Meanwhile all I've ever wanted for Christmas was a notion of size that admits (A ⊆ B) ∧ (A ≠ B) ⇒ size(A) < size(B).
There are also ordinals, which are much finer than cardinals but have the disadvantage that they only apply to well-ordered sets, and ordinal arithmetic works very different to the naturals.
And then a final one - exotic models of the real numbers like the surreals contain infinite quantities of different sizes that can be compared, divided, added just as you like and the order relation works intuitively. The disadvantage here is that they are generally harder to define and reason about than the ordinary reals. The reason people like them is that you they contain infinitesmals too, which you can use to formalise an alternative foundation for calculus.
(although strictly speaking this last one isn't about sets and sizes, it's in a similar cluster of ideas about infinite arithmetic)
But now let's divide each element of E by 2 to produce a set D. Now that set D is {1, 2, 3, ...} aka N the natural numbers. But we just applied a function to each element of E, so how can D have a different size than E? Each element of E maps to exactly one element of D: 2 -> 1, 4 -> 2, 6 -> 3, etc. So does size(D) = size(E)? Or does size(D) = size(N)?
I imagine you're right that this leads to a contradiction somewhere, but I (obviously) haven't thought it through. I just would love to see someone try to break enough axioms to make it work and see what comes out of it, or show that it contradicts either itself or something we see in the real world if we do that.
But yes, the ordinals might be more to your liking. If you equip your sets with more structure you can say more things about them.
You are assuming that this doesn't change the size and certainly that's how the normal notion of size works. But the question is whether we can create any order relationship on the sets with the desired properties.
The properties he mentioned defines a partial order and partial orders can be extended to total orders (given axiom of choice). So it is in fact possible.
But a partial order of all sets? How is this function "size" even defined? Functions don't have a domain of all sets in conventional set theory.