Can infinities have different sizes?

[DoubleMike]

I've had a discussion with two friends about infinity. I am of the opinion that infinities come in different sizes and some infinities can be bigger than others. My argument goes somewhat like this:

Between the numbers 1 and 2 there are infinite real numbers. Between 1 and infinity there are infinite integers.

However, since for any two successive integers there is an infinite number of real numbers, there are more real numbers than integers.

Despite my best efforts, neither agree with me. They say that infinity is infinite and since "it goes on forever nothing can be greater than it." But I think this definition is somewhat prosaic and has limited mathematical value.

What do you think? Can infinities be of different sizes?

 

[dextercioby]

The German mathematician Georg Cantor wrote something about the theory of infinites.

The set of reals is not COUNTABLE...

Daniel.

 

[DeadWolfe]

Generally, this discussion is more philosophical than mathematical.

In the realm of acceptable math, infinity is not really a meaningful term.

 

[Hurkyl]

What do you think? Can infinities be of different sizes?

It depends entirely on what you mean by "infinity" and "size".

 

[arildno]

Welcome to PF, DoubleMike!
Yes, there does exist different degrees of infinity, but not really in the sense you think!

The basic problem with infinities is that the distinction more/less doesn't make a good distinguishing tool.

Let us look upon the issue from a different angle:
Suppose you've got two sets of objects, A and B, and that there are finitely many objects in A and B.
We want to evaluate the relative "sizes" of these sets in the following manner:
Pair together one object from A with an object from B, and remove both from their respective sets.
Proceed in like manner.

Since you've got finitely many objects, you'll end up at last with one of the following 3 situations:
1. No objects left in A, objects left in B (we say that A had "fewer" objects to start with than B had)
2. Neither have any objects left (we say that A and B started out with equally many objects)
3. Objects left in A, none in B (we say that A had "more" objects to start with than B had)

As you can see, this cumbersome counting technique captures exaxtly what we mean in the finite case of what the words "fewer/equal/more" is supposed to mean.

This pairing-off technique is what we need to use when dealing with sets containig INFINITELY many objects!
Something very surprising happens:
Suppose A is the set of ALL natural numbers "n".
Let B consist only of the even integers.
At first, we would say there were "more" objects in A than B, but see what the pairing technique gives us.
For a given integer "n" in A, we pair that off with the EVEN integer "2n" in B.
EVERY "n" in A are thus paired to an even integer in B, and every even integer in B is paired off with some integer in A!

Is there fewer or more elements in A or B?
As you can see, for the infinite sets, that question doesn't have much meaning; what we can say is:
1) B is a subset of A (any even integer is certainly an integer, so it is contained in A somewhere)
2) There exist a way to pair off A-and B-elements (it is possible to construct a bijection)

To get back to your original idea:
It can be shown that it is impossible to construct a bijection between the set of natural numbers and the set of real numbers.

For natural reasons, a set which can be paired off with the set of natural numbers is called "countably infinite"; the set of real numbers is said to be "uncountably infinte".

So there you have it; there exist different degrees of infinities, but it is rather tricky to understand it..

 

[damoclark]

Yes, that is a very interesting question. I think there are different kinds of infinities. Let me add something if I may.

Consider what happens if you choose two numbers between 1 and 2 and multiply them together, then your answer will always be greater than the two numbers you chose to begin with.

eg 1.1*1.2 = 1.32

1.32 is bigger than 1.1 and 1.2

If you multiplied three numbers together the same principle would apply

eg. 1.1*1.2*1.3 = 1.716

If you multiplied ten number together

eg 1.1*1.2*1.3*1.4*1.5*1.6*1.7*1.8*1.9*2.0 = 67.04425728

the same thing happens.


Now imagine if we could multiply all the numbers betwwen 1 and 2 together. Since there are an infinite number of numbers between 1 and 2, when you multipy all the numbers between 1 and 2 surely your answer should be infinity.

Lets call this infinity(1..2)

Now what would your answer be if you multiplied all the numbers between 1 and let say e = 2.71828..., then surely the answer would be infinity also.
Lets call this infinity(1..e).

Now which infinity is bigger? infinity(1..2) or infinity(1..e) ?

 

[dextercioby]

How would you figure it??What would your answer be and why?Keep in mind that neither R nor any subset of it are countable...

Daniel.

 

[matt grime]

How would you figure it??What would your answer be and why?Keep in mind that neither R nor any subset of it are countable...

Daniel.

N is a subset of R, as is Q, as are (uncountably) many countable sets.

 

[matt grime]

Now which infinity is bigger? infinity(1..2) or infinity(1..e) ?

