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Weird or Wonderful?

October 3, 2012

Consider \mathbb{N} the set of natural numbers {1,2,3,4,5…} and \mathbb{E} the set of even numbers {2,4,6,8,…}. We know that \mathbb{E} \subset \mathbb{N}; the former is a proper subset of the latter. We also think that the size of a proper subset is smaller than the size of its superset.

However, we can have a mapping from \mathbb{N} to \mathbb{E} using f(n) = 2n. This function is bijective, meaning there is a one-to-one correspondence between \mathbb{N} and \mathbb{E}. That means then that the size of \mathbb{N} is equal to the size of \mathbb{E}, huh? How can this be? We just said that \mathbb{E} is a proper subset of \mathbb{N}. This is weird. No, this is wonderful. It tells us that what is obvious is not always true.

I guess this weird or wonderful phenomenon happens because we are dealing with sets that have been thought of as complete but how can they be complete when in fact the number of elements is infinite? In our discussion we have taken \mathbb{N} as an infinite set, a concept we talk about as if we have managed to contain infinite elements in it. Having done so we think we can now manipulate and do stuff with it. It is as if we have rounded the numbers including infinity,  as if we are rounding cattle.

For an alternative view why talking about an infinite set is not sensible see Prof. N. Wildberger’s videos.

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