Inner-model reflection principles

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 non-trivial 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 L\'evy-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. We also consider some questions concerning the expressibility of the principles.


Introduction
Every set theorist is familiar with the classical Lévy-Montague reflection principle, which explains how truth in the full set-theoretic universe V reflects down to truth in various rank-initial segments V θ of the cumulative hierarchy. Thus, the Lévy-Montague reflection principle is a form of height-reflection, in that truth in V is reflected vertically downwards to truth in some V θ .
In this brief article, in contrast, we should like to introduce and consider a form of width-reflection, namely, reflection to nontrivial inner models. Specifically, we shall consider the following reflection principles.
(1) The inner-model reflection principle asserts that if a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then there is a proper inner model W , a transitive class model of ZF containing all ordinals, with a ∈ W V in which ϕ(a) is true.
(2) The ground-model reflection principle asserts that if ϕ(a) is true in V , then there is a nontrivial ground model W V with a ∈ W and W |= ϕ(a).
(3) Variations of the principles arise by insisting on inner models of a particular type, such as ground models for a particular type of forcing, or by restricting the class of parameters or formulas that enter into the scheme.
(4) The lightface forms of the principles, in particular, make their assertion only for sentences, so that if σ is a sentence true in V , then σ is true in some proper inner model or ground W , respectively.
In item (1), we could equivalently insist that the inner model W satisfies ZFC, simply by reflecting the conjunction ϕ(a) ∧ AC instead of merely ϕ(a). For the rest of this article, therefore, our inner models will have ZFC. In item (2), a ground model or simply a ground of the universe V is a transitive inner model W |= ZFC over which the universe V is obtained by set forcing, so that V = W [G] for some forcing notion P ∈ W and some W -generic filter G ⊆ P. The universe V , of course, is a ground of itself by trivial forcing, but the ground-model reflection principle seeks grounds W V that are properly contained in V .
The full inner-model reflection principle is expressible in the second-order language of set theory, such as in Gödel-Bernays GBC set theory, as a scheme: with a separate statement for each formula ϕ and where the quantifier ∃W ranges over the inner models of ZFC.
The ground-model reflection principle, in contrast, is expressible as a scheme in the first-order language of set theory. To see this, consider first the ground-model enumeration theorem, which asserts that there is a definable class W ⊆ V × V for which (i) every section W r = { x | (r, x) ∈ W } is a ground of V by set forcing; and (ii) every ground arises as such a section W r (see [FHR15,theorem 12]). Thus, the collection of ground models { W r | r ∈ V } is uniformly definable and we may quantify over the grounds in a first-order manner by quantifying over the indices r used to define them. In light of the enumeration theorem, the ground-model reflection principle is expressed in the first-order language of set theory as the following scheme: ∀a ϕ(a) =⇒ ∃r W r V ∧ ϕ Wr (a) .
We may therefore undertake an analysis of the ground-model reflection principle in a purely first-order formulation of set theory, such as in ZFC. Clearly, the ground-model reflection principle strengthens the inner-model reflection principle, since ground models are inner models. Both principles are clearly false under the axiom of constructibility V = L, since L has no nontrivial inner models. Similarly, the ground-model reflection principle is refuted by the ground axiom, which asserts that there are no nontrivial grounds [Hamb,Rei06,Rei07]. In particular, the ground axiom holds in many of the canonical inner models of largecardinal set theory, such as the Dodd-Jensen core model K DJ , the model V = L[µ], and also the Jensen-Steel core model K, provided that there is no inner model with a Woodin cardinal. The reason is that these inner models are definable in a way that is generically absolute (see [JS13,Mit12]; one uses the hypothesis of no inner models with a Woodin cardinal in the case of K), and so they have no nontrivial ground models.