How are we supposed to know? You've just completely redefined multiplication to allow for infinite operands so we probably shouldn't guess what on Earth you're doing.

 

[dextercioby]

Sorry,i meant intervals... Why the heck didn't i think about N & Q?


Daniel.

 

[matt grime]

The interval [x,x] is a countable* subset of R that is also an interval.


* some people differentiate take countable to mean infinite. some, like me, do not.

 

[eNathan]

I've had a discussion with two friends about infinity. I am of the opinion that infinities come in different sizes and some infinities can be bigger than others. My argument goes somewhat like this:

Between the numbers 1 and 2 there are infinite real numbers. Between 1 and infinity there are infinite integers.

However, since for any two successive integers there is an infinite number of real numbers, there are more real numbers than integers.

Despite my best efforts, neither agree with me. They say that infinity is infinite and since "it goes on forever nothing can be greater than it." But I think this definition is somewhat prosaic and has limited mathematical value.

What do you think? Can infinities be of different sizes?

Mathamaticly, yes, your theory is correct. .999~ > .333~

 

[Gecko]

LOL! the above post doesn't make any sense because .3333~ isn't infinitly large, and neither is .9999~. if they where, then your above post wouldn't hold any credibility because .9999~=.3333~, which by your own argument means that that statement is false (because you said .99999~>.33333~ which means that .9999~=/=.3333~).

Edit: now that i look at your post again, it looks like you are arguing that .9999~ is on a greater level of infinity because .3333~ is smaller than .9999~. what you are arguing is not what the original argument was (and your argument still doesn't make sense). he was arguing that R should have a greater infinity than N (? i think its N, he was talking about the set of all integers) because N doesn't represent as many numbers as R does. the problem is that both R and N are uncountable, which means that neither one can have a greater infinity than the other.

 

[hypermorphism]

the problem is that both R and N are uncountable, which means that neither one can have a greater infinity than the other.

N is countable by definition. R is uncountable by Cantor's diagonal argument. People usually think of R as "larger" than N simply because they don't have the same cardinality, and N is a subset of R.

 

[DoubleMike]

And what is Cantor's diagonal argument?

 

[hypermorphism]

And what is Cantor's diagonal argument?

The proper form of Cantor's argument without reference to representation can be found at http://mathworld.wolfram.com/CantorDiagonalMethod.html . If this leaves you confused, there is the decimal form found in layman's and introductory texts that relies on your associating real numbers with their infinite decimal representation. This simplified argument can be found at http://planetmath.org/encyclopedia/CantorsDiagonalArgument.html .

 

[cepheid]

The interval [x,x] is a countable* subset of R that is also an interval.


* some people differentiate take countable to mean infinite. some, like me, do not.

Hi,

Just trying to improve my understanding. So if I'm getting what you are saying, a countable set is simply one that can be put in a one-one correspondence with the set of postive integers. When I saw this definition, I thought to myself: "ok, so it means you can *count* the elements in the set basically, you can say this is the first one, this is the second one...etc." So you can assign a postivie integer to each member of the set. There is a one-one mapping. I realize that nothing in there specifies that you have to use ALL of the integers. You can do this to a finite set as well. So in short, a countable set can be either finite or infinite, with at most as many elements as there are positive integers? I was just trying to express what you were saying in my own words. I hope I haven't used any terminology incorrectly.

Thanks.

 

[hypermorphism]

Hi,

Just trying to improve my understanding. So if I'm getting what you are saying, a countable set is simply one that can be put in a one-one correspondence with the set of postive integers. When I saw this definition, I thought to myself: "ok, so it means you can *count* the elements in the set basically, you can say this is the first one, this is the second one...etc." So you can assign a postivie integer to each member of the set. There is a one-one mapping. I realize that nothing in there specifies that you have to use ALL of the integers. You can do this to a finite set as well. So in short, a countable set can be either finite or infinite, with at most as many elements as there are positive integers? I was just trying to express what you were saying in my own words. I hope I haven't used any terminology incorrectly.

Thanks.

Yep. Countable is a generalization of how people count. The terminology "uncountable" just refers to the cases where you can't do what you described in order to cover the entire set.

 

[cepheid]

Ok, so isn't the existence of "countable" vs. "uncountable" sets (different cardinalities) at the heart of what the original poster was trying to say? There are infinitely-many integers, but there are more real numbers than integers...so is that a different type of infinity? Again, maybe the terminology is poor (different infinities), but the idea is sound, right? Our prof mentioned it in passing in class.

 

[hypermorphism]

Ok, so isn't the existence of "countable" vs. "uncountable" sets (different cardinalities) at the heart of what the original poster was trying to say? There are infinitely-many integers, but there are more real numbers than integers...so is that a different type of infinity? Again, maybe the terminology is poor (different infinities), but the idea is sound, right? Our prof mentioned it in passing in class.

