We now have all the tools necessary to talk about cardinal numbers. Or so you would think.
There are many competing ways to define cardinal numbers that rely on different axioms. We start with a simple definition used by Russell and Frege. This definition has problems and is not valid in ZFC set theory, but will demonstrate what we want a cardinal number to be.
Let A,B be arbitrary sets. Then the cardinal number \overline{\overline{A}} is the class of all sets equnumerous to A,
\overline{\overline{A}} = \{B \mid A \approx B\}.
We say \overline{\overline{A}} is the cardinality of A.
If we were to take this definition, we would have the following.
Using the Frege-Russell definition of cardinal numbers, we will define the cardinal numbers of the sets 0, 1 and 2.
Since we have that 0 = \varnothing, we have that
\overline{\overline{0}} = \{A \mid A \approx \varnothing\} = \varnothing.
Then \overline{\overline{0}} is the set of all sets containing no members. Similarly for 1 we have
\overline{\overline{1}} = \{A \mid A \approx \{\varnothing\} \} = \text{Set of all Singletons}.
Then