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More than one Infinity

  1. Jan 30, 2005 #1
    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?
     
  2. jcsd
  3. Jan 30, 2005 #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.
     
  4. Jan 30, 2005 #3
    Generally, this discussion is more philosophical than mathematical.

    In the realm of acceptable math, infinity is not really a meaningful term.
     
  5. Jan 30, 2005 #4

    Hurkyl

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    It depends entirely on what you mean by "infinity" and "size".
     
  6. Jan 30, 2005 #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..
     
  7. Jan 31, 2005 #6
    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) ??????
     
  8. Jan 31, 2005 #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.
     
  9. Jan 31, 2005 #8

    matt grime

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    N is a subset of R, as is Q, as are (uncountably) many countable sets.
     
  10. Jan 31, 2005 #9

    matt grime

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    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.
     
  11. Jan 31, 2005 #10

    dextercioby

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


    Daniel.
     
  12. Jan 31, 2005 #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.
     
  13. Jan 31, 2005 #12
    Mathamaticly, yes, your theory is correct. .999~ > .333~
     
  14. Jan 31, 2005 #13
    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.
     
    Last edited: Jan 31, 2005
  15. Jan 31, 2005 #14
    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.
     
  16. Jan 31, 2005 #15
    And what is Cantor's diagonal argument?
     
  17. Jan 31, 2005 #16
    Last edited: Jan 31, 2005
  18. Jan 31, 2005 #17

    cepheid

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    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.
     
  19. Jan 31, 2005 #18
    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.
     
  20. Jan 31, 2005 #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.
     
  21. Jan 31, 2005 #20
    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 .
     
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