In the rest of this paper we verify and discuss some other properties of these principles, and how they can be obtained. §2 provides a discussion of how the principles can be forced. In particular we show (in theorems 2 and 3) that both the lightface and boldface versions of the ground-model reflection principle are obtainable from models of ZFC using forcing constructions. In §3 we explain how the principles interact with large cardinal axioms, proving (in theorems 6 and 8) that the inner-model reflection principle is a consequence of sufficient large cardinals. We also discuss how these principles behave in many canonical inner models. In particular, as we shall explain in corollary 9, under the right large-cardinal assumption, the core model satisfies the inner-model reflection principle, but not the ground-model reflection principle. In §4 we show how the Maximality Principle of [SV02] and [Ham03] implies the lightface ground-model reflection principle, and how the Inner Model Hypothesis of [Fri06] implies the lightface inner-model reflection principle. Next, §5 considers the relationship with forcing axioms. We point out that while the bounded proper forcing axiom is consistent with the failure of inner-model reflection, the ground-model reflection principle is consistent with several strong forcing axioms. Finally, in §6 we discuss limitations concerning the expressibility of the inner-model reflection principle, and conclude with some open questions.

Forcing inner-model reflection
Let us begin by showing that we may easily force instances of the lightface reflection principles as follows.
Theorem 2. In the forcing extension V [c] arising by forcing to add a Cohen real, the lightface ground-model reflection principle holds, and indeed, the ground-model reflection principle holds for assertions with arbitrary parameters from V .
Proof. Suppose that an assertion ϕ(a) is true in V [c], where a ∈ V . By the homogeneity of the forcing, it follows that ϕ(a) is forced by every condition, and so it will be true in V Cohen forcing is hardly unique with the property mentioned in theorem 2, since essentially the same argument works with many other kinds of forcing, such as random forcing or Cohen forcing at higher cardinals. Indeed, let us now push the idea a little harder with class forcing so as to achieve the full principle, with arbitrary parameters, including all the new parameters of the forcing extension.
Theorem 3. Every model of ZFC has a class-forcing extension satisfying the ground-model reflection principle, with arbitrary parameters from the extension.
Proof. By preliminary forcing if necessary, we may assume that GCH holds. Let P be the proper-class Easton-support product of the forcing posets Add(δ, 1) that add a Cohen subset to every cardinal δ in some proper class of regular cardinals. Suppose that G ⊆ P is V -generic, and consider the extension V [G], which is a model of ZFC. Suppose that V [G] |= ϕ(a) for some first-order assertion ϕ and set a. So there is some condition p ∈ P forcing ϕ(ȧ) for some nameȧ withȧ G = a. Let δ be a stage of forcing that is larger than the support of p and any condition in the nameȧ, and let G 0 be just like G on all coordinates other than δ, except that on coordinate δ itself, we take only every other digit of the generic subset of δ that was added, re-indexed so as to make a generic subset of δ. Thus, the filter G 0 ⊆ P is V -generic for this forcing, p ∈ G 0 and V , where g consists of the information on the complementary digits of the subset of δ. So V [G 0 ] is a proper inner model of ZFC, and since p ∈ G 0 andȧ G0 =ȧ G , it follows that V [G 0 ] |= ϕ(a), fulfilling the desired instance of ground-model reflection.
Corollary 4. The inner-model and ground-model reflection principles are each conservative over ZFC for Π 2 assertions about sets. In other words, any Σ 2 assertion that is consistent with ZFC is also consistent with ZFC plus the ground-model reflection principle.
Proof. The point is that the previous argument, by starting the forcing sufficiently high, shows that any given model of ZFC can be extended to a model of groundmodel reflection while preserving any particular V θ and therefore the truth of any particular Σ 2 assertion. Thus, any Σ 2 assertion that is consistent with ZFC is also consistent with ground-model reflection and hence also with inner-model reflection. By contraposition, any Π 2 assertion that is provable from ZFC or GBC plus the inner-model or ground-model reflection principles is provable in ZFC alone.
The Lévy-Montague reflection principle produces for every natural number n in the meta-theory a proper class club C (n) of cardinals θ, the Σ n -correct cardinals, for which V θ ≺ Σn V . But this kind of reflection can never hold for inner models: Proof. Assume that W is a transitive class model of ZF containing all ordinals and that W ≺ Σ1 V . If W = V , then there is some set a in V that is not in W . Let θ be above the rank of a and let u = (V θ ) W . So V thinks, "there is a set of rank less than θ, which is not in u." This is a Σ 1 assertion about θ and u, witnessed by a rank function into θ. But it is not true in W , by the choice of u. So it must be that W = V .
