I tried to prove the exercise problem in Kunen (Chapter IV, problem 36.)
Problem. Show that there is a formula $\chi(x)$, such that
$\chi$ represents ZF; i.e.,$$\phi\in \mathsf{ZF}\to \mathsf{ZF}\vdash\chi(\ulcorner\phi\urcorner)\quad\text{and}\quad \phi\notin \mathsf{ZF}\to \mathsf{ZF}\vdash \lnot\chi(\ulcorner\phi\urcorner)$$
If $\ulcorner\mathsf{ZF}\urcorner$ is added via the definition $\ulcorner\mathsf{ZF}\urcorner=\{x:\chi(x)\}$, then $\mathsf{ZF}\vdash \mathsf{CON(\ulcorner ZF\urcorner)}$.
(where $\mathsf{CON}(\ulcorner T\urcorner)$ is consistency argument of $T$ which is formalized within formal theory $\mathsf{ZF}$.)
But I am not even understand the problem. Why the second incompleteness does not applied in that case? In page 144-145 in Kunen, he writes
(...) We now have for each such $S$, a sentence $\mathsf{CON}(\ulcorner S\urcorner)$ in the language of set theory asserting that $S$ is consistent. (...) The Gödel Incompleteness Theorem shows that if $S$ is consistent and extends $\mathsf{ZF}$, then $S\nvdash\mathsf{CON}(\ulcorner S\urcorner)$. (Caution: this presupports that we used a "reasonable" $\chi_S$ to represent $S$.)
(Add: $\chi_S$ represents $S$; it was explained the front of above comment.)
I also don't understand that comment. I would be thankful for any reasonable explanation.