Cardinality measures the set of even numbers to be equal in size to the set of natural numbers, which I have no problem with. Are there accepted alternative measures of size which do recognise the fact that there are two natural numbers for every even number?
For example, the even numbers and the odd numbers are two disjoint subsets equal in cardinality which can be placed in union to form the natural numbers. The direct union of the two distinct subsets into the third is surely a stronger measure of magnitude than cardinality since it preserves the index of the elements.
If we take the set of all natural numbers and remove the element $1$ from it, we can compare this new set with the natural numbers and say that the larger set contains every element which the smaller contains plus one further element which the smaller set does not. Cardinals don't allow us to say that this set is one larger. Is there some system that does?