So the situation of width-reflection is somewhat different in character from that of height reflection, where we have H κ ≺ Σ1 V for every uncountable cardinal κ and more generally V θ ≺ Σn V for all cardinals θ in the class club C (n) . Observation 5 shows that there is no analogue of this for width reflection.

Large cardinals
Next, we point out that the inner-model reflection principle is an outright consequence of sufficient large cardinals.
(1) If there is a measurable cardinal, then the lightface inner-model reflection principle holds.
(2) Indeed, if κ is measurable, then the inner-model reflection principle holds for assertions with parameters in V κ .
(3) Consequently, if there is a proper class of measurable cardinals, then the full inner-model reflection principle holds for arbitrary parameters.
Proof. The theorem is easy to prove. Suppose that ϕ(a) is true in V , where the parameter a is in V κ for some measurable cardinal κ. Let j : V → M be an ultrapower embedding by a measure on κ, with critical point κ, into a transitive class M , which must be a proper inner model, definable from the measure. Since a ∈ V κ , below the critical point, it follows that j(a) = a and consequently M |= ϕ(a) by the elementarity of j. So ϕ(a) is true in a proper inner model, thereby witnessing this instance of the inner-model reflection principle.
Using this, we can separate the inner-model reflection principle from the groundmodel reflection principle. They do not coincide.
Corollary 7. If ZFC is consistent with a proper class of measurable cardinals, then there is a model of ZFC in which the inner-model reflection principle holds, but the ground-model reflection principle fails.
Proof. One can prove this as in corollary 9 using the fact that the core model K has no nontrivial grounds; but let us give a forcing proof. If there are a proper class of measurable cardinals in V , then there is a class-forcing extension V [G] preserving them, in which every set is coded into the GCH pattern. This idea was a central theme of [Rei06,Rei07]; but let us sketch the details. After forcing GCH, if necessary, we perform a lottery iteration, which at the successor of every measurable cardinal either forces a violation of GCH or performs trivial forcing. Generically, every set is coded into the resulting pattern. The standard lifting arguments, such as those in [Ham00,Hama], show that all measurable cardinals are preserved, and so by theorem 6 it follows that V [G] satisfies the inner-model reflection principle. Meanwhile, because every set in V [G] is coded into the GCH pattern, by placing sets into much larger sets, it follows that every set is coded unboundedly often. Since set forcing preserves the GCH pattern above the size of the forcing, every ground model has this coding. So V [G] can have no nontrivial grounds and consequently does not satisfy the ground-model reflection principle.
We can improve the large cardinal hypothesis of the preceding results by using the work of Vickers and Welch [VW01].
Theorem 8. If Ord is Ramsey, then the inner-model reflection principle holds.
Proof. Following [VW01, definition 2.2; see also definition 1.1], we say that Ord is Ramsey if and only if there is an unbounded class I ⊆ Ord of good indiscernibles for V, ∈ . This is a second-order assertion in GBC. One can arrange set models of this theory with first-order part V κ , if κ is a Ramsey cardinal, and so the hypothesis "Ord is Ramsey" is strictly weaker in consistency strength than ZFC plus the existence of a Ramsey cardinal, which in turn is strictly weaker in consistency strength than ZFC plus the existence of a measurable cardinal.
Meanwhile, the argument of [VW01, theorem 2.3] shows how to construct from the class I a transitive class M with a nontrivial elementary embedding j : M → V , where the critical point of j can be arranged so as to be any desired element of I. Note that M V in light of the Kunen inconsistency. If ϕ(a) holds in V , then there is such a j : M → V with critical point above the rank of a and therefore with a ∈ M and j(a) = a. It follows by elementarity that M |= ϕ(a), thereby fulfilling the inner-model reflection principle.
In particular, any statement that is compatible with Ord being Ramsey is also compatible with the inner-model reflection principle, which makes a contrast to corollary 4. One can use theorem 8 to weaken the hypothesis of corollary 7 as follows, where we now use the core model rather than forcing. Note that if "Ord is Ramsey" holds, then the hypothesis of corollary 9 holds in an inner model.
