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Who is right here? Are the amount of numbers between both 0 to 1 and 0 to 2 the same?by tahayassen
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#1
Dec2912, 12:02 PM

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#2
Dec2912, 12:19 PM

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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.



#3
Dec2912, 01:21 PM

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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. 


#4
Dec2912, 01:50 PM

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


#5
Dec2912, 01:53 PM

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As for your question:
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: http://www.physicsforums.com/showthread.php?t=507003 (also check out the sequels whose link is at the bottom of the thread). 


#6
Dec2912, 04:00 PM

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#7
Dec2912, 04:16 PM

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Thanks everyone. The FAQ was very helpful. I also found this helpful: http://www.youtube.com/watch?v=AQoutHCu4o



#8
Dec3112, 06:44 AM

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Two sets have the same cardinaility if there exists a 11 correspondence between them.
The function f(x) = 2x establishes such a 11 correspondence between the rationals in [0,1] and [0,2]. It also establishes a 11 correspondence between the reals in [0,1] and [0,2]. 


#9
Dec3112, 11:51 AM

P: 1,302

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. 


#10
Dec3112, 12:59 PM

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#11
Dec3112, 01:02 PM

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Edit: I take my thank you back. 


#12
Jan213, 03:21 PM

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[quote]amount of numbers[quote]
^ This is the crux of the problem. Talking about the "amount of numbers" in a set or which set is "bigger" (or "smaller" or "equal in size" or the notion of "size" at all) is an imprecise claim. When you're comparing two sets, there are many equally legitimate ways to do it. You can for example... Designate a count to each set, where the count is either the number of elements (if that number is finite) or infinity. Under this definition, the reals have the same count as the natural numbers because both are infinite. Designate a cardinality to each set. We say two sets have the same cardinality if there is a a bijection between them. We order the cardinalities with a relation, saying that A <= B when there is a bijection between A and some subset of B. In this case, card N < card R. However, card [0, 1] = card [0, 2] The notion of subset, too, gives us a way to compare sets. People take offense to the idea that [0, 1] is in some sense "the same" as [0, 2] because they are conflating two different senses of "same". One sense where [0, 1] and [0, 2] are clearly different is via a subset relation. Since [0, 1] is a subset of [0, 2], we can claim that [0, 1] is "less than" [0, 2] in this sense. However, this notion is different than the notions of cardinality or count above because it is a partial order (the other two are total orders). This means that you can't compare certain pairs of sets this way: [0, 1] has no relation to [2, 3]. There's one last common one that only works for special kinds of classes (including the real numbers), and that is measure. The sets [0, 1] has a certain property that, when written down on a number line, it gives us a certain length: 1  0 = 1. Similarly, the set [2, 5] also has a length: 5  2 = 3. This leads us to the notion of a measure. However, this is a very restricted notion, as it applies only to sets endowed with a measure space structure. And some sets, such as {0} in R or Q in R, have a measure of 0, despite having an infinite count. I post this because I feel too many people quickly jump on the idea that cardinality is the most important way to define a "size" of a set. I don't believe this is the case. In every day problem solving, count and the subset relation are just as important (if not more). 


#13
Jan213, 03:58 PM

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#14
Jan213, 04:36 PM

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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 onetoone 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? 


#15
Jan213, 04:40 PM

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P^n (N) (=P(P(P...P(N)))) for arbitrary n has the same cardinality with N.



#17
Jan213, 04:55 PM

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Who's P(N) ? Write an explicit formula using the 3 symbols: ..., { and }. Then calculate its cardinality.



#18
Jan213, 05:10 PM

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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 nonproper 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. 


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