Actually the proof is not so easy. It is not technically hard, but you
have to be very careful about definitions.
First, of course, you need to know what countable means, and that was
given in @Raphael's answer.
The proof relies on the fact that the set of finite sequences of elements of
a countable set is itself countable. You may try to prove this. Look
at the proof that rational numbers are countable.
You can read that more intuitively as implying that anything that has
a finite description using a finite set of symbols is countable.
Then a possible way to answer your question is to check whether this
is the case for pushdown automata.
We know from the definition that they are all finitely described. Then
the remaining question is whether the set of symbols used is itself
countable. But short of defining that set (how?), we cannot answer
The simple answer is to state that PDAs are defined up to an
isomorphism. Actually a very simple isomorphism, which is a simple
renaming of symbols used in the description, which has to be finite for each PDA. Then it is always
possible to take the symbols in a given countable set, for example the