More than one Infinity

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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?
 

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  • #2
dextercioby
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The German mathematician Georg Cantor wrote something about the theory of infinites.

The set of reals is not COUNTABLE...

Daniel.
 
  • #3
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Generally, this discussion is more philosophical than mathematical.

In the realm of acceptable math, infinity is not really a meaningful term.
 
  • #4
Hurkyl
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What do you think? Can infinities be of different sizes?
It depends entirely on what you mean by "infinity" and "size".
 
  • #5
arildno
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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..
 
  • #6
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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) ??????
 
  • #7
dextercioby
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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.
 
  • #8
matt grime
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dextercioby said:
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.
 
  • #9
matt grime
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damoclark said:
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.
 
  • #10
dextercioby
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Sorry,i meant intervals... :blushing: Why the heck didn't i think about N & Q? :grumpy:


Daniel.
 
  • #11
matt grime
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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.
 
  • #12
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DoubleMike said:
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~
 
  • #13
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LOL! the above post doesnt make any sense because .3333~ isnt infinitly large, and neither is .9999~. if they where, then your above post wouldnt 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 doesnt 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 doesnt 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.
 
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  • #14
Gecko said:
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.
 
  • #15
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And what is Cantor's diagonal argument?
 
  • #16
DoubleMike said:
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 [Broken] .
 
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  • #17
cepheid
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matt grime said:
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 realise 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.
 
  • #18
cepheid said:
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 realise 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.
 
  • #19
cepheid
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Ok, so isn't the existance 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.
 
  • #20
cepheid said:
Ok, so isn't the existance 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 .
 
  • #21
Galileo
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In the sense of cardinal numbers, there are different kinds of infinity.
In general, if A is a set, then the powerset of A: [itex]\rho(A)[/itex] (which is the set containing all subsets of A) is 'bigger' than A in the sense that there exists no surjection [itex]f:A \to \rho(A)[/itex].

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 [itex]\mathbb{C}[/itex] resulting in the extended complex plane. This is a totally different kind of infinity ofcourse.
 
  • #22
Hurkyl
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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!
 
  • #23
Galileo
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Hurkyl said:
Yet more proof that there does not exist a set of all sets!
Hurkyl! Just the man I need. :smile:

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?
 
  • #24
matt grime
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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 thoery 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.
 
  • #25
saltydog
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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
 

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