Your tangent about big O seems out of place, I'm not sure I understand what point you're trying to make.

I'm proving that showing the difference in growth rates(cardinality) can be shown in a finite way.

Cardinality most certainly does not have to do with growth rate.

Except it literally does. When you compare two sets, and they grow, especially at infinity, you compare how fast they grow. Otherwise cardinality would have NO USE. Meaning it would ALWAYS be cardinality = infinity. Oh sorry that's it. No more.

You should get {1,2,3,4,5} and {2,4,6,8,10}. Same cardinalities, right?

Fatal error. You added 10 to the set, which was originally not in the set. Therefore the correct way to do this is the following:

Starting with the sets:

{1,3,5,7} and {2,4,6,8}

Swapping one odd and one even, exactly 1 iteration. So we switch 1 and 2 we get the following:

{2,3,5,7} and {1,4,6,8}

However, we swapped TWO total numbers. As in we swapped {1,2} which is where you get 2n. Now do this to infinity, and your result is that for every even number you swap, you actually swap 2 numbers. Because of the odd number you swap.