The unit of measure is elements in a set. Some sets are finite, and we can assign a counting number to them. For example, the set of states in the United States has 50 elements. However, other sets are infinite, which means that we can’t assign a counting number to them. For example, the set of counting numbers is infinite, because if we assigned a counting number to it, you could always add one more to that counting number, and get a contradiction.
I’m not sure if “set” has a strict mathematical definition, but I don’t see anything you’ve written here that I take issue with.
First, IANAM and I can’t remember anything I may have learned about the different types of infinity. But I can amplify this reasoning.
Integers are discrete. You can count them. You would be counting forever, but you can count them. Just start with 0 and keep adding 1 in the positive direction and subtract 1 in the negative direction. You’ll never miss any.
Real numbers fall on a continuum. No matter what two real numbers you can identify, you can always identify one more that falls in between them. You can’t count them. While you can go from 1 to 2 and say, “OK, there are two integers, let’s keep going,” there are an infinite number of real number between 1 and 2. There are an infinite number of real numbers between 1.000000000000000001 and 1.000000000000000002.
Now, whether that means that one infinity is bigger than another infinity, I’m not prepared to defend.
Blue and green don’t have sizes. The size of a set is the number of elements that it contains. You can say that one set A is larger than another set B if:
(1) You can set up a one-to-one correspondence between the elements of set B and the elements of a subset of set A, and
(2) You cannot set up a one-to-one correspondence between the elements of set B and the elements of set A.
So, 3 is larger than 2, because if you take a set with 3 elements (e.g., {X,Y,Z}) and a set with two elements (e.g., {P,Q}), you can set up a one-to-one correspondence between any 2 elements of {X,Y,Z} and {P,Q}, but you can’t do that with the whole set {X,Y,Z}.
So counting is a bit like putting a ruler next to a length that you want to measure, and seeing how far the ruler is in on-to-one correspondence with the length.
This can be extended to infinite sets, with the premier example being the set of counting numbers.
Yeah, see I understand the concept you are explaining, but I just don’t see how that’s a “proof” that we can treat infinity as thought it’s a number, or quantity.
There is a problem with the intuitive approach that CookingWithGas took, because of this paradox. We definite “rational numbers” as the fractions that you can get by dividing one counting number by another – not allowing division by zero, and treating A/B and C/D as equal if A times D equals B times C.
Now, if you take two rational numbers, you can find an infinite number of different rational numbers between them – unlike the counting numbers, where there are no counting numbers between consecutive counting numbers N and N+1.
However, you can still set up a one-to-one correspondence between the counting numbers and the rational numbers (fractions). So, even though there’s an infinite number of fractions between any two counting numbers, the infinity of the two sets is the same.
I suppose that at some point it becomes as much philosophy as math. There may be some technical mathematical applications where you need to deal with infinity, and may need to define different degrees of infinity. I don’t know if mathematicians debate this as an “angels on the head of a pin” debate, or if there is some more important upshot to how we define and treat infinity.
But you can certainly make an argument that if there are an infinite number of real numbers that correspond to every integer, there must be more real numbers than integers. And if there are an infinite number of integers, then there must be a “bigger infinity” of real numbers. That’s essentially what Cantor said. So that enhances the concept of infinity beyond being trivially defined as “without end.”
Oh, goody—another thread about infinity.
Here’s one really good online introduction to infinity and some of the ideas being discussed here.
If you want to explore a little deeper, this page: What Is Infinity? has links to various subtopics related to infinity (like infinity as a limit, cardinal and ordinal numbers, and paradoxes of infinity).
CookingwithGas, what you’re illustrating is that the reals are dense. (An ordered set is dense when for any pair of members A > B, there’s another member C such that A > C and C > B.) But the rationals are also dense–and the cardinality of the rationals turns out to be the same as the cardinality of the integers.
Here’s a way to explain what is meant when mathematicians say the set of reals is larger than the set of integers.
You can’t list all the integers on a sheet of paper, of course. But you can put them in order, such that for every integer x, there is an integer n such that we can say that x is in the nth place on the list. A very trivial way to do this is just to put them in “counting order.” The number one, we can say, will appear first on the list. The number seven will appear seventh. And so on.
You also can’t list all the reals of course, but it also turns out you can never put them in order the same way you can put the integers in order. I.e., you can never put them in order in such a way that, for every real, there is an integer n such that that real will appear in the nth place on that list. Whatever order you try to put the reals in, you will always end up with real numbers that have no place in that order–there will be no n such that that real is in the “nth place” in that order.
Well that came out looking more complicated than I intended. Sorry!
Again, that’s not a good argument for the (true) conclusion that there are more reals than integers. For there are an infinite number of rational numbers correpsonding to every integer–namely, correspond each rational less than one with one, each rational between one and two with two, and so on–yet there are no more rational numbers than integers.
