I tried to prove the exercise problem in Kunen (Chapter IV, problem 36.)

Problem. Show that there is a formula $\chi(x)$, such that

  1. $\chi$ represents ZF; i.e.,$$\phi\in \mathsf{ZF}\to \mathsf{ZF}\vdash\chi(\ulcorner\phi\urcorner)\quad\text{and}\quad \phi\notin \mathsf{ZF}\to \mathsf{ZF}\vdash \lnot\chi(\ulcorner\phi\urcorner)$$

  2. 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.

  • 1
    $\begingroup$ I can't explain why the second incompleteness theorem doesn't apply, but the following is (I think) a $\chi$ satisfying the two conditions: $\chi(x)$ iff $x\in ZF$ and $Con(\{y\in ZF: y \leq x\})$. $\endgroup$ – GME Dec 1 '14 at 11:51
  • 2
    $\begingroup$ GME's proposed representation works. This weird example is due, I believe, to Feferman. The reason that the second incompleteness theorem doesn't apply is that it assumes that the representation of the theory is effective (i.e. computable) and the proposed one isn't. $\endgroup$ – Miha Habič Dec 1 '14 at 12:07
  • $\begingroup$ @Miha, GME: Thanks for your comment. It is really helpful for me. $\endgroup$ – Hanul Jeon Dec 2 '14 at 10:58
  • $\begingroup$ This is an awesome problem. (+1 for Kunen). $\endgroup$ – hot_queen Dec 16 '14 at 19:17

Thanks to GME and Habič's comments, I understand and can prove that problem. This answer is just filling in the details of their comments.

Let define $$\chi(n)\leftrightarrow n\in \mathsf{ZF}\land \mathsf{CON}\{m\in\mathsf{ZF}:m\le n\}.$$

We regard $\mathsf{ZF}$ as the recursive set of Gödel numberings of the axioms of ZF. If $\phi\in \mathsf{ZF}$, then $\mathsf{ZF}\vdash (\ulcorner\phi\urcorner\in \mathsf{ZF})$. Also, if $\psi_1$, $\psi_2$, $\cdots$, $\psi_k$ are axioms of ZF whose Gödel number is less than the Gödel number of $\phi$ then by reflection, the consistency of $\{\psi_1,\cdots,\psi_k,\phi\}$ is provable from ZF; that is, $\mathsf{ZF}\vdash \mathsf{CON}\{m\in\mathsf{ZF}:m\le \ulcorner\phi\urcorner\}$. Therefore $\mathsf{ZF}\vdash\chi (\ulcorner\phi\urcorner)$. If $\phi\notin \mathsf{ZF}$ then $\mathsf{ZF}\vdash (\ulcorner\phi\urcorner\notin \mathsf{ZF})$ so $\mathsf{ZF}\vdash\lnot\chi(\ulcorner\phi\urcorner)$

We will prove that, if $\ulcorner\mathsf{ZF}\urcorner$ is added via the definition $$\ulcorner\mathsf{ZF}\urcorner=\{n:\chi(n)\}$$ then $\mathsf{ZF}\vdash \mathsf{CON(\ulcorner ZF\urcorner)}$. We will use the reduction to absurdity within the formal theory $\mathsf{ZF}$.

If $\lnot\mathsf{CON(\ulcorner ZF\urcorner)}$ holds, then there is a proof $\pi$ from some assumptions $a_1,a_2,\cdots,a_n\in \ulcorner \mathsf{ZF}\urcorner$ to contradiction. But we already know that $\mathsf{CON}\{a_1,a_2,\cdots a_n\}$ holds. So we cannot derive a contradiction from $a_1$, $\cdots$, $a_n$. From this, we get $\mathsf{ZF+}\lnot\mathsf{CON(\ulcorner ZF\urcorner)}$ derives contradiction so $\mathsf{ZF}\vdash \mathsf{CON(\ulcorner ZF\urcorner)}$.

It does not contradict with second imcompleteness theorem since $\chi$ is at least $\Pi_1^0$, so is not recursive.


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