The Definability of Cardinal Numbers

One says that the sets x and y are equinumerous (in symbols, x ≈ y) if there is a 1 -- 1 function mapping x on y. The notion of the cardinal |x| of x is obtained from equinumerosity by abstraction. The use of |x| does usually not require any special apparatus. E.g., when we say |x| --- ℵ0 we mean to say that there is a 1 -- 1 function mapping x on the set of all natural numbers; when we say |x| < |y| we mean that there are 1 -- 1 functions mapping x into y, but none of them is onto y; etc. As is common in mathematics, one tends to pass from the abstract notion of cardinal numbers to real cardinal numbers, i.e., one wants to regard the cardinal numbers as objects of the mathematical system. This is where one encounters the problem of how to define the cardinal | x | of x as an object of set theory.