Cardinal numbers

We already know how to measure the size (synonym: cardinality) of a finite set. For example, $\char93 \{3,4,5,6\}=4$, $\char93 \emptyset = 0$, and $\char93 \{a,\{b,c\}\}=2$. For centuries, people believed that there was no meaningful way to compare the sizes of infinite sets, but in the late 1800's Cantor developed a system for doing exactly this.

In his system, every set $S$ has a cardinality $\char93 S$ (alternative notation: $\vert S\vert$). If $S$ is finite, then $\char93 S$ is an ordinary nonnegative integer, as above. But if $S$ is infinite, then $\char93 S$ is a new kind of ``number,'' called a cardinal number or simply a cardinal. New symbols are needed: for instance, the cardinal numbers $\aleph_0$ (pronounced aleph-zero, aleph-nought, or aleph-null) and ${\mathfrak{c}}$ (the ``cardinality of the continuum'') are defined by

$$\aleph_0 := \char93 {\mathbb{N}},$$

$${\mathfrak{c}} := \char93 {\mathbb{R}}.$$