> What drives me bananas is when anybody starts using the words "size" or "larger" or "smaller". I will insist to my dying day that while the cardinality of eve

> What drives me bananas is when anybody starts using the words "size" or "larger" or "smaller". I will insist to my dying day that while the cardinality of even integers is the same as that of integers, the set of even integers is still smaller than the set of integers. That's it's still half the size. Note that size of a set has nothing to do with individual set members and subset relationship. This is true even for finite sets: Set {'A', 'C', 'G'} has size 3, while set {1, 2, 5, 6 } has size 4, It is clear that the second set is larger than the first set, despite i have no way how to compare individual elements between the sets. But what does even mean that set {'A', 'C', 'G'} has size 3? We get such result by counting the elements, which is just process of defining one-to-one mapping to {0, 1, 2} (or traditionally {1, 2, 3}, it does not matter), canonical representation of size 3. From this point of view, the concept of infinite cardinality is just a natural extension of this.