I have heard that there are infinities of various sizes. I was wondering what that actually means-how do we compare their cardinalities? I have just started real analysis and I am slowly coming to terms with notions of countability,Cantor's diagonalization method and limit points.Can anyone please explain, in simple words,what that actually means?
What does the notion of different sizes of infinity really mean?
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real-analysis
infinity
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1Have you looked around the site? I am fairly certain this question was answered before. – 2012-12-14
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1You might want to look at this: http://math.stackexchange.com/questions/5378/types-of-infinity?rq=1 , and http://math.stackexchange.com/questions/1/different-kinds-of-infinities – 2012-12-14
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0And also http://math.stackexchange.com/questions/182171/are-all-infinities-equal – 2012-12-14
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1If you had two finite piles of beans, and you didn't know how to count, how would you tell which pile had more beans? – 2012-12-14
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0@ThomasAndrews You'd find two pots of water of equivalent size. You put the bean piles in separate pots. Whichever displaces more water either has more beans or almost surely has more food value, and that way you've started soaking the beans so you can cook them later. Alright, so what's a better example where one-to-one pairing might work out as a little more relevant? – 2012-12-14
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0@DougSpoonwood That doesn't work, because you don't know each bean is the same size. You don't want the weight or volume, you want to know they have the same number. (Might be easier to imagine if they were coins of the same denomination, but possibly different weights/sizes due to when they were minted.) Hint: When I said "you can't count," assume you can count to $1$ - that is, you know how to take one bean off a pile. – 2012-12-14
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0@ThomasAndrews I edited my comment and added "or almost surely has more food value." You're right that doesn't take into account weight or volume and that makes using pots irrelevant for determining the number of beans. But, it doesn't make such a method irrelevant for determining how much value the piles of beans have in terms of nutrition... why else would we want to know how many beans we have in a pile? – 2012-12-14
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0@DougSpoonwood You are being obtuse, since in practicality, you can also count them, so what is the value of not knowing how to count? The point is to notice that we can determine which pile has more beans without having an ability to "count," so that the relative size of finite sets is independent of our ability to count. Then we later apply that same idea, with variations, to infinite sets. – 2012-12-14
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0http://www.youtube.com/watch?v=UPA3bwVVzGI – 2012-12-14
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0@ThomasAndrews For finite sets if we pair off *each* element of set A with an element of set B which has no more than one element of A which it got paired with, and there exists at least one more element of set B which did not get paired off with an element of set A, then set B has more members. This does NOT work for the idea of cardinality in Cantorian set theory. If you pair the natural number 1 with the rational number 1, 2 with 2, and so on, the above method valid for finite sets would indicate the rationals as having a larger size than the naturals. That is NOT Cantorian cardinality. – 2012-12-14
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0@DougSpoonwood As I said, "with variations." The point is to start with the idea of comparing sizes without counting in the finite case, before jumping into the complexity of Cantor. (Without choice, two arbitrary sets aren't even necessarily comparable. Indeed, the "pairing up one at a time" is, in the infinite case, also assuming we can well-order the sets of beans, on some level, and we are really comparing the ordinals if we were to really extend my idea to infinite sets.) – 2012-12-14