Proof We prove the lemma by induction on $n\geq 1$ . The base case $n=1$ follows by combining Magidor’s theorem (cf. Theorem 2.1) with the well-known fact that supercompact cardinals are $\Sigma _2$ -correct [Reference Kanamori18, Proposition 22.3]. Thus, we shall assume by induction that the lemma holds for n and prove it for $n+1$ .

Let $\delta $ be a $\Sigma _{n+1}$ -supercompact cardinal, and let $\varphi (x,y)$ be a $\Pi _{n+1}$ formula with $a\in V_\delta $ . Suppose first that $V_\delta $ satisfies the sentence $\exists x\varphi (x,a)$ and let $b\in V_\delta $ be a witness for it. Since $\delta $ is $\Sigma _n$ -supercompact, the induction hypothesis guarantees that $\delta \in C^{(n+1)}$ , so that $\varphi (b,a)$ is true.

Conversely, suppose that $\exists x\varphi (x,a)$ is true. Let $\lambda <\delta $ be such that $a\in V_\lambda $ . Let b be a witness for this formula and let $\mu \in C^{(n+2)}\setminus \delta ^+$ be such that $b\in V_\mu $ . Then, $V_\mu \models \exists x\varphi (x,a)$ . By the $\Sigma _{n+1}$ -supercompactness of $\delta $ we may find $\bar {\lambda },\bar {\delta }<\bar {\mu }<\delta $ , $\bar {\mu }\in C^{(n+2)}$ , $\bar {a}\in V_{\overline {\lambda }}$ , and $j:V_{\bar {\mu }}\rightarrow V_\mu $ such that $\mathrm {crit}(j)=\bar {\delta }$ , $j(\bar {\delta })=\delta $ , $j(\bar {\lambda })=\lambda $ , and $j(\bar {a})=a$ . In particular, $V_{\bar {\mu }}\models \exists x\varphi (x,\bar {a})$ . Notice that, by elementarity, $\bar {\lambda }<\bar {\delta }$ , hence $a=j(\bar {a})=\bar {a}$ , and thus $V_{\bar {\mu }}\models \exists x\varphi (x,a)$ . Finally, since $\bar {\mu }<\delta $ and $\bar {\mu }\in C^{(n+2)}$ , our induction hypothesis yields $V_{\bar {\mu }}\prec _{\Sigma _{n+1}}V_\delta $ . From this latter assertion it is clear that $V_\delta \models \exists x\varphi (x,a)$ , as wanted.

Proof Suppose that $\delta $ is $C^{(n)}$ -extendible. Fix any $\lambda>\delta $ in $C^{(n+1)}$ and $a\in V_\lambda $ . By $C^{(n)}$ -extendibility, let $\mu>\lambda $ in $C^{(n+1)}$ , and let $j: V_\mu \longrightarrow V_\theta $ be such that $\mathrm {crit}(j)=\delta $ , $j(\delta )>\mu $ and $j(\delta )\in C^{(n)}$ , for some ordinal $\theta $ . Notice that $j\upharpoonright V_{\lambda }\in V_\theta $ .

Claim 2.4.1. $V_\theta $ satisfies the following sentence $:$

$$ \begin{align*} \exists \bar{\lambda}<j(\delta)\;\exists \bar{\delta}<\bar{\lambda}\;\exists \bar{a}\in V_{\bar{\lambda}}\;\exists j^*: V_{\bar{\lambda}}\longrightarrow V_{j(\lambda)}\nonumber\\ (j^*(\bar{a})=j(a)\;\wedge\; j^*(\bar{\delta})=j(\delta)\;\wedge\; V_{\bar{\lambda}}\prec_{\Sigma_{n+1}} V_{j(\lambda)}).\nonumber \end{align*} $$

By elementarity, $V_\mu $ satisfies the above sentence with the parameters $j(\delta )$ and $j(\lambda )$ replaced by $\delta $ and $\lambda $ , respectively. Hence, since $\mu \in C^{(n+1)}$ , the sentence is true in the universe. Since $\lambda $ was arbitrarily chosen in $C^{(n+1)}$ , this implies that $\delta $ is a $\Sigma _{n+1}$ -supercompact cardinal.

For the converse implication, let $\lambda $ be greater than $\delta $ and let us show that there exists an elementary embedding $j: V_\lambda \longrightarrow V_\theta $ , for some ordinal $\theta $ , such that $\mathrm {crit}(j)=\delta $ , $j(\delta )>\lambda $ , and $j(\delta )\in C^{(n)}$ . Take $\mu>\lambda $ in $C^{(n+1)}$ and let $\bar {\delta },\bar {\lambda }<\bar {\mu }$ and $j:V_{\bar {\mu }}\longrightarrow V_\mu $ be such that $\mathrm {crit}(j)=\bar {\delta }$ , $j(\bar {\delta })=\delta $ , $j(\bar {\lambda })=\lambda $ , and ${\bar {\mu }}\in C^{(n+1)}$ . Now notice that the sentence

