[neonskies]

> Take the infinities of all numbers > 0 and then all even numbers > 0.

Define your number first, then we'll work it out from there. From the looks of it, you are considering the integers.

> So you have

> 1,2,3,4,5,6,.... 2,4,6,8,........

> Why can't we just consider both infinities to be the same size (they go on forever), but the item in a given position simply differs.

These sets have the same size. Two sets have the same size if you can find a function from one to the other which is bijective meaning if two sets match exactly element for element(elements don't have to be the same ones), they have the same size. For example, the sets {1, 2, 3} and {a, b, c} have the same size because we can match 1 to a, 2 to b and 3 to c. Or we can match 1 to c, 2 to a and 3 to b. So these two "matching" examples constitute two bijective functions. Going back to your example, the function f from {1,2,3,4,5,6,....} to {2,4,6,8,........} given by f(x) = 2x for x in {1,2,3,4,5,6,....} is bijective and therefore lines up the two sets in one to one correspondence. To prove f is bijective, we can find the inverse for f, multiply f and its inverse and get an identity or show f is surjective and injective. These are slightly technical, but not too bad. Any intro to discrete math textbook contains this material.