A Second Puzzle

Now, let's move to the new alphabet

$$\sim \: P \: M \: 1 \: 0$$

and identify natural numbers with their correlates in binary notation.

To each expression we assign the Gödel number of the expression. We do this according to the following scheme:

$\sim$	P	M	1	0
10	100	1000	10000	100000

The mirror of an expression $\phi$ is the expression $\phi$ followed by its Gödel number. A sentence is an expression having one of the following four forms (where $\nu$ is any number): $P \nu$, $P M \nu$, $\sim P \nu$, $\sim P M \nu$.

Naturally, $P \nu$ is true iff $\nu$ is the Gödel number of a printable expression, $P M \nu$ is true iff $\nu$ is the Gödel number of an expression whose mirror is printable, and so on.