I want a method of attributing sizes with sets with the following properties (Assume AC):
- The collection of all sizes can be well-ordered in a poset. Let us call the relation in this poset $<$.
- If $A\subseteq B$ then the size of $A$ is at most $B$
- Every set in $V$ is given exactly one size
- Given a size, the collection of all sets with that size is a set*
- There is a proper class of sets that aren't their own size.
*This is optimal but may not be possible.
Cardinality satisfies 1, 2, 3, and 5, but not 4. Lebesgue measure satisfies 1,2,4, and 5, but not 3. Order type only satisfies 1, 2 and 5.
Are there any other methods of attributing sizes with sets known? They may not even satisfy all of these properties, I just want to know. However, if it is possible, please create such a poset of sizes well-ordered by $<$.
EDIT: Yes there is, it's called the rank of a set. I really should've thought about this more. Because of this, I now require that the size of a set (assuming AC) is an ordinal, and if the set is finite, its size is its cardinality.