(1) $$ \begin{align} \exists \alpha\;\exists j^*: V_{\bar{\lambda}}\longrightarrow V_\alpha\,(\mathrm{crit}(j^*)=\bar{\delta}\;\wedge\; j^*(\bar{\delta})>\bar{\lambda}\;\wedge\; {j^*(\bar{\delta})}\in C^{(n)}) \end{align} $$

is $\Sigma _{n+1}$ -expressible. Moreover, it is true in $\mathrm {V}$ as witnessed by $\lambda $ and j because $j(\bar {\delta })=\delta>\bar {\lambda }$ and $\delta \in C^{(n)}$ (cf. Lemma 2.3). Thus, since $V_{\bar {\mu }}$ is $\Sigma _{n+1}$ -correct and contains $\bar {\delta }$ and $\bar {\lambda }$ , it is also true in $V_{\bar {\mu }}$ . By elementarity, $V_\mu $ thinks that the sentence

$$ \begin{align*} \exists \alpha\;\exists j^*: V_{{\lambda}}\longrightarrow V_\alpha\,(\mathrm{crit}(j^*)={\delta}\;\wedge\; j^*({\delta})>{\lambda}\;\wedge\; {j^*(\delta)}\in C^{(n)}) \end{align*} $$

is true. Since $\mu \in C^{(n+1)}$ , the above displayed sentence is true in $\mathrm {V}$ and so $\delta $ is $\lambda $ - $C^{(n)}$ -extendible. As $\lambda $ was arbitrarily chosen, $\delta $ is a $C^{(n)}$ -extendible cardinal.

Proof Let $\kappa $ be an accumulation point of K. Let $p\in \mathbb {P}$ and $\tau \in V^{\mathbb {P}}$ be such that $p\Vdash _{\mathbb {P}}\tau \in V[\dot {G}]_\kappa $ . Since $\kappa $ is a limit cardinal there is $q\leq _{\mathbb {P}} p$ and $\lambda <\kappa $ such that $q\Vdash _{\mathbb {P}} \tau \in V[\dot {G}]_\lambda $ . Extending q if necessary, we may find $\theta \in K\cap \kappa $ above $\lambda $ such that $q\Vdash _{\mathbb {P}}\tau \in V[\dot {G}]_\lambda \subseteq V_\theta [\dot {G}_\theta ]$ . Thus, $\kappa \in K$ , as wanted.

Proof On the one hand, by induction on the rank and using the fact that $\mathbb {P}_{\lambda }$ is $\lambda $ -cc and preserves the inaccessibility of $\lambda $ , one can easily show that $V[G_\lambda ]_\lambda \subseteq V_{\lambda } [G_\lambda ]$ .

On the other hand, as $\mathbb {P}_\lambda $ preserves the inaccessibility of $\lambda $ , $|V[G_\lambda ]_\lambda |=\lambda $ . Hence, since $\Vdash _{\mathbb {P}_{\lambda }}\text {"}\,\dot {\mathbb {Q}} \text { is } \lambda \text {-distributive},\text {"}$ and so $i_{G_\lambda }(\dot {\mathbb {Q}})$ does not add any new subsets of $V[G_\lambda ]_\lambda $ , we have

$$ \begin{align*} V[G]_\lambda \subseteq V[G_\lambda]_\lambda. \end{align*} $$

Hence, $V[G]_\lambda \subseteq V_\lambda [G_\lambda ]$ .

Proof Note first that the only atomic formulas in the language $\mathcal {L}$ are of the form “ $x\in y$ ,” “ $x=y$ ,” or “ $x\in \mathbb {P}\,,$ ” where x and y are variable symbols. Hence, if $\mathbb {P}$ is $\Sigma _m$ (resp. $\Pi _m$ ) definable, then the formula $\text {"}x \in \mathbb {P}\,\text {"}$ is equivalent to a $\Sigma _m$ (resp. $\Pi _m$ ) formula of the language of set theory. It follows that a Boolean combination of atomic formulas of the language $\mathcal {L}$ is equivalent to a Boolean combination of $\Sigma _m$ and $\Pi _m$ formulas in the language of set theory. Hence, the truth predicate for Boolean combinations of atomic formulas in the language $\mathcal {L}$ is $\Delta _{m+1}$ definable. The same applies to formulas with bounded quantifiers. By induction on k one can now show (as in [Reference Jech16, Section 13]) that $\vDash _{\Sigma _k^{ \mathcal {L}}}$ (resp. $\vDash _{\Pi _k^{ \mathcal {L}}}$ ) is $\Sigma _{m +k}$ (resp. $\Pi _{m+k}$ ) definable.

If $\mathbb {P}$ is $\Delta _m$ -definable, with $m\geq 1$ , then “ $x\in \mathbb {P}$ ” is both $\Sigma _m$ and $\Pi _m$ -expressible. It easily follows that $\vDash _{\Sigma _0^{\mathcal {L}}}$ is $\Delta _m$ -definable, and by induction on k one readily shows that $\vDash _{\Sigma _k^{ \mathcal {L}}}$ (resp. $\vDash _{\Pi _k^{ \mathcal {L}}}$ ) is $\Sigma _{m +k-1}$ (resp. $\Pi _{m+k-1}$ ) definable.

Notation 3.8. For $m<\omega $ , we will denote by $\Gamma _m$ (resp. $\mathbf {\Gamma }_m$ ) the collection of all the formulas in the language of set theory that are either $\Sigma _m$ or $\Pi _m$ (resp. with parameters). In the sequel, expressions such as “X is $\Gamma _m$ ” should be read as “X is definable by a formula in $\Gamma _m$ .”

Proof The class $C^{(0)}_{\mathbb {P}}$ is $\Pi _0$ , being the class of all ordinals. If $k\geq 1$ , then we have that $\alpha \in C^{(k)}_{\mathbb {P}}$ if and only if $\alpha \in C^{(k-1)}_{\mathbb {P}}$ and

$$ \begin{align*}\forall X,Y (X=V_\alpha \wedge Y=\mathbb{P}\cap X \to \forall \bar{a}\in X\,\forall\varphi \in\Sigma^{\mathcal{L}}_k(\,\vDash_{\Sigma_k^{\mathcal{L}}}\varphi(\bar{a})\to\\[-36pt]\end{align*} $$

$$ \begin{align*} \langle X,\in,Y\rangle\vDash \varphi(\bar{a}))). \end{align*} $$

Now using induction on k and Proposition 3.6 the complexity of the class $C^{(k)}_{\mathbb {P}}$ is easily seen to be as desired.

