I have trouble working out a proof in the second part of
> Jean-Pierre Ressayre and Alex Wilkie. <b>Modèles non standard en arithmétique et théorie des ensembles</b>. Publications Mathématiques de l'Université Paris VII, 1987.
On page 140, Ressayre writes:
> <b>4.6 Théorème</b> […] – b – En revanche, l'énoncé plus faible “$\forall\gamma\ \exists(\mathrm L_{\alpha_i})_{i\leqslant\gamma}$ chaîne faiblement $\Pi_n$-élémentaire” n'est pas $\omega$-conséquence de $\Sigma_{n+1}$-$(\text{collection}+\text{fondation})$.
> <b>Preuve de (b)</b>: on applique 4.5 dans un $\omega$-modèle non standard $M$.
This refers back to page 139 on which he writes:
> <b>4.5 Corollaire</b> – Pour tout modèle dénombrable non standard $M$ de $\Pi_n$-collection, et tout ordinal non standard $\rho$ de $M$, il existe $I\subset^{\rm e}\mathrm{On}^M$ tel que $\rho\in I$ et $\mathrm L_I^M\models\Pi_n\text{-collection}+{}$“il n'existe pas de chaîne $\Pi_n$-élémentaire $(\mathrm L_{\alpha_i})_{i\leqslant\rho+\rho}$”
Perhaps I should explain some of the terms used here.
- $I\subset^{\mathrm e}\mathrm{On}^M$ means $I$ is a proper initial segment of $\mathrm{On}^M$ and $\mathrm{On}^M\setminus I$ has no minimum element.
- If $I\subset^{\mathrm e}\mathrm{On}^M$, then $\mathrm L_I^M=\bigcup_{\alpha\in I}\mathrm L_\alpha^M$.
- _$\Pi_n$-collection_ denotes the scheme consisting of (extensionality, pair, union, foundation, $\Delta_0$-separation, and) all sentences of the form
$$\forall a,\bar c\bigl(\forall x\in a\ \exists y\ \theta(x,y,\bar c)\rightarrow
\exists b\ \forall x\in a\ \exists y\in b\ \theta(x,y,\bar c)\bigr)$$
where $\theta\in\Pi_n$.
- _$\Gamma$-foundation_ is the scheme saying “every nonempty parametrically $\Gamma$-definable class/set has an $\in$-minimal element”.
- A chain $(\mathrm L_{\alpha_i})_{i\leqslant\gamma}$ is _(weakly) $\Pi_n$-elementary_ if $\mathrm L_{\alpha_i}\prec_{\Pi_n}\mathrm L_{\alpha_\gamma}$ for all $i<\gamma$.
Corollary 4.5 visually gives a model of $\Pi_n$-collection, or equivalently, $\Sigma_{n+1}$-collection. It is also easy to verify that this model satisfies $(\Sigma_n\cup\Pi_n)$-foundation by elementarity. It is, however, not clear to me how to get a model of $\Sigma_{n+1}$-foundation out of 4.5, and I see no reason why $(\Sigma_n\cup\Pi_n)$-foundation should imply $\Sigma_{n+1}$-foundation. From how Ressarye writes about it, the proof is apparently straightforward (if not immediate).
Does anyone have any idea of how an argument showing $\Sigma_{n+1}$-collection in Theorem 4.6(b) above can go?
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**Edit:** With the help of Google Translate, I made a rough translation of Ressayre's statements quoted above:
> <b>4.5 Corollary</b> – For every nonstandard denumerable model $M$ of $\Pi_n$-collection, and every nonstandard ordinal $\rho$ of $M$, there exists $I\subset^{\rm e}\mathrm{On}^M$ such that $\rho\in I$ and $\mathrm L_I^M\models\Pi_n\text{-collection}+{}$“there does not exist a $\Pi_n$-elementary chain $(\mathrm L_{\alpha_i})_{i\leqslant\rho+\rho}$”
> <b>4.6 Theorem</b> […] – b – On the other hand, the weaker assertion “$\forall\gamma\ \exists(\mathrm L_{\alpha_i})_{i\leqslant\gamma}$ that is weakly $\Pi_n$-elementary” is not an $\omega$-consequence of $\Sigma_{n+1}$-$(\text{collection}+\text{foundation})$.
> <b>Proof of (b)</b>: one applies 4.5 to a nonstandard $\omega$-model $M$.
Please feel free to edit the text for any improvements on the translation.