I would put "more" in parentheses, but yes. The cardinal number for the naturals is called aleph_null while the cardinal for the reals is bet, or the continuum. There are other concepts of infinity as well: http://mathworld.wolfram.com/CardinalNumber.html .

 

[Galileo]

In the sense of cardinal numbers, there are different kinds of infinity.
In general, if A is a set, then the powerset of A: $\rho(A)$ (which is the set containing all subsets of A) is 'bigger' than A in the sense that there exists no surjection $f:A \to \rho(A)$.

The above can be quite easily and elegantly proved, but for some reason it doesn't seem to work when you take A to be the set of all sets...


Another kind of infinity is 'the point at infinity' in the complex plane. You can add this point to $\mathbb{C}$ resulting in the extended complex plane. This is a totally different kind of infinity ofcourse.

 

[Hurkyl]

for some reason it doesn't seem to work when you take A to be the set of all sets...

Yet more proof that there does not exist a set of all sets!

 

[Galileo]

Yet more proof that there does not exist a set of all sets!

Hurkyl! Just the man I need.

Is the nonexistance of the set of all sets related to Bertrand Russell's paradox? (When A is the set of all sets which do not contain themselves).
How would you show it?

 

[matt grime]

Russel's paradox states that, if we define sets with unrestricted rules, then there is a problem

Namely define X:= { Y | Y is a set and Y is not an element of Y}

Then X is a member of itself and not a memeber of itself, contradiction.

Cantor shows that given a set A there is no bijection with P(A), thus if A were the set of all sets, then, as P(A) is a set, containing A, it must actually equal A, and thus there would be a bijection if there were indeed a set of all sets.

Exercise. Attempt to actually think of a set that contains itself. If you do so, you'll quickly realize how unimportant this actually is.

You should be careful about saying "existence" in such things. We tend to say that given a model of a set theory, the class of all objects that are sets in that theory is not a set in that theory, though it may be a set in some larger universe - that is some other model of the set theory. A set theory is a collection of rules, a model of a set theory is a class of objects that satisfy those rules. Nothing a priori comes with the absolute label "set" round its neck.

This idea may be familiar from other axiomatic systems. For instance, what is a vector (please don't say something with length and direction, we aren't talking physics here)? It is an element of a vector space. What is a vector space? It's a class of objects satisfying certain rules. Is the vector space a vector inside itself? No. Same with set theory.

 

[saltydog]

Density

Infinity and the Real Number System: Very interesting. May I make a proposal?

Why does Mathematics fit nature so well?

Because the Real Number System is dense!

Works for me,
Salty

 

[matt grime]

The reals are dense in what? That makes no sense at all. Denseness is a property of a subset with respect to an ambient space.

 

[mathwonk]

the original work by georg cantor is still appealing on the topic of different sizes of infinity, though about 100 years old by now:

Contributions to the Founding of the Theory of Transfinite Numbers
Cantor, Georg
Price: US$ 4.45 [Convert Currency]

Book Description: Dover. Trade Paperback. 211 p. Very good condition. Light wear to extremities. * * * Selling books of merit since 1988. * * * Prompt, Professional Service. Satisfaction Guaranteed. * * *. Bookseller Inventory #388390

Bookseller: Harvest Book Company (Fort Washington, PA, U.S.A.)



here is another even more valuable book, covering many more topics:

SET THEORY
Hausdorff, Felix
Price: US$ 22.00 [Convert Currency]

Book Description: New York: Chelsea Publishing, 1962. Hard Cover. Very Good/No Jacket. 8vo - over 7¾" - 9¾" tall. Ex-library. Some wear on cover, mostly on corners and spine. Bookseller Inventory #001841

Bookseller: Ivan A Luka (Brentwood, MD, U.S.A.)

 

[robert Ihnot]

Now imagine if we could multiply all the numbers betwwen 1 and 2 together. Since there are an infinite number of numbers between 1 and 2, when you multipy all the numbers between 1 and 2 surely your answer should be infinity.

Double Mike:Lets call this infinity(1..2)

Now what would your answer be if you multiplied all the numbers between 1 and let say e = 2.71828..., then surely the answer would be infinity also.
Lets call this infinity(1..e).

Cantor had the concept of cardinality, relating one set to another. THERE ARE AS MANY NUMBERS BETWEEN 0 AND 1 AS ON THE WHOLE LINE! This can be illustrated by a simple diagram,use your imagination.

