TITLE: Is the propositional set countably infinite?
QUESTION [6 upvotes]: Recently I'm learning logic. Here is the definition from the book "Logic For Computer Science":
A countable set $PS$ of proposition symbols: $p_0,p_1,\dots$
The set $\text{Prop}$. propositions is the smallest that satisfies:

Every proposition symbol $p_i$ is in
$\text{Prop}$.
Whenever $\varphi$ is in $\text{Prop}$, $\neg \varphi$ is also in
    $\text{Prop}$.
Whenever $\varphi, \psi$ are in $\text{Prop}$, $(\varphi \vee \psi), (\varphi\wedge \psi)$ and $(\varphi\rightarrow\psi)$ are also in $\text{Prop}$.
A string is in $\text{Prop}$ only if it is
formed by applying the rules $1, 2$ and  $3$.


Now I want to know whether $\text{Prop}$ is countably infinite, and if so how to prove that.

REPLY [3 votes]: Just how quick and dirty a proof is acceptable? Take your favourite Gödel-numbering style coding system for strings of symbols, and take the function that takes $n$ to the wff with code $n$ (if there is one) and to $P_0$ otherwise. And voilà, a surjective function from $\mathbb{N}$ to the set of wffs, which makes the latter countable by definition.