I already asked if there are more rationals than integers here...
However, there is one particular argument that I didn't give before which I still find compelling...
Every integer is also a rational. There exist (many) rationals that are not integers. Therefore there are more rationals than integers.
Obviously, in a sense, I am simply choosing one particular bijection, so by the definition of set cardinality this argument is irrelevant. But it's still a compelling argument for "size" because it's based on a trivial/identity bijection.
EDIT please note that the above paragraph indicates that I know about set cardinality and how it is defined, and accept it as a valid "size" definition, but am asking here about something else.
To put it another way, the set of integers is a proper subset of the set of rationals. It seems strange to claim that the two sets are equal in size when one is a proper subset of the other.
Is there, for example, some alternative named "size" definition consistent with the partial ordering given by the is-a-proper-subset-of operator?
EDIT clearly it is reasonable to define such a partial order and evaluate it. And while I've use geometric analogies, clearly this is pure set theory - it depends only on the relevant sets sharing members, not on what the sets represent.
Helpful answers might include a name (if one exists), perhaps for some abstraction that is consistent with this partial order but defined in cases where the partial order is not. Even an answer like "yes, that's valid, but it isn't named and doesn't lead to any interesting results" may well be correct - but it doesn't make the idea unreasonable.
Sorry if some of my comments aren't appropriate, but this is pretty frustrating. As I said, it feels like I'm violating some kind of taboo.