Proof Assume $p\Vdash _{\mathbb {P}} \varphi (\tau )$ . Suppose, for a contradiction, that $q\in \mathbb {P}_\kappa $ is such that $q\leq _{\mathbb {P}_\kappa } p$ and $M_\kappa \models \text {"}q\Vdash _{\mathbb {P}_\kappa } \neg \varphi (\tau ).\text {"}$ Suppose G is $\mathbb {P}$ -generic over V, with $q\in G$ . Then

$$ \begin{align*}V[G]\models \varphi (i_G(\tau)).\end{align*} $$

Note that, since $\mathbb {P}^{M_\kappa } =\mathbb {P}_\kappa $ , a direct limit is taken at $\kappa $ (for otherwise some conditions in $\mathbb {P}_\kappa $ would be proper classes in $M_\kappa $ ), and therefore being a dense subset of $\mathbb {P}_\kappa $ is absolute between M and V. Thus, $G_\kappa :=G\cap \mathbb {P}_\kappa $ is $\mathbb {P}_\kappa $ -generic over $M_\kappa $ . Also, since $\tau \in M_\kappa $ , hence $\tau \in M_\kappa ^{\mathbb {P}_\kappa }$ , $i_{G_\kappa }(\tau )\in M_\kappa [G_\kappa ]$ and $i_G(\tau )=i_{G_\kappa }(\tau )$ . Since $\varphi $ is $\Delta _0$ , hence absolute for transitive classes, $M_\kappa [G_\kappa ] \models \varphi (i_{G_\kappa }(\tau )).$ But as $q\in G_\kappa $ , this contradicts $M_\kappa \models \text {"}q\Vdash _{\mathbb {P}_\kappa }\neg \varphi (\tau ).\text {"}$

A similar argument proves the converse.

Proof The proof of Proposition 3.11 shows that “ $p\Vdash _{\mathbb {P}} \varphi (\tau )$ ” holds if and only if

$$ \begin{align*} \forall M \, \forall \kappa ( M \mbox{ is a transitive set } \wedge \kappa \in M \wedge M_\kappa\models ZFC^\ast \wedge \mathbb{P}^{M_\kappa} =\mathbb{P}_\kappa \, \wedge \\[-36pt]\end{align*} $$

$$ \begin{align*}p,\tau \in M_\kappa \to M_\kappa \models \text{"}p\Vdash_{\mathbb{P}_\kappa}\varphi(\tau)\text{"})\end{align*} $$

and also if and only if

$$ \begin{align*} \exists M \exists \kappa (M \mbox{ is a transitive set } \wedge \kappa \in M \wedge M_\kappa\models ZFC^\ast \wedge \mathbb{P}^{M_\kappa} =\mathbb{P}_\kappa \, \wedge \\[-36pt]\end{align*} $$

$$ \begin{align*} p,\tau \in M_\kappa \wedge M_\kappa \models \text{"}p\Vdash_{\mathbb{P}_\kappa}\varphi(\tau)\text{"}). \end{align*} $$

Now the two displayed sentences above are easily seen to be $\Pi _1^{\mathcal {L}}$ and $\Sigma _1^{\mathcal {L}}$ , respectively, with p and $\tau $ as parameters.

Items (2) and (3) now follow easily from (1) and Proposition 3.6.

Proof First, note that the class $V^{\mathbb {P}}$ of $\mathbb {P}$ -names is $\Delta _{1}^{\mathcal {L}}$ . Now given a $\Sigma _k$ formula $\exists x \forall y \exists z \ldots \psi (x,y,z,\ldots ,t)$ of the language of set theory, and given a condition $p\in \mathbb {P}$ and $\tau \in V^{\mathbb {P}}$ ,

$$ \begin{align*}p\Vdash_{\mathbb{P}} \exists x \forall y \exists z \ldots \psi (x,y,z,\ldots ,\tau)\end{align*} $$

if and only if

$$ \begin{align*}\exists \sigma (\sigma \in V^{\mathbb{P}} \wedge \forall \sigma' (\sigma'\in V^{\mathbb{P}} \to \exists \sigma"(\sigma"\in V^{\mathbb{P}} \wedge \ldots p\Vdash_{\mathbb{P}} \psi (\sigma ,\sigma ', \sigma", \tau)))).\end{align*} $$

Since $p\Vdash _{\mathbb {P}} \psi (\sigma ,\sigma ', \sigma ", \tau )$ is $\Delta _1^{\mathcal {L}}$ (Corollary 3.12), the last displayed sentence is $\Sigma _{k}^{\mathcal {L}}$ , as wanted.

Items (2) and (3) follow easily from (1) and Proposition 3.6.