(0,1) (1/2,1/2)

(0,0) (1/2,0)

We draw two parallel lines at X=0 and X=1/2 of height 1. Then we construct a line to connect (0,1) and (1/2,0) (If you can visualize that.) This gives us the triangle (0,0), (0,1), (1/2,0). Now consider triangle hypothesis moving along the X axis from 1/2 to infinity.

Then at any point in the advance, the hypothesis cuts the line between (1/2,0),(1/2,1) AT EXACTLY ONE POINT, yet the hypothesis sweeps across the whole positive X axis starting at 1/2. (Well, O.K. move the triangle back 1/2 to get rid of that problem). SO THERE ARE AS MANY POINTS BETWEEN 0 AND INFINITY AS BETWEEN 0 AND 1, so there is no difference in your calculations!

The diagonal argument deals with the difference between the rational and the irrational.

 

[mathwonk]

here is one of cantors diagonal arguments as i recall it from high school, about 45 years ago:

assume there is the same number of real numbers between 0 and 1, even those having only zeroes and 1's in their decimal expansion, as there are positive integers.

that means we canj write down a list (infinitely long) containing all those real numbers.

so we write down first the decimal corresponding to the integer 1, then under it we write down the real decimal corresponding to 2, etc...

this yields a doubly infinite array of the digits 0 and 1. i.e. we have an infinitely long list of decimals, and in each row we have an infinitely long decimal, possibly ending in all zeroes after a while.


Then we will construct another decimal that is actually not in the list as follows:

Just run along the "diagonal" of the list. i.e. the "diagonal decimal" is the decimal whose first entry is the first entry of the first decimal in the list. its second entry is the second entry of the second decimal in the list. its third entry is the third entry of the third decimal in the list...etc...


now with this decimal in hand, go down it entry by entry and change every single entry to the opposite choice. i.e. for each entry which is a 1, change it to a zero, and vice versa.


the result will be a decimal which does not equal any of the decimals in the list, since to differ from a decimkal in the list it suffices to differ in only one entry, and this new decikmal differs from the nth decimal in the list by its nth entry.

now since every list of decimals gives rise to a decimal not in the list, it follows that no such list can contain them all. hence it is not true that there are the same number of such decimals as positive integers.




to abstract this argument, consider the set of positive integers as a given set N, and think about all subsets S of that set. To describe one subset S of N, means for each element of N, i.e. for each positive integer, we must decide whether or not it belongs to our subset S. If it does assign a 1, if not assign a 0. this gives us an infinite sequence of 0's and 1's, lo and behold, precisely an infinite decimal containing only 0's and 1's.



hence if there were the same number of subsets of N as elements of N< then we could write down a list of those subsets and hence of those decimals, each numbered by a positive integer.

we are back where we were before. since there is no such list of decimkals there is no such list of subsets, and so in fact there are mroe subsets of N than there are elements of N.


more abstractly, if N is any set at all, and we find a map f from N to the set of all subsets of N, we can construct a subset S of N, using the map f, as follows: let x belong to S if and only if x does not belong to f(x).

i.e. the "diagonal subset" contains x if and only if x does belong to f(x), and we change it to the opposite subset, where for every x in S, x does not belong to f(x).


then this new subset S is not f of any x, since if S = f(x), then x does not belong to f(x), i.e. x does not belong to S. But by definition of S, if x does not belong to S, then x did belong to f(x). Since we are assuming S = f(x), this is a contradiction.

this argument proves there is no surjection from N to the set of all subsets of N, for any set N at all. hence any set, even an infinite one, has more subsets than elements.

 

[shrumeo]

Hmm, I didn't even bother to read all the posts but it seems to simple even to a non-mathematician like myself.

Infinity is a quantity like "2."
There is only one "infinity" like there is only one "2."
There aren't different kinds of 2 because 2 is just a quantity.

There are infinity Real numbers between 1 and 2.
There are infinity integers between 1 and forever.

An analogy could be made to say:
There are 2 cars in the garage.
There are 2 testicles in my scrotum.

Are there 2 types of "2"?
Nope.

Infinity is just a quantity (albeit an uncountable one).

 

[matt grime]

Well, thankfully you're wrong. But don't let that stop you arguing from a point of ignorance.

 

[hypermorphism]

Hmm, I didn't even bother to read all the posts but it seems to simple even to a non-mathematician like myself.

Infinity is a quantity like "2."

This statement is false. Infinity does not behave like any real number.

 

[mathwonk]

that's it. I'm outta here. (after reading shrumeos post.)

 

[shrumeo]

ok geniuses, explain it to me

why isn't infinity a quantity and 2 is?

 

[Hurkyl]

2 is a real number (or rational number, integer, et cetera) by definition. And, by definition, there is no real number (or rational number, integer, et cetera) called infinity.