Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the same?

  1. This is a discussion we had in another part of the forum and I'm wondering who is correct. The discussion is becoming increasingly confusing and annoyingly (regardless of the posts in between the discussion), no one with a "Science Advisor", "Homework Helper", or "PF Mentor" title is stepping in to end the argument, so I would appreciate it if you would end the argument.
     
  2. jcsd
  3. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    The cardinalities of the intervals [0 1] and [0 2] are the same. In fact, there are as many numbers in the interval [0 1] as there are real numbers.

    This is a pretty poor explanation of what's happening. Taking limits in calculus is a process; we examine what happens as a quantity increases without bound. The limit of that function as x goes to infinity is zero because x! increases faster as x increases, not because it is a "larger quantity". "Infinity" in this context has a completely different meaning than the "infinity" used to describe the size of sets.
     
  4. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    there is an infinite, but countably infinite amount of rational numbers between 0 and 1. this is also the case for the interval from 0 to 2.

    but there is an uncountably infinite amount of real numbers between 0 and 1. this is also the case for the interval from 0 to 2.
     
  5. micromass

    micromass 18,566
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Jack212222 is wrong. Cantor's theory is nothing like what he describes. He seems to be confusing things with limits.

    And yes, the cardinality of [0,1] is exactly the same as the cardinality of [0,2]. So both sets have the same size.

    EricVT is wrong too, since he assumes that [itex]\frac{+\infty}{+\infty}[/itex] is defined when it is not. So the ratio doesn't even make sense.
     
  6. micromass

    micromass 18,566
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    As for your question:

    Yes, this is true. But the example of [0,1] and [0,2] is not a good example since both infinities are the same here.

    However, we can look at the sets [itex]\mathbb{N}[/itex] and [itex]\mathbb{R}[/itex]. Those are infinite sets, but the latter set is much larger than the former.

    See the following FAQ post: https://www.physicsforums.com/showthread.php?t=507003 (also check out the sequels whose link is at the bottom of the thread).
     
  7. HallsofIvy

    HallsofIvy 40,314
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    You also have to be careful how you define "larger". Micromass is completely correct about cardinality and that will be appropriate if 'larger' means "has more numbers in the set" as was originally asked. But one can also argue that the interval [0, 2] is twice as long, and so twice as large, as the interval [0, 1]. It depends upon what you are comparing.
     
  8. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Thanks everyone. The FAQ was very helpful. I also found this helpful:
     
    Last edited by a moderator: Sep 25, 2014
  9. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Two sets have the same cardinaility if there exists a 1-1 correspondence between them.
    The function f(x) = 2x establishes such a 1-1 correspondence between the rationals in
    [0,1] and [0,2]. It also establishes a 1-1 correspondence between the reals in [0,1] and [0,2].
     
  10. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Some infinities are larger than others. Namely, uncountable infinities are larger than countable infinities and that is all there is to it.

    The number of real numbers between 0 and 1 is uncountably infinite. The number of real numbers between 0 and 2 is uncountably infinite. The number of real numbers period is uncountably infinite. These all describe the same cardinality.

    The natural numbers are countably infinite, the real numbers are uncountably infinite. Therefore the cardinality of natural numbers is smaller than that of the real numbers.

    There is no differentiation between any two uncountably infinite, or two countably infinite sets, that I am aware of, and such an idea doesn't really make sense to me.
     
  11. micromass

    micromass 18,566
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Sure there is. There are many types of uncountably infinite sets. For example, the set [itex]\mathcal{P}(\mathbb{R})[/itex] (power set of the reals) has a strictly larger cardinality than [itex]\mathbb{R}[/itex]. and [itex]\mathcal{P}(\mathcal{P}(\mathbb{R}))[/itex] is even larger!! This process continues indefinitely.
     
  12. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Ah, thanks for pointing that out.

    Edit: I take my thank you back.
     
    Last edited: Dec 31, 2012
  13. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

     
  14. dextercioby

    dextercioby 12,304
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Just as a curiosity, what's the connection between the discrete/countable infinite sets (sequences) [itex]\mathcal{P}^{n}(\mathbb{R})[/itex] and [itex] \aleph_{n} [/itex] ?
     
    Last edited: Jan 2, 2013
  15. Ah.. I failed to think of deriving a set from an uncountable set in such a way that the cardinality must be greater. It doesn't make much sense at all to say that a set and it's power set have the same cardinality. That is easy to see.

    Thank you.

    Is this limited to the idea of power sets? It feels like this could only occur when you "build" a "higher order" uncountable set from a previously uncountable set.

    Also- the power set of the naturals would certainly be of higher cardinality than the naturals and thus not at a one-to-one correspondence with the naturals and therefore uncountable, right?

    Does that make P(N) have the same cardinality as R, and P(P(N)) have the same cardinality as P(R)?

    Or am I oversimplifying the idea?
     
    Last edited: Jan 2, 2013
  16. dextercioby

    dextercioby 12,304
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    P^n (N) (=P(P(P...P(N)))) for arbitrary n has the same cardinality with N.
     
  17. I don't understand how that could be the case.
     
  18. dextercioby

    dextercioby 12,304
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    Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Who's P(N) ? Write an explicit formula using the 3 symbols: ..., { and }. Then calculate its cardinality.
     
  19. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Forgive me if I'm not understanding but,

    The number of elements of P(N) is the number of possible subsets of natural numbers that can be formed.

    That is surely uncountable, as I can combine any number of any natural numbers I want to form some subset. If I call one subset, the non-proper subset, the entire set of natural numbers, one subset, what's another subset? The subset of 1 million of any natural numbers? 2 million any natural numbers? Considering all singleton sets of each natural number is already a countably infinite number of subsets, and there are uncountably many subsets besides those, it doesn't work in my brain to call P(N) a set of countable cardinality.

    EDIT:

    I found this

    http://www.earlham.edu/~peters/writing/infapp.htm#thm3

    Which I guess is a much better and more formal version of what I'm thinking.
     
    Last edited: Jan 2, 2013
  20. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    You're right, P(N) is uncountable. An easy way to see this is to identify each element of P(A) with an infinite binary string (easy to show that this is a bijection) and note that the set of all such strings is uncountable.
     
  21. Re: Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the s

    Okay, I'm not sure how to construct these infinite binary strings but I do understand why P(N) is of uncountable cardinality. I have a few more questions.

    I have been reading about "beth numbers" and this seems to be the name for the "order of infinity."

    So here are my questions:

    Is the beth number the sole indicator of cardinality of uncountable sets? If so, this would imply that P(N) and R have the exact same cardinality, right?
    Is the power set concept the only source of constructing a "higher order" infinity?

    Are there any sets that we naturally consider that aren't derived from taking the power set of some other set with a beth number greater than two?

    I do realize that this question is pretty much identical to my previous one, but I guess what I'm asking is if there is a set with a beth number greater than or equal to 2 that we could describe in some other way than a power set of some other set?

    I'm sorry for kind of derailing the topic but I find this really cool, as a grad, what focuses on this type of thing? Is it just set theory?
     
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