What is the formal definition of cardinality of sets? I know the intuition behind the concept of cardinality (the number of elements of a given set) and the formal definition of sets with equal cardinalities, but what's the formal definition of the cardinality of sets itself?
Thanks
 A: Formally, we can define cardinality in the following way.
A bijection $f:E\to F$ is a function from $E$ to $F$ which is both surjective and injective. We say that two sets are equivalent if there exists a bijection between them. Finally, a cardinality can be defined as an equivalence class for this equivalence relation. This means the cardinal $3$ will be the class of all sets of size $3$ (any two of these sets are in bijection with each other). Similarly, the cardinal $\aleph_0$ is the class containing the sets $\mathbb N, \mathbb N^2,\mathbb Q, \mathbb Z,\dots$
A: The cardinality of $A$ is the equivalence class of $A$ under the equivalence relation "equal cardinalities" (the existence of bijections).
Since in most cases this equivalence class is a proper class, we often identify it with a particular representative when we can. For example, finite ordinals or $\aleph$ numbers. 
To see why this makes sense, we return to the finite case, where in our language and intuition we don't separate the notions of "cardinal" from "cardinality". That is, $5$ is the cardinality of fingers I have attached to my left hand, but it's also the cardinal number of the fingers I have attached to my left hand.
