where mathematical ideas such as this come from). As far as I’m concerned, Cantor died in a plane crash when he was three years old (which is pretty impressive, actually, seeing as how planes wouldn’t be invented for another 55 years or so). I don’t know what the hell aleph-naught means).
Anyway, back to my point. Let’s try that–if A and B are infinite sets, we’ll say that A is bigger than B means that B is a proper subset of A.
Does this fit our usual notion of what “bigger” means? Well, one thing that occurs to me is that “bigger” (in the usual sense) is transitive; in other words, if someone tells me that a bear is bigger than a mosquito, and an elephant is bigger than a bear, then I can conclude that an elephant is bigger than a mosquito.
Can I use this same logic for my definition of “bigger” with respect to infinite sets? If A, B, and C are infinite sets, A is bigger than B, and B is bigger than C, can I conclude that A is bigger than C? If not, then I’m a little troubled by our new definition–it certainly doesn’t fit any preconceived notions of what “bigger” should mean.
Turns out we’re OK here, though. B is a proper subset of A (since A is bigger than B), and C is a proper subset of B (since B is bigger than C), so I can rightfully conclude that A is bigger than C (since a proper subset of a proper subset is still a proper subset of the original set).
What about these two sets:
A = {1, 2, 3, 4, 5,…}
B = {1.5, 2.5, 3.5, 4.5,…}
Hmmm, looks like there could be a problem here. Is A a proper subset of B? Certainly not, and neither is B a proper subset of A. So A isn’t bigger than B, and B isn’t bigger than A.
Now, does this mean there’s anything “wrong” with our definition of “bigger”? Certainly not–it’s my word, I made up this word, I made up this definition, dammit, and I can define my own word any damn way I please, thankyouverymuch! As far as I’m concerned, “A is bigger than B” means B is a subset of A, and that’s that. Period.
.
.
.
But you know something?
.
.
.
Now that I think about it, maybe I should be a little more humble.
.
.
.
My definition of “bigger” certainly has some usefulness, I guess, but really, now that I think about it, it’s pretty damn limited, isn’t it? Why the hell can’t I even compare those two sets? I mean, it’s an okay definition and all, but I think I can do better. You know what I’d really like? I’d really like to be able to take any two sets, and always be able to say, “This is bigger than that.” Or that they’re of equal size, or whatever. That’s really what I’m used to when saying one thing is bigger than other–I can always compare bigness, can’t I?
So looking back on it, maybe I should scrap this definition of “bigger” and try again, maybe I can find one that works even better.
.
.
.
.
Hello…(taps mike}…is this thing even on? I’ve been rambling again, haven’t I? I’m not sure if you’ll find anything worth a damn in what I just said, I think maybe I’ve gotten off on a completely wrong tangent and nobody knows what the hell my point is, anymore. Hell, I’m not even sure what my point is. I guess my point is to get you to open your eyes to how mathematics is invented/discovered (no, I don’t won’t to go there). The point being that, there’s never been a silver platter offered to us with a buffet of quadratic equations, mean value theorems, and whatever else. Math that you routinely learn in college has taken centuries of development to become what you find in the text book. You don’t see the false starts that different mathematicians have had. There’s no mathematical deity telling us “It must be done this way!” In reality, we ask “How can we do this?”, try various ways, and see which way works best–see which way seems more natural, and more satisfying or complete, in some way. Like our definition of “bigger”–nothing really wrong with it, in some sense, but it’s still somewhat unsatisfying, for the reasons I mentioned above, and I think we can do better.
Jesus Christ, I can’t believe I rambled on like this over one simple quote.
