In first-order set theory, the principle of complete reflection is the following.

$\tag{ CR} \forall \alpha\exists \beta>\alpha\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{x}))$

where $\phi\in \mathcal L_\in$ with free variables among $\vec{x}$ and where $\phi^\beta$ is the result of binding all the quantifiers in $\phi$ to $V_\beta$.

It is well-known that over ZFC minus replacement and infinity, CR implies replacement and infinity. I'm interested in the following principle:

$\tag{ CR*} \forall \alpha\exists \beta>\alpha\exists j\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{j(x)}))$

Essentially, I want to know if CR* implies replacement over the other axioms of ZFC plus the claim that every set is in some $V_\alpha$: that is, over ZFC - replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

However, since $j$ might be a proper class, the question is whether CR* implies first-order replacement in ZFC2 - second-order replacement + $\forall x\exists \alpha(x\in V_\alpha)$. (By ZFC2 I mean the axioms of ZFC with the axiom schemas of replacement and separation replaced by the corresponding second-order axioms together with the full comprehension schema (i.e. $\exists X\forall x(x\in X\leftrightarrow \phi)$, for any $\phi$ in the language)).

Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$, where $V_\gamma$ is an elementary substructure of $V_\alpha$. In other words, $V_\alpha$ is a limit of its $V_\beta$ elementary substructures. Then it is a simple result that $V_\alpha$ models replacement.

My question: Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$ and an elementary embedding from $V_\gamma$ to $V_\alpha$. Does it follow that $V_\alpha$ models replacement?

In first-order set theory, the principle of complete reflection is the following.

$\tag{ CR} \forall \alpha\exists \beta>\alpha\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{x}))$

where $\phi\in \mathcal L_\in$ with free variables among $\vec{x}$ and where $\phi^\beta$ is the result of binding all the quantifiers in $\phi$ to $V_\beta$.

It is well-known that over ZFC minus replacement and infinity, CR implies replacement and infinity. I'm interested in the following principle:

$\tag{ CR*} \forall \alpha\exists \beta>\alpha\exists j\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{j(x)}))$

Essentially, I want to know if CR* implies replacement over the other axioms of ZFC plus the claim that every set is in some $V_\alpha$: that is, over ZFC - replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

However, since $j$ might be a proper class, the question is whether CR* implies first-order replacement in ZFC2 - second-order replacement + $\forall x\exists \alpha(x\in V_\alpha)$. (By ZFC2 I mean the axioms of ZFC together with the second-order axioms of replacement and separation and the full comprehension schema (i.e. $\exists X\forall x(x\in X\leftrightarrow \phi)$, for any $\phi$ in the language)).

In first-order set theory, the principle of complete reflection is the following.

$\tag{ CR} \forall \alpha\exists \beta>\alpha\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{x}))$

where $\phi\in \mathcal L_\in$ with free variables among $\vec{x}$ and where $\phi^\beta$ is the result of binding all the quantifiers in $\phi$ to $V_\beta$.

It is well-known that over ZFC minus replacement and infinity, CR implies replacement and infinity. I'm interested in the following principle:

$\tag{ CR*} \forall \alpha\exists \beta>\alpha\exists j\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{j(x)}))$

Essentially, I want to know if CR* implies replacement over the other axioms of ZFC plus the claim that every set is in some $V_\alpha$: that is, over ZFC - replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

However, since $j$ might be a proper class, the question is whether CR* implies first-order replacement in ZFC2 - second-order replacement + $\forall x\exists \alpha(x\in V_\alpha)$. (By ZFC2 I mean the axioms of ZFC with the axiom schemas of replacement and separation replaced by the corresponding second-order axioms together with the full comprehension schema (i.e. $\exists X\forall x(x\in X\leftrightarrow \phi)$, for any $\phi$ in the language)).

In first-order set theory, the principle of complete reflection is the following.

$\tag{ CR} \forall \alpha\exists \beta>\alpha\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{x}))$

where $\phi\in \mathcal L_\in$ with free variables among $\vec{x}$ and where $\phi^\beta$ is the result of binding all the quantifiers in $\phi$ to $V_\beta$.

It is well-known that over ZFC minus replacement and infinity, CR implies replacement and infinity. I'm interested in the following principle:

$\tag{ CR*} \forall \alpha\exists \beta>\alpha\exists j\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{j(x)}))$

Essentially, I want to know if CR* implies replacement over the other axioms of ZFC plus the claim that every set is in some $V_\alpha$: that is, over ZFC - replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

However, since $j$ might be a proper class, the question is whether CR* implies first-order replacement in ZFC2 - second-order replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

In first-order set theory, the principle of complete reflection is the following.

$\tag{ CR} \forall \alpha\exists \beta>\alpha\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{x}))$

where $\phi\in \mathcal L_\in$ with free variables among $\vec{x}$ and where $\phi^\beta$ is the result of binding all the quantifiers in $\phi$ to $V_\beta$.

It is well-known that over ZFC minus replacement and infinity, CR implies replacement and infinity. I'm interested in the following principle:

$\tag{ CR*} \forall \alpha\exists \beta>\alpha\exists j\forall\vec{x}\in V_\beta(\phi^\beta(\vec{x}) \leftrightarrow \phi(\vec{j(x)}))$

Essentially, I want to know if CR* implies replacement over the other axioms of ZFC plus the claim that every set is in some $V_\alpha$: that is, over ZFC - replacement + $\forall x\exists \alpha(x\in V_\alpha)$.

However, since $j$ might be a proper class, the question is whether CR* implies first-order replacement in ZFC2 - second-order replacement + $\forall x\exists \alpha(x\in V_\alpha)$. (By ZFC2 I mean the axioms of ZFC together with the second-order axioms of replacement and separation and the full comprehension schema (i.e. $\exists X\forall x(x\in X\leftrightarrow \phi)$, for any $\phi$ in the language)).