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Question from Brian:

I was reading an answer to a question on your site regarding infinite sets (, and I think they may have got the answer wrong.

I his example, he claims that the set of real numbers BETWEEN 0 AND 1 is larger than the set of positive integers.

Please correct me if I am wrong, but I believe those two sets are -- pardon the expression -- equally infinite. For any integer, there is a corresponding real number between 0 and 1, and vice versa.

For instance, using the decimal as a "mirror", you can create a mirror image of any real number between 0 and 1 as an integer (i.e. 0.1234 gets mirrored as the integer 4321 -- I could write it out algebraically, if you want, but you get my point)

Am I wrong?



Hi Brian,

Infinity is a very confusing concept. I understand your statement about both sets being infinite. They are. And one could stop there and never wonder if there was a way to talk about the "size" of an infinite set. Mathematicians say that two sets have the same size when they can be put into 1-1 correspondence. There isn't a 1-1 correspondence between the positive integers and the real numbers in (0, 1). The answer to which you refer demonstrates that. The argument is really slippery: to prove there is no 1-1 correspondence, you first assume there is one and then you find a real number in (0,1) that doesn't correspond to any integer. That contradicts the assumption that to every real in (0,1) there was a corresponding integer. This argument shows that the set of positive integers and the set of reals in (0,1) don't have the same size (using this particular concept of two sets having the same size). Since it is possible to put the positive integers into 1-1 correspondence with a proper subset of the reals in (0,1), there is a sense in which it is reasonable to say that (0,1) is "bigger" than the set of positive integers.

You actually make only two assertions that aren't true. One is the statement about 1-1 correspondences. The other is the statement about mirror images. Under your scheme, which integer would correspond to 1/3 = .3333333...? The string of threes does not terminate. The "mirror image across the decimal point" is not an integer because integers have a finite (therefore terminating) collection of digits.

I hope this makes some sense.

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