I should have been more careful in choosing my words. I agree that {1,2,3,...} and {2,4,6,...} are both countably infinite and have the cardinality aleph-0. When I said they are not the same size, I did not mean to invoke the idea of set cardinality. Rather, I was thinking more in terms of set difference: if set A contains all the elements of B and set A contains other elements also, then set A is bigger than set B.
With that said, the reasoning that you two are employing seems mysterious. In my way of thinking, the cardinality of a set of dollar bills and the quantity of dollars are not the same thing. If you start with $20 and lose $20, then you lost $20, and likewise, if you start with an infinite quantity of dollars and lose $20, then you lost $20. Whether the set of dollars before and after gambling have the same cardinality is quite beside the point: $20 never equals $0, so you were $20 richer before you gambled and $20 less rich after you gambled.
With that said, the reasoning that you two are employing seems mysterious. In my way of thinking, the cardinality of a set of dollar bills and the quantity of dollars are not the same thing. If you start with $20 and lose $20, then you lost $20, and likewise, if you start with an infinite quantity of dollars and lose $20, then you lost $20. Whether the set of dollars before and after gambling have the same cardinality is quite beside the point: $20 never equals $0, so you were $20 richer before you gambled and $20 less rich after you gambled.