Thinking of it in terms of “putting them in a certain kind of order” as I discussed in my last post, it turns out you can put the rationals in order in a way such that for every rational, there’s an integer n such that that rational is in the nth place in that order.
Here’s how to do it:
1/1
1/2
2/1
1/3
2/2
3/1
1/4
2/3
3/2
4/1
and so on. The pattern is to list every rational with numerator and denomenator totally to two first. Then list every one with them totalling three. Then list every one with them totalling four. And so on. Within each of these groups, start with a one in the numerator. Then for the next element in the list, add one to the numerator and subtract one from the denominator. And so on til you’ve exhausted every rational with that total.
It’s fairly easy to see that you’ll hit every rational, eventually, doing things this way. You’ll also repeat some of them (reduce the fractions…) but that isn’t really relevant. If you don’t see why, feel free to ask.
Let’s start with Zeno’s Paradox. The archer shoots an arrow at the target. The arrow first has to go half the distance to the target, then half the remaining distance, then half the remaining distance, and so on. It can never do this because there are a infinite number of “and so on’s”. Yet it does reach the target. Hence the “paradox.” Mathematicians used your thinking and could not explain this.
Eventually Newton and Leibniz came along with the calculus, a device that could handle infinities by the use of infinitesimals and limits. Limits are an extremely crucial concept for handling infinities, which pop up everywhere in math. Every area under a curve can be divided into a infinity of smaller parts and then added up. All of calculus deals with infinities. Math deals with infinities more than non-infinities, in that sense. They are not abstract or unobtainable. They are everywhere and extremely basic. What is 1/2 + 1/4 + 1/8 + 1/16 … equal to? There is no end to the sequence. It is infinite. But it equals 1. Exactly. You can ask about 9/10 + 9/100 + 9/1000 = … That also equals 1 exactly. (Or you can express it as 0.9999999~. That’s also exactly 1, just as 0.3333~ is exactly 1/3.) How does a mathematician calculate pi? You take any of a number of infinite sequences and add them up. You can never get to the last digit and yet you can prove that they exactly equal pi.
Even so, mathematicians couldn’t see any way of handling other forms of infinities. The rational numbers, 1,2,3,4,5,6,… are obviously infinite. But what about fractions? What about irrational numbers? What about transcendentals? Complex numbers? Equations of curves? How do they compare to one another?
Then came Georg Cantor in the 19th century. He saw that the way to handle infinities was to put them into one-to-one correspondence. The even integers can be placed into a one-to-one correspondence with the odd integers. 1 : 2; 3 : 4; … And the squares can as well. 2 : 4; 3 : 9; 4 : 16… And all the powers. You don’t have to count them all or even say them out loud for this to work.
But what about irrationals? Between 0 and 1 is .0087374948994949… and .0873749383893938… and 0.00873749492928892828… Can you put these into a one-to-one correspondence with the integers? The answer turns out to be no. Cantor did it more rigorously, of course, with the Cantor set. Set theory is the heart of all formal mathematics. It is what makes mathematics rigorous. If you don’t know that sets have the strictest of definitions, from which everything, and I mean everything including 0 and 1 and from them all the rest of the numbers, is built from, then you are missing the last century’s worth of formalized math. That negates all your arguments.
Once you introduce sets and define sets by the number of members in a set, you can play with infinities to your heart’s content. With sets you can add or multiply infinities, or you can define what happens if you take 2 and raise it to an infinity, or even if you raise an infinity to an infinity. Everything becomes possible, and rigorously so. Why? Because it takes math away from the slippery words that you are using. Your words have no rigor and no formal definition. But mathematical set notation does. They are different worlds.
I am not a mathematician, but I play one on the internet.
My 4 and a half year old said to me the other day “I can count to infinity!”. I said “go ahead” and he said “Infinity-minus-two, infinity-minus-one, Infinity!”.
Four and a half, and already he’s better at math than me.
All the stuff about set theory is fine. But, like I said, keep in mind that’s not the only context in which someone, even a mathematician, might want to talk about “infinity”.
For example, here’s another system of some interest: consider the arithmetic of rational functions (one polynomial divided by another). We can do basic arithmetic (addition, subtraction, multiplication, division) on these pointwise; for example, if F(x) = (x^2 + 3)/(x - 8) and G(x) = x^4/5, then (F * G)(x) = (x^6 + 3x^4)/(5x - 40). We can even think of any particular finite number as just a special case of a “generalized number” of this form; for example, 5 corresponds to the constant function which always returns 5.
We’ll also call any F positive just in case F(x) is always positive for sufficiently large x. And with this, we can define an ordering: we’ll say F > G just in case F - G is positive. And these definitions play nicely with the arithmetic, with all the familiar rules; if F is positive, then F + G > G, if you multiply a positive by a positive you get a positive, etc. We get a simple (and, as it happens, mathematically important) theory of the ordered arithmetic of these generalized numbers.
