this doesn't make sense to me, smart people? (infinity)

[daviddem]

Hey surviva, if it helps, Galileo had the same problem you have. Only instead of using the example of the set of natural numbers and the set of even natural numbers, he used the set of natural numbers and the set of the squares of natural numbers {0,1,4,9,16,25,36,...}. This problem is called "Galileo's paradox", and the solution to it is just as I said: dump the old notion of "size" and replace it with cardinality. Or rather, extend, generalize the notion of size to that of cardinality.

See here:

http://en.wikipedia.org/wiki/Two_New_Sciences#Infinity

and here:

http://en.wikipedia.org/wiki/Galileo%27s_paradox

From the first article, Galileo's conclusion:

We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," greater," and "less," are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.

And following this, the remark of the author of the article:

This conclusion, that ascribing sizes to infinite sets should be ruled impossible, owing to the contradictory results obtained from these two ostensibly natural ways of attempting to do so, is certainly a consistent resolution to the problem but less powerful than that used nowadays. In contemporary mathematics, the problem is resolved instead by only considering Galileo's first definition of what it means for sets to have equal sizes; that is, the ability to put them in to one-to-one correspondence. This turns out to yield a way of comparing the sizes of infinite sets that is free from contradictory results.