cardinal numbers

The number of elements of a set is said to be the cardinal number. The cardinal number of a set $A$ is written by $\mbox{card}A$. If a set is finite, the cardinal number of the set becomes the number of elements. That is, if $A=\left\{-1,2,\sqrt{3},4,15 \right\}$, then $\mbox{card}A=5$.

Of course, $\mbox{card}\left\{0,1 \right\}=2$. We are able to define the cardinal number of the direct product of sets, too. If $A=\left\{0,1 \right\}\times \left\{0,1 \right\}\times \left\{0,1 \right\}$, or $A=\left\{(x_1, x_2, x_3)|x_i\in \left\{0, 1\right\}, i=1, 2, 3 \right\}$, then $\mbox{card}A=2^3=8$. If a set is finite, it is very simple.

In the case of an infinite set, you may not to be able to define the cardinal number, or may think that there is just only one cardinal number which is defined. However, we have understood an infinite set could be countable or uncountable. Hence, there are two types of infinite sets.

We define $\aleph_0$ as the cardinal number of a countably infinite set. For example,
\[ \mbox{card}\mathbb{Q}=\aleph_0  \]
Next, we define $\aleph $ as the cardinal number of real numbers.
\[ \mbox{card}\mathbb{R}=\aleph  \]
As real numbers are rational numbers plus irrational numbers, the number of elements of real numbers is more than those of rational numbers. Therefore,
\[ \aleph_0<\aleph \]
It was an astonishing fact discovered by G. Cantor that the number of elements of an infinite set also had a magnitude relationship.