\label{def:goedel-numbering-as-nf-address}
Let $\mathrm{ZFC\textrm{-}VM} = (\mathrm{Sent}_{\mathrm{ZFC}},\, {\vdash}_{\mathrm{ZFC}},\, \ulcorner\cdot\urcorner,\, \mathrm{Con}(\mathrm{ZFC}))$ be the ZFC virtual machine (Definition~\ref{def:zfc-as-e2-vm}).  The \textbf{NF-address system of the ZFC-VM} is the G\"odel numbering function
\begin{equation}\label{eq:ch64-goedel-nf}
\ulcorner\cdot\urcorner \colon \mathrm{Sent}_{\mathrm{ZFC}} \;\longrightarrow\; \mathbb{N},
\end{equation}
satisfying the following properties:
\begin{enumerate}
\item\emph{(Injectivity.)} Distinct sentences receive distinct codes: $\ulcorner\varphi\urcorner = \ulcorner\psi\urcorner$ implies $\varphi = \psi$.

\item\emph{(Decodability.)} There exists a primitive recursive function $\mathrm{dec} \colon \mathbb{N} \to \mathrm{Sent}_{\mathrm{ZFC}} \cup \{\bot\}$ such that $\mathrm{dec}(\ulcorner\varphi\urcorner) = \varphi$ for every sentence $\varphi$, and $\mathrm{dec}(n) = \bot$ if $n$ is not the G\"odel number of any sentence.

\item\emph{(Self-reference.)} For every primitive recursive predicate $P$ on $\mathbb{N}$, there exists a sentence $\sigma_{P}$ such that
$\mathrm{ZFC} \vdash \sigma_{P} \leftrightarrow P(\ulcorner\sigma_{P}\urcorner)$.
This is the diagonal lemma, which closes the self-referential loop of the NF-address system.
\end{enumerate}
The G\"odel number $\ulcorner\varphi\urcorner$ is the NF address of the sentence $\varphi$ in the ZFC-VM's code space.