The concept of cardinality comes from the notion of equinumerosity of sets. To define the cardinality $|A|$ of a set $A$ , one desirable property is that $A$ is equinumerous to $B$ precisely when $|A|=|B|$ . The first attempt, due to Frege and Russel, is to define a relation $\sim$ on the class $V$ of sets so that $A\sim B$ iff there is a bijection from $A$ to $B$ . This relation is an equivalence relation on $V$ . Then we can define $|A|$ as the equivalence class containing the set $A$ . However, $|A|$ is not a set, so we can't do much with $|A|$ in ZF.

The second attempt, due to Von Neumann, defines $|A|$ to be the smallest ordinal $\card(A)$ equinumerous to $A$ . Now, $\card(A)$ exists if $A$ is well-orderable. But in general, we do not know if $A$ is well-orderable unless the well-ordering principle is applied, which is just another form of the axiom of choice. Thus, this definition depends on AC, and, in everyday mathematical usage (which assumes ZFC), $|A|:=\card(A)$ suffices.

The third way, due to Scott, of looking at $|A|$ , without AC, is to modify the first attempt somewhat, so that $|A|$ is a set. Recall that the rank of a set $A$ is the least ordinal $\alpha$ such that $A\subseteq V_{\alpha}$ in the cumulative hierarchy. A set having a rank is said to be grounded. By the axiom of foundation, every set is grounded. For any set $A$ , let $R(A):=\lbrace \rho(B)\mid B\sim A\rbrace$ . Then $R(A)$ , as a class of ordinals, has a least element $r(A)$ . So $r(A)\le \rho(A)$ . Next, we define (borrowing the terminology used in the first reference below) $$\kard(A):=\lbrace B \mid B\sim A\mbox{ and }\rho(B)=r(A)\rbrace,$$ and set $|A|:=\kard(A)$ . Since every element in $\kard(A)$ is a subset of $V_{r(A)}$ , $\kard(A)\subseteq V_{r(A)^+}$ , so that $|A|$ is a set. This method is known as Scott's trick. It can also be used in defining other isomorphism types on sets. It is easy to see that $|A|=|B|$ iff $A\sim B$ . However, with this definition, $\kard(n)\ne n$ in general, where $n$ is a natural number.

Nevertheless, it is known that every finite set is well-orderable, and so we come to the fourth definition of the cardinality of a set: given a set $A$ :

Bibliography

1

H. Enderton, Elements of Set Theory, Academic Press, Orlando, FL (1977).

2

T. J. Jech, Set Theory, 3rd Ed., Springer, New York, (2002).

3

A. Levy, Basic Set Theory, Dover Publications Inc., (2002).