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Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.

Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ with parameters from $\mathbf{V}$, $P$ decides $\phi(\hat{\overline{a}})$. Hence it makes sense to ask whether $(HC, \in)$ is an elementary substructure of $(\mathbf{V}[G], \in)$ for some or any $P$-generic extension $\mathbf{V}[G]$.

Consider the axiom schema $\Phi$ that asserts that $(HC, \in)$ is elementary in $(\mathbf{V}[G], \in)$.

Question: Is $\Phi$ consistent relative to large cardinals, and if so, what is its strength? (Or at least get an upper bound)

As an example, $\Phi$ implies that $\omega_1$ is inaccessible to the reals. To see this, fix a real number $x$; then $\mathbf{L}[x]$ satisfies the powerset axiom, so $HC \models `` \mathbf{L}[x] \mbox{ satisfies the powerset axiom}"$, so $\mathbf{L}_{\omega_1}[x]$ satisfies the powerset axiom.

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$\newcommand\HC{\text{HC}}$Allow me to denote your theory $\Phi$ by "$\HC\prec V[G]$". As you mention, this is not a single statement, since we have no truth predicate able to express truth in the extension $V[G]$, but rather it is expressed as a scheme, asserting of each formula $\varphi$ with parameters $z\in\HC$ that $\HC\models\varphi[z]$ if and only if $\Vdash\varphi(\check z)$, where the forcing relation is over the Levy collapse of all ordinals.

Theorem. The following theories are equiconsistent:

  1. ZFC plus your theory, $\HC\prec V[G]$, expressed as a scheme.

  2. ZFC plus the axiom scheme Ord is Mahlo, which asserts that every definable (from parameters) closed unbounded class $C$ of ordinals contains a regular cardinal.

  3. ZFC plus the assertion scheme that $\kappa$ is an inaccessible reflecting cardinal. That is, $V_\kappa\prec V+\kappa$ is inaccessible, expressed as a scheme.

Proof. The equiconsistent of 2 and 3 is explained on the Cantor's attic page for Ord is Mahlo, to which I linked.

If your axiom holds, then we have $\HC\prec V[G]$. If $\kappa=\omega_1^V$, it follows easily that $L_\kappa\prec L$, since $L_\kappa=L^{\HC}$ and $L=L^{V[G]}$. And since also $\kappa$ is inaccessible in $L$ as you observed, this gives us a model of statement 3.

Conversely, if statement $3$ holds, then consider the Levy collapse $V[G]$ of all ordinals. Let $g=G\upharpoonright\kappa$, so that $V[g]$ is the usual Levy collapse of $\kappa$, and $\HC^{V[g]}=V_\kappa[g]$. Since $V_\kappa\prec V$, it is not difficult to see that $V_\kappa[g]\prec V[G]$, as follows. If $V_\kappa[g]\models\varphi[x]$, then by taking names, we get that some $p\in g$ forces $\varphi(\dot x)$ in $V_\kappa$, and so $p$ forces $\varphi(\dot x)$ in $V$, and so $V[G]\models\varphi[x]$. So $\HC^{V[g]}\prec V[G]$ holds, verifying your theory in $V[g]$. QED

In particular, the axiom is stronger than just an inaccessible cardinal. For example, you could have also continued your observation about inaccessibility to reals to notice that $\omega_1$ must be a limit of inaccessible cardinals in $L$, since there can be no bound below $\omega_1$, as that bound would not work in $L^{V[G]}$. Similarly, it must be $\alpha$-inaccessible for every countable $\alpha$, and $\alpha$-hyperinaccessible and so on. Ultimately, of course, you get that $\kappa$ is an inaccessible fully reflecting cardinal in $L$, as I argued.

The axiom Ord is Mahlo is stronger than inaccessible and $\alpha$-inaccessible and hyperinaccessible and various strong hyper-degrees of inaccessibility, but strictly weaker than the existence of a Mahlo cardinal.

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    $\begingroup$ One needs to verify that the Levy collapse forcing exhibits the definability lemma (the definability of the forcing relation), which is does when suitably formalized, although there are subtle issues. There is an excellent new paper discussing such issues: math.uni-bonn.de/people/pluecke/pub/classforcing.pdf. $\endgroup$ – Joel David Hamkins Nov 5 '15 at 0:16

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