A predicate $P$ is expressible (in PA) if there exists a formula $\phi(x_1,\ldots, x_n)$ of $L_A$ such that for all $m_1,\ldots, m_n$ elements of $\mathbb{N}$, we have that $P(m_1,\ldots, m_n)$ holds in $\mathbb{N}$ iff for all $M\models PA$, $M\models\phi(m_1,\ldots, m_n)$.
How do we show that that the following properties of strings of symbols are expressible?
- Being a term.
- Being a formula.
- Being a sentence.
- Being a proof in PA.
With a lot of work (not difficult, but a bit tiresome!)
There are two routes to showing those properties are expressible. You can show, relative to a fixed sensible scheme of Gödel-numbering, that they are primitive recursive properties and then show (quite generally) that any primitive recursive property is expressible in the language of PA, and expressible by a $\Sigma_1$ wff.
Or, second route, you can proceed more directly, and directly produce $\Delta_0$ wffs which (relative to a fixed sensible scheme of Gödel-numbering of course) express those properties. (So this is a slightly better result as it gives us $\Delta_0$ rather than $\Sigma_1$ expressibility, though the difference matters little for most purposes.)
For the first route, see e.g. my Introduction to Gödel's Theorems (Chs 15 and 19). For the second direct route, see e.g. Christopher Leary, A Friendly Introduction to Mathematical Logic (Ch. 4).
[The detailed stories are, however, far too long to give here. Leary takes 30+ pages. I take 12 pages, cutting corners, in my book. I cut even more corners in Ch. 6 of the freely downloadable http://www.logicmatters.net/resources/pdfs/gwt/GWT.pdf ]