alternative definition of cardinality

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 $\mathrm{card}(A)$ equinumerous to $A$. Now, $\mathrm{card}(A)$ 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), $|A|:=\mathrm{card}(A)$ 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 $R(A):=\{\rho (B)\mid B\sim A\}$. 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)

$$\mathrm{kard}(A):=\{B\mid B\sim A\text{ and }\rho (B)=r(A)\},$$

and set $|A|:=\mathrm{kard}(A)$. Since every element in $\mathrm{kard}(A)$ is a subset of $V_{r(A)}$, $\mathrm{kard}(A)\subseteq V_{r{(A)}^{+}}$, 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, $\mathrm{kard}(n)\ne n$ 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 $A$:

$$|A|:=\{\begin{array}{cc}\mathrm{card}(A)\text{ if }A\text{ is well-orderable},\hfill & \\ \mathrm{kard}(A)\text{ otherwise }.\hfill & \end{array}$$

The one big advantage of this definition is clear: it does not require AC, and with AC, it is identical to the second definition above. At the same time, it also resolves the conflict with our intuitive notion about cardinality: the cardinality of a finite set is the number of elements in the set. However, the one big disadvantage in this definition is that we do not have $A\sim |A|$ in general (of course, $A$ is infinite). There is no way, without AC, to find a definition of $|A|$, such that $A\sim B$ iff $|A|=|B|$, and $A\sim |A|$ at the same time.

Title	alternative definition of cardinality
Canonical name	AlternativeDefinitionOfCardinality
Date of creation	2013-03-22 18:50:11
Last modified on	2013-03-22 18:50:11
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03E10