The idea is that if there is any bijection at all between the two sets then they are the same "size". So you can make all kinds of relations between two sets t

The idea is that if there is any bijection at all between the two sets then they are the same "size". So you can make all kinds of relations between two sets that are not bijections but once you find one you have satisfied the definition of same size in this sense. Note that it is a way of defining size that corresponds to what we mean with finite sets and then extends that definition in a consistent way. So yes you could define a "malandrew size" however you like and see what kind of insights you get.