Proof Note that $\mathbb {P}$ forces $V[\dot {G}]_\kappa \subseteq V_\kappa [\dot {G}_\kappa ]$ if and only if the $\mathcal {L}$ -sentence

$$ \begin{align*} \forall p, \tau \,(\tau \in V^{\mathbb{P}} \wedge p\Vdash_{\mathbb{P}}\text{"}rk(\tau)<\kappa\text{"}\to \\[-36pt]\end{align*} $$

$$ \begin{align*} \exists \sigma, q\, (\sigma \in V^{\mathbb{P}}\wedge q\leq p\wedge rk(\sigma)<\kappa\wedge q\Vdash_{\mathbb{P}}\text{"}\sigma=\tau\text{"})) \end{align*} $$

holds in $\langle V,\in ,\mathbb {P}\rangle $ . Thus, since $V^{\mathbb {P}}$ is a $\Delta _1^{\mathcal {L}}$ -definable class, and the expressions “ $p\Vdash _{\mathbb {P}} rk(\tau )<\kappa $ ” and “ $q\Vdash _{\mathbb {P}}\sigma =\tau $ ” are also $\Delta _1^{\mathcal {L}}$ (Corollary 3.12), it is easily seen that the sentence “ $\kappa $ is a $\mathbb {P}$ -reflecting cardinal” is $\Pi _{2}^{\mathcal {L}}$ , with $\kappa $ as an additional parameter.

Items (2) and (3) follow easily from (1) using Proposition 3.6.

Proof The claim is clear for $k=0$ . So, assume inductively that $\mathbb {P}$ forces $V[\dot {G}]_\kappa \prec _{\Sigma _{k-1}} V[\dot {G}]$ . Notice that, as $\kappa $ is $\mathbb {P}$ -reflecting, any member of $V[\dot {G}]_\kappa $ is given by a $\mathbb {P}$ -name in $V_\kappa $ . Let $\varphi ({x})$ be a $\Sigma _k$ formula in the language of set theory and let $\tau \in V_\kappa $ be a $\mathbb {P}_\kappa $ -name such that $p\Vdash _{\mathbb {P}}\varphi (\tau )$ , for some $p\in \mathbb {P}$ . We only need to show that $p\Vdash _{\mathbb {P}} \text {"}V[\dot {G}]_\kappa \models \varphi (\tau ).\text {"}$

Proof of claim Suppose, aiming for a contradiction, that $p' \leq _{\mathbb {P}_\kappa } p\restriction \kappa $ is such that $q\not \Vdash _{\mathbb {P}}\varphi (\tau )$ , for all $q\leq _{\mathbb {P}_\kappa } p'$ . Since $\langle V_\kappa ,\in ,\mathbb {P}_\kappa \rangle \prec _{\Sigma _k}\langle V,\in ,\mathbb {P}\rangle $ , and for every $q\in \mathbb {P}_\kappa $ the sentence “ $q \not \Vdash _{\mathbb {P}} \varphi (\tau )$ ” is $\Pi _{k}^{\mathcal {L}}$ (Proposition 3.13), we have that “ $q\not \Vdash _{\mathbb {P}_\kappa } \varphi (\tau )$ ” holds in $\langle V_\kappa ,\in , \mathbb {P}_\kappa \rangle $ . Therefore,

$$ \begin{align*} \langle V_\kappa ,\in , \mathbb{P}_\kappa\rangle\models \text{"}p' \Vdash_{\mathbb{P}_\kappa} \neg \varphi(\tau).\text{"} \end{align*} $$

Again, since $\langle V_\kappa ,\in ,\mathbb {P}_\kappa \rangle \prec _{\Sigma _{k}}\langle V,\in ,\mathbb {P}\rangle $ , and the quoted displayed sentence is $\Pi _{k}^{\mathcal {L}}$ ,

$$ \begin{align*} \langle V,\in, \mathbb{P}\rangle \models \text{"}p' \Vdash_{\mathbb{P}} \neg \varphi(\tau),\text{"} \end{align*} $$

which yields the desired contradiction to the fact that $p\Vdash _{\mathbb {P}}\varphi (\tau )$ .

By the claim, $p\restriction \kappa \Vdash _{\mathbb {P}}\varphi (\tau )$ , and since $\langle V_\kappa ,\in ,\mathbb {P}_\kappa \rangle \prec _{\Sigma _{k}}\langle V,\in ,\mathbb {P}\rangle $ and the above sentence is $\Sigma ^{\mathcal {L}}_{k}$ we have

$$ \begin{align*} \langle V_\kappa,\in,\mathbb{P}_\kappa\rangle\vDash \text{"}p\restriction \kappa \Vdash_{\mathbb{P}_\kappa}\varphi (\tau).\text{"} \end{align*} $$

Thus, if G is $\mathbb {P}$ -generic over V, with $p\in G$ , then $V_\kappa [G_\kappa ]\models \varphi (i_{G_\kappa }(\tau ))$ . Hence, since $i_{G_\kappa }(\tau )=i_G(\tau )$ , and $\kappa $ is $\mathbb {P}$ -reflecting, $V[G]_\kappa \models \varphi (i_G(\tau ))$ , as wanted.

Proof First, the assertion “ $\mathbb {P}\cap V_\kappa =\mathbb {P}_\kappa $ ” is $\Pi _{m+1}$ (in the parameter $\kappa $ ) if $\mathbb {P}$ is $\Pi _m$ -definable, and $\Pi _m$ if $\mathbb {P}$ is $\Sigma _m$ -definable. For the quoted sentence holds if and only if:

$$ \begin{align*}\forall x (x\in \mathbb{P} \to ({\mathrm{rk}}(x)<\kappa \leftrightarrow {\mathrm{lh}}(x)<\kappa)).\end{align*} $$

Second, by Proposition 3.14, the sentence “ $\kappa $ is a $\mathbb {P}$ -reflecting cardinal” is $\Pi _{m+2}$ if $\mathbb {P}$ is $\Gamma _m$ -definable; and it is $\Pi _{m+1}$ if $\mathbb {P}$ is $\Delta _m$ -definable. Finally, as shown in Proposition 3.9, for $k\geq 0$ the class $C^{(k)}_{\mathbb {P}}$ is $\Pi _{m+k}$ if $\mathbb {P}$ is $\Pi _m$ or $\Sigma _m$ -definable; and it is $\Pi _{m+k-1}$ if $\mathbb {P}$ is $\Delta _m$ -definable.