Corollary 9. If the core model K exists and satisfies "Ord is Ramsey", then K satisfies the inner-model reflection principle, but not the ground-model reflection principle.
Proof. If K |= Ord is Ramsey, then it satisfies the inner-model reflection principle by the previous theorem. And since K is definable in a way that is generically absolute, it has no nontrivial grounds and therefore cannot satisfy the groundmodel reflection principle.
It is thus natural to ask whether fine structural extender models can satisfy the ground-model reflection principle. The following theorem, which was observed by Ralf Schindler and which we include with his permission, provides an affirmative answer to this question. It builds on the methods of Fuchs and Schindler [FS16]. Proof. The argument uses some techniques from [FS16], adapted to the present context. One key ingredient is the "δ generator version" of Woodin's extender algebra see [SS09, Lemma 1.3]. If M is a normally (ω, κ + + 1)-iterable premouse and δ is a Woodin cardinal in M, then this forcing notion P = P M|δ has the property that for every subset A ⊆ κ + , there is a normal nondropping iteration tree on M with a last model N such that if π is the iteration embedding, then A ∩ π(δ) is π(P)-generic over N .
The other key ingredient is the P-construction of [SS09]. If M is a premouse, δ is a cutpoint of M (that is, δ is not overlapped by any extender on the M sequence), P is a premouse of height δ + ω, δ is a Woodin cardinal inP,P |δ is definable in M|δ, andP[G] = M|(δ + 1), for some G which is generic overP for its version of the δ generator version of Woodin's extender algebra, then P(M,P, δ) is the result of the maximal P-construction overP with respect to M, above δ. Essentially, this construction appends to the extender sequence ofP the restrictions of the extenders on the extender sequence of M that are indexed beyond δ, as long as this results in a structure in which δ is still a Woodin cardinal. We will define P(M,P, δ) also in the case that δ is not a cutpoint of M, by letting α be least such that α ≥ δ, E M α = ∅ and κ = crit(E M α ) ≤ δ, letting ζ ≤ ht(M), α ≤ ζ be maximal such that κ +M|α = κ +M|ζ , and setting P(M,P, δ) = P(ult n (M||ζ, E M α ),P, δ), where n is least such that ρ n+1 (M||ζ) ≤ κ, if such an n exists, and n = 0 otherwise. See the discussion after the statement of Lemma 3.21 in [SS09] for details.
An iteration tree T on an extender model W which is definable in L[E] is guided by P-constructions if the branches in T at limit stages are determined by Q-structures that are pullbacks of Q-structures obtained in L[E] by maximal P-constructions, see the discussion after Definition 3.22 in [FS16] for details.
We modify [FS16,Definition 3.25] to say that an extender model W definable in L[E] is minimal for L[E] if it has a proper class of Woodin cardinals, and if for every δ that is Woodin in W , whenever T ∈ L[E] is a normal iteration tree on W which is guided by P-constructions in L[E] and uses only extenders indexed above the supremum of the Woodin cardinals of W below δ, and is based on W |δ, then the following holds true: if T has limit length, then T lives strictly below δ iff P(L[E], M(T ) + ω, δ(T )), if defined, is not a proper class, and if T has successor length and [0, ∞] T does not drop, then P(L[E], M T ∞ |i T 0,∞ (δ)+ω, i T 0,∞ (δ)), if defined, is a proper class.
