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Why define cardinality to distinguish between $|\mathbb{Z}|$ and $|\mathbb{R}|$? They are completely different objects. One is countable, the other has the least upper bound property. In my mind it's like comparing isometries to permutations. I just don't see a motivation to distinguish between the two in terms of cardinality.

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closed as off-topic by Andres Caicedo, Lord_Farin, Dominic Michaelis, no identity, azimut Nov 21 '13 at 8:53

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$\mathbb{Z}$ has the least upper bound property too. :) –  Hurkyl Nov 20 '13 at 20:56
They are both sets. Why not consider whether each is comparable to the other? Comparing $\Bbb Z$ with $\Bbb Q$ is favorable... There's only a Galois algebraic completion or two left before the completion is complete... –  abiessu Nov 20 '13 at 20:56
Because we can. –  copper.hat Nov 20 '13 at 20:59
Much of mathematics involves summing over everything in a set. When that set is countable (or only countably many of them give a non-zero contribution), then the summation can make sense. Any larger cardinality and its nonsense (unless you're a physicist, maybe). So knowing that $\Bbb R$ isn't countable is very important. –  zibadawa timmy Nov 20 '13 at 21:00
You ask something, and then note that one is countable and the other isn't, which is precisely why we distinguish them in cardinality. A simple example of why this might be useful: Increasing functions on the reals can have at most countably many discontinuities. –  Thomas Andrews Nov 20 '13 at 21:01

2 Answers 2

Yes, you can differentiate the objects themselves, but what happens when you strip down the structure? You don't care about order, addition, multiplication, whatever. You just want to know, is there or is there not a bijection between the two sets?

Why is it interesting? Well, Cantor himself wrote to Dedekind that he cannot believe that there is a bijection between $[0,1]$ and $[0,1]^2$. The math was on the paper, he saw the result and its correctness, but he didn't believe it.

You are living in an age where these sort of results are deeply engraved into our collective mind, so it doesn't surprise us as much as it did surprise Cantor. But the fact is that this is a very nontrivial fact, that $\Bbb R$ is uncountable, while $\Bbb Z$ and $\Bbb Q$ are.

Finally, you can notice that cardinality is not only used to show that $\Bbb Z$ and $\Bbb R$ are completely different objects. Cardinality arguments are used through and through in set theory, in topology and even in analysis and algebra sometimes. Tis true, the further you get from set theory the less likely you are to see cardinality arguments, but they are there from time to time. If you feel that set theory has no merits as a field of research, it's your right to do so, but let me disagree and point out that it is a very rich field of research. If you do agree that there some merit to working on set theory, then cardinality arguments are very important there, and should not be taken lightly.

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If you don't distinguish between $|\mathbb{Z}|$ and $|\mathbb{R}|$ then you can't state a correct version of the Baire category theorem. The real line is not the union of $|\mathbb{Z}|$ many closed nowhere dense subsets, but it is the union of $|\mathbb{R}|$ many closed nowhere subsets, namely singletons.

Also, I don't see why "comparing isometries to permutations" would be bad, because isometries of a metric space with itself are permutations of the underlying set. I think you are too quick to conclude that two things are "completely different objects."

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Originally I wanted to appeal to the Baire category theorem, but I didn't do that very well. So I started over again and stayed somewhat vague about cardinality arguments. I'm glad that you came along and posting a proper answer about that! :-) –  Asaf Karagila Nov 20 '13 at 23:02

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