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Principle of counting says that

"the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall."

This does not match with my common sense (I am not a mathematician, but a CS student).

Can some people here could help me to reach a mathematicians level of thinking for this problem. I have searched net a lot (Wikipedia also)

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    You probably have never counted them all yet. :)2012-11-21
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    In the inifinite case, it is no longer true that if $A$ is a proper subset of $B$, then the "size of $A$" is strictly less than the "size of $B$." That intuition is true in finite sets, but would prove useless when dealing with infinite sets.2012-11-21
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    Note, most mathematicians don't say "the number of odd integers" because that is confusing the word "number." Rather, we talk about the "cardinality of the set of odd integers."2012-11-21
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    The thing is, even if you think you do, you have very little *common sense* regarding «counting infinite sets». You, like most people, are just extrapolating your life-long experience with manipulating *finite* sets, and there is no reason —if you think about it— to expect things to work the same.2012-11-21
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    One way to think about "cardinality" is that it is just one way to characterize the size of infinite sets. There are other ways - for example, measure theory gives you a sense of two sets being "different sizes" that is different from the result of cardinality. Or you could define the notion of "density" of sets of natural numbers. What you'll find as you mature as a mathematician is that cardinality is a "primary" measure in a lot of ways that might not initially seem obvious.2012-11-21
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    To add on Thomas' last comment, the idea behind cardinality is to discard internal structure, because there are plenty of very large sets which don't have any internal structure which makes "intuitive sense" for us.2012-11-21

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