So what? Well, now, look: consider the function W(x) = x. Not only is W positive, but so are W - 1, W - 2, W - 3, and so on. That is, W > 0, W > 1, W > 2, W > 3, and so on. W is greater than every finite number. In that sense, W is infinite.
But! 1 + W > W. So even though W is infinite, 1 + W is an even higher infinite quantity still. And 2 + W is even higher still. And W * W is way higher than those. And so on…
So here’s a very simple theory of the ordered arithmetic of infinite quantities which has many different levels of infinities, but has nothing to do with cardinalities, and, indeed, differs radically from cardinality arithmetic in some ways [e.g., with cardinalities, adding 1 to an infinite quantity doesn’t increase it]. And for many applications, this system is far more relevant to what one is interested in than cardinalities. Everything always depends on what you’re doing. It’s up to you to pick what particular math-game of the infinite will be of interest to you to study; the only thing forcing your hand is that once you’ve figured out the rules you care about, they have consequences.
Wikipedia has a decent summary of Cantor’s Diagonal Argument, which demonstrates that the reals can’t be put into one-to-one correspondence with the counting numbers, and are therefore uncountable.
And for the nonmathematicians, a quick glossary:
A set is finite if there is some integer N so that the elements of that set can be put into one-to-one correspondence with the integers from 1 to N.
(Windy digression: as Giles said earlier, this is how mathematicians tell whether sets have the same number of elements: can they be put into one-to-one correspondence with each other? If yes, then they have the same number of elements (or cardinality, as we math geeks would say); if not, then one set can be put into one-to-one correspondence with a proper subset of the other. The one set is the smaller, and the other is the larger of the two.
In the case of finite sets, it’s pretty trivial: a set with 5 objects can’t be put into one-to-one correspondence with a set with 7 objects, but the 5-object set can be put into 1-to-1 correspondence with a 5-object subset of the 7-object set, so the 5-object set is smaller than the 7-object set.
But we shortcut this by implicitly putting the 5-object set into 1-to-1 correspondence with the numbers [1,2,3,4,5] and saying it has 5 elements in it, while the other similarly corresponds to the set [1,2,3,4,5,6,7] hence has 7 objects in it, and 5 < 7. That’s what “5 < 7” means, and how it relates to this one-to-one correspondence stuff.)
A set is countably infinite if (a) it’s not finite, and (b) its elements can be put into one-to-one correspondence with the counting numbers, i.e. the integers from 1 on up.
A set is uncountably infinite or simply uncountable if its elements cannot be put into one-to-one correspondence with the counting numbers.
Cantor’s diagonal argument demonstrates the uncountability of the real numbers, so there are more reals than there are counting numbers (and other sets of the cardinality of the counting numbers, such as the integers and rationals).
I’m not getting the ‘ordered’ part here. Partially ordered, perhaps?
Take H(x) = x + 2 sin(x). None of H - W, W - H, H - (W + 1), or (W + 1) - H, are positive. As best as I can tell, H is incomparable with W and W + 1.
I said to use rational functions, not arbitrary functions. Then you get a total order.
Even if you use arbitrary functions, the result produces a B-valued “total” order, where B is the Boolean algebra of subsets of the reals (or whatever ordered field you’re working over) modulo the ideal of downwards closed proper subsets of R. If one then further quotients by an ultrafilter on this Boolean algebra, one can collapse the result into a “true” total order again; this is just an example of the ultrapower construction of Abraham Robinson’s nonstandard analysis.
But none of that is necessary for the example I was using with just rational functions.
You can’t measure infinities, but you can make statements about them based on their characteristics. An infinite duration can be thought of as a series of hours - and we know that there are 3600 seconds in every hour, therefore there must be more seconds in an infinite duration than there are hours, even if we can’t count either.
Mathematical proofs are nearly always like that - they’re not usually based upon a complete, brute-force sampling of the real world, but rather, upon a set of consistent, logical, demonstrable statements.
Note: The sense in which there are more seconds in an infinite duration than there are hours is NOT that of cardinalities. A countably infinite collection of hours comprises just a countably infinite collection of seconds, for example.
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You could just as well say that the infinity of real numbers is fixed in size, if you like. As Frylock said about CookingWithGas, it sounds like you are simply pointing out that the real numbers are dense; between any two, there’s another. But this is true of the rational numbers as well, yet the rational numbers have the same cardinality as the integers.
That having been said, Cantor’s original argument for the uncountability of the reals was not the famous diagonal argument, but rather one along these lines, but using a somewhat stronger density property.
(I am not a mathematician either, but I watched a brilliant TV documentary about them the other day)
You’re right :smack: - because we could define a one-to-one relationship between seconds and hours. I was grasping for an easy analogy, but this one fails.