An easy calculation now shows that the predicate “ $\kappa $ is a $\mathbb {P}$ - $\Sigma _k$ -reflecting cardinal” has the claimed complexity.

Proof (1): Assume $\delta $ is $\mathbb {P}$ - $\Sigma _{n}$ -supercompact. Let $\lambda>\delta $ be a $\Sigma _{n}$ -correct cardinal and let $a\in V_\lambda $ . Let $\kappa>\lambda $ be a $\mathbb {P}$ - $\Sigma _n$ -reflecting cardinal. Notice that $V_\kappa \vDash \text {"}\lambda \in C^{(n)}.\text {"}$ By $\mathbb {P}$ - $\Sigma _n$ -supercompactness, there are $\bar {\delta } < \bar {\kappa }<\delta $ , with $\bar {\kappa }$ being $\mathbb {P}$ - $\Sigma _n$ -reflecting, and there are $\bar {a}\in V_{\bar {\kappa }}$ and $\bar {\lambda } <\bar {\kappa }$ , together with an elementary embedding $j:V_{\bar {\kappa }}\longrightarrow V_\kappa $ such that $\mathrm {crit}(j)=\bar {\delta } \ j(\bar {\delta })=\delta $ , $j(\bar {a})=a$ , and $j(\bar {\lambda })=\lambda $ . By elementarity, $V_{\bar {\kappa }}$ thinks that $\bar {\lambda }$ is a $\Sigma _{n}$ -correct cardinal. Since $\bar {\kappa }\in C^{(n)}$ , it follows that $\bar {\lambda }\in C^{(n)}$ .

(2) and (3): Assume $\delta $ is $\Sigma _3$ -supercompact (in case (2), i.e., $m=n=1$ ) or $\Sigma _{m+n}$ -supercompact, in case (3). Let $\lambda>\delta $ be $\mathbb {P}$ - $\Sigma _{n}$ -reflecting, and let $\kappa>\lambda $ be a $\Sigma _{m+n}$ -correct cardinal. Since being a $\mathbb {P}$ - $\Sigma _{n}$ -reflecting cardinal is a $\Pi _{3}$ property (case (2)), and a $\Pi _{m+n}$ property otherwise (case (3)), $V_\kappa $ thinks that $\lambda $ is $\mathbb {P}$ - $\Sigma _{n}$ -reflecting (cf. Proposition 3.17). By our assumption, there exist $\bar {\delta }<\bar {\lambda }<\bar {\kappa }<\delta $ with $\bar {\kappa }\in C^{(3)}$ (case (2)), or $\bar {\kappa }\in C^{(m+n)}$ , otherwise, and there exists an elementary embedding $j:V_{\bar {\kappa }}\longrightarrow V_\kappa $ such that $\mathrm {crit}(j)=\bar {\delta }$ , $j(\bar {\delta })=\delta $ , and $j(\bar {\lambda })=\lambda $ . By elementarity, $V_{\bar {\kappa }}$ thinks that $\bar {\lambda }$ is $\mathbb {P}$ - $\Sigma _{n}$ -reflecting. Since $V_{\bar {\kappa }}\prec _{\Sigma _{3}}V$ (case (2)), or $V_{\bar {\kappa }}\prec _{\Sigma _{m+n}}V$ , otherwise, $\bar {\lambda }$ is $\mathbb {P}$ - $\Sigma _{n}$ -reflecting in $\mathrm {V}$ . Thus, the restriction $j\upharpoonright V_{\bar {\lambda }}$ witnesses the $\mathbb {P}$ - $\Sigma _n$ -supercompactness of $\delta $ .

Proof Closedness is obvious. As for unboundedness, fix any $\lambda $ and build inductively a sequence $\{\theta _n\}_{n\in \omega }$ of ordinals greater than $\lambda $ such that for all $\eta \geq \theta _{n+1}$ ,

$$ \begin{align*} \Vdash_{\mathbb{P}_{\eta}}\text{"}\dot{\mathbb{Q}}_{\eta} \text{ is } \theta_n\text{-directed closed."} \end{align*} $$

Notice now that $\theta ^*:=\sup _n\theta _n$ is an element of C.

Proof Suppose G is $\mathbb {P}$ -generic over $\mathrm {V}$ . By Corollary 2.7 and Lemma 3.16, it is sufficient to take an arbitrary $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinal $\lambda>\delta $ and any $\alpha <\lambda $ , and find a $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinal $\bar {\lambda }$ , ordinals $\bar {\delta }, \bar {\alpha }<\bar {\lambda }$ , and an elementary embedding $j: V[G]_{\bar {\lambda }}\longrightarrow V[G]_\lambda $ such that $\mathrm {crit}(j)=\bar {\delta }$ , $j(\bar {\delta })=\delta $ , and $j(\bar {\alpha })=\alpha $ .

So pick a $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinal $\lambda>\delta $ and any $\alpha <\lambda $ . Since $\delta $ is $\mathbb {P}$ - $\Sigma _{n+1}$ -supercompact there exist $\bar {\delta }<\bar {\lambda }< \delta $ and $\bar {\alpha }<\bar {\lambda }$ , together with an elementary embedding $j: V_{\bar {\lambda }}\longrightarrow V_{\lambda }$ such that $:$

1. (i) $\mathrm {crit}(j)=\bar {\delta }$ and $j(\bar {\delta })=\delta $ .

