# Capturing the $\omega_1^{\mathrm{CK}}$-th stage of Gödel's constructible hierarchy

For an ordinal $$\alpha$$, let $$L_\alpha$$ be the $$\alpha^{th}$$ set of Gödel's constructible hierarchy and let $$\omega_1^{\mathrm{CK}}$$ be the first non-recursive ordinal or the first admissible ordinal after $$\omega$$.

Question : does there exists a $$\Sigma_2$$ formula able to capture that we reached, in Gödel's construction, the $$\omega_1^{\mathrm{CK}}$$-th stage? That is, is there some $$\Sigma_2$$ formula $$\phi$$ such that

• For all $$\alpha < \omega_1^{\mathrm{CK}}$$, $$L_\alpha \not\models \phi$$
• $$L_{\omega_1^{\mathrm{CK}}} \models \phi$$

If no, is $$\Pi_2$$ powerful enough?

This couldn't be achieved even by $$\Sigma_3$$ sentences. First note that $$L_{\omega_1^{CK}}$$ (as any other model of $$\mathsf{KP}\omega+L=V$$) satisfies the scheme of $$\Sigma_3$$-reflection: $$\varphi(\vec{p})\to \exists a \;(\mathsf{Trans}(a)\land \vec{p}\in a\land (\varphi(\vec{p}))^a),\text{ where \varphi is \Sigma_3}.$$ And note that there is a $$\Pi_2$$ sentence $$F$$ such that for each transitive set $$a$$, the sentence $$F$$ is true in $$a$$ iff $$a$$ is of the form $$L_{\omega(1+\alpha)}$$. Henceforth for any $$\Sigma_3$$ sentence $$\varphi$$, if $$L_{\omega_1^{CK}}\models\varphi$$ then there is a transitive set $$a\in L_{\omega_1^{CK}}$$ such that $$a\models \varphi\land F$$ which means that $$a$$ is of the form $$L_{\alpha}$$, where $$\alpha<\omega_1^{CK}$$ and $$L_{\alpha}\models \varphi$$.
Note that $$\omega_1^{CK}$$ is the least $$\alpha$$ such that $$L_{\alpha}$$ is a model of $$\mathsf{KP}\omega-\mathsf{Foundation}$$. And the theory $$\mathsf{KP}\omega-\mathsf{Foundation}$$ could be axiomatized by a single $$\Pi_3$$ sentence (the only axiom that isn't $$\Pi_2$$ is the axiom of $$\Sigma_1$$-collection). Note that everywhere in this answer the classes $$\Pi_n$$ were understood as consisting of formulas that start with an unbounded quantifier prefex $$\vec{\forall}\vec{x}_1\ldots \vec{Q}\vec{x}_n$$ followed by a $$\Delta_0$$ formula. However if we switch to the classes $$\hat\Pi_n$$ defined in terms of alternation depth of unbounded quantifiers (bounded quantifiers could appear anywhere) the answer changes. The axiom of $$\Sigma_1$$-collection is a $$\hat\Pi_2$$-sentence and hence $$\mathsf{KP}\omega-\mathsf{Foundation}$$ is $$\hat\Pi_2$$-axiomatizable.
However I don't know whether there is a $$\hat\Sigma_2$$ sentence that "captures" $$L_{\omega_1^{CK}}$$. Non-existence of $$\hat\Pi_1$$ sentence is trivial due to downward-absoluteness. And non-existence of $$\hat\Sigma_1$$ sentence follows from the fact that $$\mathsf{KP}\omega$$ proves $$\Sigma$$-reflection (the class $$\Sigma$$ is exactly $$\hat\Sigma_1$$).
• Thank you for your answer ! Could one say, in the same spirit, that there is $a \in L_{\omega_1^{CK}}$ witnessing $L_{\omega_1^{CK}} \models \exists \alpha \, \forall\beta \, \psi_0$ i.e. $L_{\omega_1^{CK}} \models \exists \alpha \in a \, \forall\beta \, \psi_0$ and that since $\omega_1^{CK}$ is a limit ordinal and can be written as a supremum of ordinals, $\omega_1^{CK} = \bigcup_i \alpha_i$, we have $a \in \alpha_k$ for some $k$ and eventually $L_{\alpha_k} \models \varphi$ as $a \in L_{\alpha_k} \subset L_{\omega_1^{CK}}$ ? Is the reflection principle hidden in the witness argument ? Jun 14 '19 at 18:35
• @Johan I am not sure whether I understood your question. What you are describing seems to be a proof of the fact that $L_{\omega_1^{CK}}$ (and actually $L_{\alpha}$ for any limit $\alpha$) satisfies $\Sigma_2$-reflection. And this is enough to provide an answer to your initial question about "capturing" $\omega^{CK}_1$ by a $\Sigma_2$-sentence. Jun 14 '19 at 19:24
• Oh right, this wasn't so clear. I meant indeed that this was enough for the $\Sigma_2$-case. It is however not enough $\Sigma_3$ or even $\Pi_2$ sentences, do you know a reference about this sentence $F$ used in your proof ? Jun 16 '19 at 15:00
• @Johan I don't know the refrence for exactly the construction of $F$. In Barwise book "Admissible sets and structures" he proved that the $\alpha\longmapsto L_{\alpha}$ is a $\Sigma$-function over $\mathsf{KP}$ (and hence it have $\Sigma_1$-definable graph). Using this definition we construct $\Pi_2$ sentence $F$: $\forall x\exists \alpha\;(x\in L_\alpha)$ and observe that $F$ holds in $L_{\alpha}$ for admissible $\alpha$ and every transitive model of $F$ is a level of $L$. Actually this is enough for the purposes I used $F$ in my comment. Jun 16 '19 at 18:49
• @Johan Nevertheless to obtain $F$ with the property that I described one need to guarantee that $\Sigma_1$ formula $\exists y\; \varphi(x,y,\alpha)$ ($\varphi$ is $\Delta_0$) that defines the relation $x\in L_{\alpha}$ have the property $\mathsf{KP}\vdash \exists y\; \varphi(x,y,\alpha)\mathrel{\leftrightarrow}\exists x\in L_{\alpha+\omega}$. Which could be done by an examination of how the definitions work. Jun 16 '19 at 18:49