What is the consistency strength of "width reflection"?

Question

Lots of people will be familiar with the second-order principle asserting that certain properties are reflected to $V_\alpha$. For a second-order formula $\phi$, parameter $A$, and relativisation of quantifiers and parameters to $V_\alpha$, we can have:

$\phi(A) \rightarrow \exists \alpha V_\alpha \models \phi^{V_\alpha} (A^{V_\alpha})$

This yields lots of small large cardinals (e.g. inaccessible, Mahlo, etc.).

Now there are some 'width-like' reflection principles, for example Friedman's Inner Model Hypothesis. For first-order $\phi$ we say:

"If $\phi$ is true in an inner model $I^{V*}$ of an outer model $V*$ of $V$, then $\phi$ is true in an inner model $I^V$ of $V$"

This principle has surprisingly high consistency strength (the known proof shows that it is consistent relative to the existence of a Woodin cardinal with an inaccessible above). Here though, there are significant metamathematical issues (for one, you have to code extensions if you think there's a "real" $V$).

I'm therefore wondering about the following (greater than first-order) principle (for first-order $\phi$ with/without parameters—I'm interested in both):

"If $\phi$ is true in $V$, then $\phi$ is true in a proper inner model of $V$."

I'm guessing this is either incredibly weak or inconsistent. I'm guessing more confidently the former (you can certainly get $V\not=L$ out of it, just by virtue of the fact that you get a single proper inner model), but without further information about what holds in $V$ you just don't know what more (hence why the strength flies up once we have extensions).

EDIT: Added after Joel Hamkins' very nice answer: I'm also interested in any consequences this principle has, as well as the outright consistency strength. I'm pretty confident that a slight modification of his proof shows that apart from $V\not=L$ and the existence of infinitely many inner models, there's not that much (by running something similar to the Hamkins argument below, I'm guessing that we can arrange a lot of possibilities for $V$ with a suitable forcing).

Answer 1

Update. Neil, Andrés, Gunter, myself and Jonas have written a paper on the topic of this question, exploring the matter further, including and extending the various comments and answers here. It available at:

Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz, Inner-model reflection principles, manuscript under review, arXiv:1708.06669, blog post.

Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH.

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Original answer. Following a generalization of Andres's idea to higher cardinals, I claim that your width-reflection principle is equiconsistent with ZFC, even when parameters are allowed in the scheme.

To see this, start with any model $V\models\newcommand\ZFC{\text{ZFC}}\ZFC+\text{GCH}$, and then perform the proper class forcing $\mathbb{P}$, which is the Easton product of the forcing to add a Cohen subset at every regular cardinal. I claim that the resulting forcing extension $V[G]$ satisfies your width-reflection principle, even with arbitrary parameters.

To see this, suppose that $\varphi(a)$ is true in $V[G]$, where $a\in V[G]$. Fix a name $\dot a$ for $a$, and let $p$ be a condition forcing $\varphi(\dot a)$. Let $\theta$ be a regular cardinal large enough that it is above the support of $p$ and above the support of any condition appearing in $\dot a$. Since the forcing at coordinate $\theta$ is adding a subset to $\theta$, we can view it as having adding two, and then let $G^-$ be the generic filter obtained by removing one of those factors. But removing one factor gives rise to forcing that is isomorphic to $\mathbb{P}$ again, and so $G^-$ can be viewed as $V$-generic for $\mathbb{P}$. Since $\theta$ was above $p$ and $\dot a$, we still have $\varphi(a)$ being true in $V[G^-]$, which is a strictly smaller inner model, but still containing the object $a$. So this fulfills width-reflection, as desired.

Answer 2

Concerning Jonas's comment: That suggested strengthening of the original principle follows from the maximality principle (saying that if phi is forceably necessary, then it is true). Namely, if phi is true in V, then after nontrivial forcing, it is necessarily true in a proper ground. So the statement "phi is true in a proper ground" is forceably necessary, hence true in V by the maximality principle. So phi is true in a proper ground. This all concerns the parameter free version. And the strength of the lightface maximality principle is also just ZFC.