2. (ii) $j(\bar {\alpha })=\alpha $ .

3. (iii) $\bar {\lambda }$ is $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting.

It will then suffice to show that j can be lifted to an elementary embedding $j:V_{\bar {\lambda }}[G_{\bar {\lambda }}]\longrightarrow V_\lambda [G_\lambda ]$ , for then, since both $\lambda $ and $\bar {\lambda }$ are $\mathbb {P}$ -reflecting, we have that $V_\lambda [G_\lambda ]=V[G]_\lambda $ and $V_{\bar {\lambda }}[G_{\bar {\lambda }}]=V[G]_{\bar {\lambda }}$ .

The iterations $\mathbb {P}_{\bar {\lambda }}$ and $\mathbb {P}_{{\lambda }}$ factorize as follows:

1. (i) $\mathbb {P}_{\bar {\lambda }}\cong \mathbb {P}_{\bar {\delta }}\ast \mathbb {Q}$ with $|\mathbb {Q}|\leq \bar {\lambda }$ .

2. (ii) $\mathbb {P}_{{\lambda }}\cong \mathbb {P}_{{\delta }}\ast \mathbb {Q}^*$ with

   $$ \begin{align*} \Vdash_{\mathbb{P}_{\delta}} \text{"}\mathbb{Q}^* \text{ is weakly homogeneous and } \delta\text{-directed closed."} \end{align*} $$

Indeed, (i) is clear since $\bar {\lambda }$ is a $\beth $ -fixed point. For (ii), since $\mathbb {P}$ is weakly homogeneous and $\lambda $ is $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting, $\mathbb {P}_\lambda $ is weakly homogeneous in $V_\lambda $ , and therefore $\mathbb {P}_\delta $ forces that $\mathbb {Q}^\ast $ is weakly homogeneous. Thus, we only need to see that $\Vdash _{\mathbb {P}_{\delta }}$ “ $\mathbb {Q}^*$ is $\delta $ -directed closed.”

Recall from Proposition 5.2 that the class

$$ \begin{align*} C(\mathbb{P}):=\{\mu : \forall\eta\geq\mu,\, \Vdash_{\mathbb{P}_\eta}\text{"} \dot{\mathbb{Q}}_{\eta} \text{ is } \mu\text{-directed closed"}\} \end{align*} $$

is a club class. Thus, it will be sufficient to show that $\delta $ is a limit point of $C(\mathbb {P})$ , and therefore it belongs to $C(\mathbb {P})$ . So, let $\mu <\delta $ and notice that since $\mathbb {P}$ is a suitable iteration, the sentence $\varphi (\mu )$ asserting:

$$ \begin{align*} \exists \theta>\mu\, \forall\eta\geq\theta \; ( \; \Vdash_{\mathbb{P}_\eta}\text{"}\dot{\mathbb{Q}}_\eta \text{ is } \mu\text{-directed closed"})\end{align*} $$

holds in $\mathrm {V}$ . Since $\mathbb {P}$ is $\Delta _{m+1}$ -definable, $\varphi (\mu )$ is easily seen to be equivalent to the $\Sigma _{m+2}$ sentence

$$ \begin{align*} \exists \theta>\mu \, \forall\eta\geq \theta\, \forall \alpha >\eta \, (\alpha \in C^{(m)}\to \\[-36pt]\end{align*} $$

$$ \begin{align*} V_\alpha\vDash \text{"}\Vdash_{\mathbb{P}_\eta}`\,\dot{\mathbb{Q}}_\eta \text{ is } \mu\text{-directed closed}\, ' \text{"}). \end{align*} $$

Since $\delta $ is a $\Sigma _{n+2}$ -correct cardinal (by Lemma 2.3 and Proposition 4.2), and $m\leq n$ , there must be a witness for $\varphi (\mu )$ below $\delta $ . Arguing inductively, we define an $\omega $ -sequence of ordinals above $\mu $ with limit in $C(\mathbb {P})$ . This shows that $C(\mathbb {P})$ is unbounded in $\delta $ , as wanted.

Since $\delta \in C^{(n+2)}$ , and j is elementary with $j(\bar {\delta })=\delta $ , we have that $j(\mathbb {P}_{\bar {\delta }})=\mathbb {P}_\delta $ . Also, since $\bar {\delta }$ is the critical point of j, we have that $j\text {"}G_{\bar {\delta }}=G_{\bar {\delta }} \subseteq G_\delta $ , and so $j\restriction V_{\bar {\lambda }}$ can be lifted to an elementary embedding

$$ \begin{align*} j:V_{\bar{\lambda}}[G_{\bar{\delta}}]\longrightarrow V_{\lambda}[G_\delta]. \end{align*} $$

Let us denote by $G_{[\bar {\delta },\bar {\lambda })}$ and $G_{[{\delta },{\lambda })}$ the filters $G\cap \mathbb {Q}$ and $G\cap \mathbb {Q}^*$ , respectively. Notice that these filters are generic for $\mathbb {Q}$ and $\mathbb {Q}^*$ over $V_{\bar {\lambda }}[G_{\bar {\delta }}]$ and $V_{\lambda }[G_\delta ]$ , respectively. In order to lift the embedding j to the further generic extension $V_{\bar {\lambda }}[G_{\bar {\lambda }}]=V_{\bar {\lambda }}[G_{\bar {\delta }}][G_{[\bar {\delta },\bar {\lambda })}]$ , notice first that $j"G_{[\bar {\delta },\bar {\lambda })}$ is a directed subset of $\mathbb {Q}^\ast $ of cardinality $\leq \bar {\lambda }$ . Also, since $j\upharpoonright V_{\bar {\lambda }}\in V_\lambda [G_\delta ]$ , and $j"G_{[\bar {\delta },\bar {\lambda })}$ can be computed from $j\upharpoonright V_{\bar {\lambda }}$ and $G_{[\bar {\delta },\bar {\lambda })}$ , we have that $j"G_{[\bar {\delta },\bar {\lambda })}\in V_\lambda [G_\delta ]$ . Therefore, since $\mathbb {Q}^*$ is a $\delta $ -directed closed forcing notion in $V_{\lambda }[G_\delta ]$ , there is some condition $p\in \mathbb {Q}^*$ such that $p\leq q$ , for every $q\in j"G_{[\bar {\delta },\bar {\lambda })}$ . Thus, p is a master condition in $\mathbb {Q}^*$ for the embedding j and the generic filter $G_{[\bar {\delta },\bar {\lambda })}$ . So, if $H\subseteq \mathbb {Q}^*$ is a generic filter over $V_{\lambda }[G_\delta ]$ containing p, then j can be lifted to an elementary embedding

$$ \begin{align*} j: V_{\bar{\lambda}}[G_{\bar{\lambda}}]\longrightarrow V_{\lambda}[G_\delta\ast H]. \end{align*} $$

By taking $H\subseteq \mathbb {Q}^*$ as in the claim above, we thus obtain a lifting

$$ \begin{align*}j:V_{\bar{\lambda}}[G_{\bar{\lambda}}]\longrightarrow V_{\lambda}[G_\lambda],\end{align*} $$

as wanted.

Proof Since $\mathbb {P}$ is adequate, $\Delta _2$ -definable, and there exists a proper class of $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinals, Corollary 4.3 implies that $\delta $ is $\mathbb {P}$ - $\Sigma _{n+1}$ -supercompact. Now, Theorem 5.4 applies to get the desired result.

Proof We begin with the following auxiliary claim:

Claim 5.7.1. For each $\alpha \in \operatorname {\mathrm {ORD}}$ there is a $\mathbb {P}$ -name $\tau _\alpha $ such that

$$ \begin{align*} \Vdash_{\mathbb{P}} \text{"}\tau_\alpha=V[\dot{G}]_{\alpha}.\text{"} \end{align*} $$

Proof By induction on $\alpha $ . If $\alpha =0$ , then the result is clear. So, assume that $\langle \tau _\beta \mid \beta <\alpha \rangle $ has already been found, and let us find $\tau _\alpha $ . If $\alpha $ is a limit ordinal, then let $\tau _\alpha :=\bigcup _{\beta <\alpha }\tau _\beta $ . Clearly, $\tau _\alpha $ is as desired.

Now suppose that $\alpha $ is a successor ordinal; say, $\alpha =\bar {\alpha }+1$ . Let $\varphi \colon \delta \rightarrow \tau _{\bar {\alpha }}$ be a bijection, some cardinal $\delta $ . Thus, $\Vdash _{\mathbb {P}}\text {"}\check {\varphi }:\check {\delta }\to \tau _{\bar {\alpha }} \text { is onto.}\text {"}$ Since by the induction hypothesis, $\Vdash _{\mathbb {P}}\text {"}\tau _{\bar {\alpha }}=V[\dot {G}]_{\bar {\alpha }},\text {"}$ we have that $\Vdash _{\mathbb {P}}\text {"}\check {\varphi }:\check {\delta }\to V[\dot {G}]_{\bar {\alpha }} \text { is onto.}\text {"}$ As $\mathbb {P}$ is suitable we can find a cardinal $\theta $ such that $\tau _{\bar {\alpha }}$ is a $\mathbb {P}_\theta $ -name and, moreover,

$$ \begin{align*} \Vdash _{\mathbb{P}_\theta}\text{"}\dot{\mathbb{P}}_{[\theta,\operatorname{\mathrm{ORD}})} \text{ is } \delta^+\text{-closed}.\text{"} \end{align*} $$

As the remaining part $\mathbb {P}_{[\theta ,\operatorname {\mathrm {ORD}})}$ of the iteration does not add any new subsets of $\delta $ , hence no new elements of $V[\dot {G}]_\alpha $ , we have that

$$ \begin{align*} \Vdash _{\mathbb{P}}\text{"}V[\dot{G}]_\alpha =V[\dot{G}_\theta]_\alpha.\text{"} \end{align*} $$

So we can just take $\tau _\alpha $ to be a $\mathbb {P}_\theta $ -name for $V[\dot {G}_\theta ]_\alpha $ , where $\dot {G}_\theta $ is the standard $\mathbb {P}_\theta $ -name for the generic.

We are now in conditions to show that the class K of $\mathbb {P}$ -reflecting cardinals is unbounded. To this aim fix any ordinal $\alpha $ and define inductively a sequence of ordinals $\langle \alpha _n :n<\omega \rangle $ in D, with $\alpha _0=\alpha $ and $\Vdash _{\mathbb {P}}\text {"}V[\dot {G}]_{\alpha _n}\subseteq V_{\alpha _{n+1}}[\dot {G}_{\alpha _{n+1}}],\text {"}$ as follows: Suppose that $\alpha _n$ has already been defined. By the above claim there is a $\mathbb {P}$ -name $\tau $ such that $\mathbb {P}$ forces “ $\tau =V[\dot {G}]_{\alpha _n}$ .” Let $\beta $ be an ordinal such that $p\in \mathbb {P}_\beta $ and $\mathbb {P}_\beta =\mathbb {P}\cap V_\beta $ . Then, $\Vdash _{\mathbb {P}}\text {"}\tau \in V_\beta [\dot {G}_\beta ]\text {"}$ and thus $\Vdash _{\mathbb {P}} \text {"}V[\dot {G}]_{\alpha _n}\subseteq V_{\alpha _{n+1}}[\dot {G}_{\alpha _{n+1}}],\text {"}$ where $\beta :=\alpha _{n+1}$ . Finally, let $\lambda $ be the supremum of the $\alpha _n$ ’s. In this case we have that $\Vdash _{\mathbb {P}} \text {"}V[\dot {G}]_\lambda \subseteq V_\lambda [\dot {G}_\lambda ],\text {"}$ and so $\lambda $ is $\mathbb {P}$ -reflecting.

Proof Clearly, $\mathcal {K}_{n+1}$ is a club class. Also, by the above discussion, $\mathcal {K}_{n+1}$ is $\Pi _{m+n+1}$ , and $\mathcal {K}_1$ is $\Pi _{m+2}$ . Let $\kappa \in \mathcal {K}_{n+1}\cap \mathrm {Reg}$ . To show that $\kappa $ is $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting it is enough to verify that $\kappa \in D$ .

Since $\kappa $ is an accumulation point of D, $\mathbb {P}\cap V_\kappa \subseteq \mathbb {P}_\kappa $ . Conversely, $\kappa $ is inaccessible, hence $\mathbb {P}_\kappa $ is the direct limit of the previous stages, and so $\mathbb {P}_\kappa \subseteq \mathbb {P}\cap V_\kappa $ . Hence $\kappa \in D$ . The proof for $\mathcal {K}_1$ is the same.

Proof Let $\mathcal {C}$ be a $\mathbf {\Pi _{m+n+1}}$ club proper class of ordinals. By Proposition 5.8, $\mathcal {C}\cap \mathcal {K}_{n+1}$ is a $\mathbf {\Pi _{m+n+1}}$ club proper class such that $\mathcal {C}\cap \mathcal {K}_{n+1}\cap \mathrm {Reg}$ is contained in the class of $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinals. Thus, the class of $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinals is $\mathbf {\Pi _{m+n+1}}$ -stationary. To show the class is unbounded, given any cardinal $\kappa $ , let $\mathcal {C}:=C^{(n+m)}\setminus \kappa ^+$ . Then $\mathcal {C}$ is a $\mathbf {\Pi _{m+n+1}}$ -definable club proper class which, as before, contains a $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinal.

Proof Since $\mathbb {P}$ is $\Delta _{m+1}$ -definable, $\mathbb {P}^{V_\kappa }=\mathbb {P}\cap V_\kappa $ for every $\kappa \in C^{(m+1)}$ . Since $\mathrm {ORD}$ is $\mathbf {\Pi _{m+1}}$ -Mahlo, and direct limits are taken at inaccessible stages of the iteration, the class of inaccessible cardinals $\kappa $ such that $\mathbb {P}^{V_\kappa }=\mathbb {P}_\kappa $ is a proper class. Thus, $\mathbb {P}$ is adequate.

Proof Let us prove the lemma for $n>1$ . The case $n=1$ is similar, using the fact that $\mathrm {VP}(\mathbf {\Pi _1})$ is equivalent to the existence of a proper class of supercompact cardinal, and that every supercompact cardinal belongs to $C^{(2)}$ . So, let $n>1$ and assume that $\mathrm {VP}(\mathbf {\Pi _{n}})$ holds.

Let $\mathcal {C}$ be a $\mathbf {\Sigma _{n+1}}$ -definable club proper class of ordinals and let $\varphi (x,\vec {a})$ be some $\Sigma _{n+1}$ -formula defining it. Let $\kappa $ be a cardinal and $\lambda $ be a $C^{(n-1)}$ -extendible cardinal with $\vec {a}\in V_\lambda $ and $\kappa <\lambda $ . Note that this $\lambda $ exists by virtue of Theorem 6.2. We claim that $\mathcal {C}\cap \lambda $ is unbounded in $\lambda $ . For if $\alpha <\lambda $ , then the sentence “ $\exists \beta>\alpha (\beta \in \mathcal {C})$ ” is $\Sigma _{n+1}$ (with $\vec {a}$ as parameters), hence it is true in $V_\lambda $ because $\lambda $ is $C^{(n-1)}$ -extendible and so it belongs to $C^{(n+1)}$ [Reference Bagaria1, Proposition 3.4]. Since $\mathcal {C}$ is closed, $\lambda \in \mathcal {C}$ . Since $\lambda $ is $\lambda $ -Mahlo, hence also $\kappa $ -Mahlo, the result follows.

Proof $(1) \Rightarrow (2)$ : Let $n\geq 1$ and let $\mathbb {P}$ be $\Delta _{m+1}$ -definable weakly homogeneous suitable iteration, some $m\geq 1$ . By Lemma 5.12, $\mathbb {P}$ is adequate. By Lemma 6.3, ${\mathrm {ORD}}$ is $\mathbf {\Pi _{m+n+1}}$ -Mahlo and so Proposition 5.11 yields a proper class of $\mathbb {P}$ - $\Sigma _{n+1}$ -reflecting cardinals. Also, by Corollary 4.3 and Theorem 6.2 we infer that there is a proper class of $\mathbb {P}$ - $\Sigma _n$ -supercompact cardinal.

$(2)\Rightarrow (1)$ : Since (2) holds, by Corollary 4.3 there exists a proper class of $C^{(n)}$ -extendible cardinals, for every $n\geq 1$ , hence by Theorem 6.2, $\mathrm {VP}$ holds.