It follows that L[E] itself is minimal in this sense. For example, if δ, T are as above, where the length of T is a limit ordinal and T lives strictly below δ, then if P = P(L[E], M(T ) + ω, δ(T )) were a proper class, then it would be an iterable extender model with a proper class of Woodin cardinals that's below L[E] in the canonical pre-well-ordering of iterable extender models. The point is that T would be according to the iteration strategy of L[E], and so, there is a cofinal well-founded branch such that M T b is iterable. But since T lives strictly below δ, it would follow that π T 0,b (δ) > δ(T ), and it would follow that δ(T ) is not Woodin in M T b , but it is Woodin in P, which implies that P is below L[E]; see the proof of [FS16,Lemma 3.23], which shows that there is no L[E]-based sequence of length 2, in the terminology introduced there. If the length of T is a limit ordinal and T does not live strictly below δ, then one can argue as in the proof of [FS16, Lemma 3.26] to show that the relevant P-construction yields a proper class model, and similarly in the case that T has successor length. Now, to show that L[E] satisfies the ground-model reflection principle, let ϕ(a) be a statement true in L[E]. By assumption, there is a δ > rnk(a) that is Woodin in L[E]. Let η > δ be a cutpoint of the extender sequence E. LetẼ code E as a class of ordinals in some natural way. Form an iteration tree on L[E] as follows (the construction is much as in the proof of [FS16, Lemma 3.29]): first, hit some total extender on the E-sequence, indexed below δ, but above every Woodin cardinal less than δ, with critical point greater than rnk(a), η many times. After that, at successor stages, choose the least total extender in the current model with an index greater than the supremum of the Woodin cardinals below the current image δ ′ of δ that violates an axiom of the extender algebra with respect toẼ ∩ δ ′ . Since such extenders suffice to witness the Woodinness of δ, one can work with the so restricted extender algebra. If there is no such extender, or if a limit stage λ is reached such that P = P(L[E], M(T ↾ λ) + ω, δ(M(T ↾ λ))) is a proper class, then the construction is complete. Otherwise, as in the proof of [FS16, Lemma 3.29], it follows that P can serve as a Q-structure, and the branch for T ↾ λ given by the iteration strategy in V can be found inside L[E], allowing us to extend the iteration tree in this case. Further, as in that proof, it follows that this process terminates at a limit stage λ = η +L Since L[E] is iterable in V , it follows that T has a cofinal well-founded branch b in V , such that M T b is iterable in V . Since η is a cutpoint of E, it follows that it is a cutpoint of P, and it follows further that P is iterable above η +L [E] . The coiteration of P and M T b has to result in a common (proper class) iterate Q. Let π : L[E] −→ M T b , σ : M T b −→ Q and τ : P −→ Q be the iteration embeddings. Note that a is not moved by π. Since the coiteration between M T b and P is above , a is not moved by σ or τ either. Hence, we get: π(a))) ⇐⇒ Q |= ϕ(τ (a)) ⇐⇒ P |= ϕ(a).
Thus, P is a nontrivial ground of L[E] which reflects the truth of ϕ(a), as desired.
An immediate question is whether the ground-model reflection principle suffices to characterize such an L[E]. We do not yet know how to show this. It is possible that the techniques of [FS16] could be adapted for this purpose, but there seem to be some challenges. For example, it might be that there are issues with the extender sequence not being definable from ordinal parameters inside L[E] (in the appropriate argument from [FS16], this difficulty is dealt with by assumption). We therefore ask the following: A positive answer to this question would be interesting for a couple of reasons. First, it would show that the ground-model reflection principle, while weak, can have large cardinal requirements when stipulated to hold in a particular context. This would represent an unusual situation, since normally we do not expect the strength of a set-theoretic statement to vary wildly depending on whether or not the model in which it is instantiated is L-like. Second, it would show that the groundmodel reflection principle provides a way of characterizing a certain canonical finestructural inner model, highlighting a possible further application of the principle.

The Maximality Principle and Inner Model Hypothesis
Consider next the maximality principle of [SV02] and [Ham03], which asserts that whenever a statement is forceably necessary, which is to say that it is forceable in such a way that it remains true in all further extensions, then it is already true. This is expressible in modal terms by the scheme ϕ → ϕ, the principal axiom of the modal theory S5, where the modal operators are interpreted so that ψ means ψ is true in some set-forcing extension and ψ means ψ is true in all set-forcing extensions.
Theorem 12. The maximality principle implies the lightface ground-model reflection principle.
Proof. Suppose that a sentence σ is true in V . Consider the statement, "σ is true in some nontrivial ground." In light of the ground-model enumeration theorem, this supplementary statement is expressible in the first-order language of set theory. Furthermore, it becomes true in any nontrivial forcing extension V [G], since V is a nontrivial ground of V [G], and the statement remains true in any further forcing extension. Thus, the supplementary statement is forceably necessary in V , and therefore by the maximality principle it must already be true in V . So there must be a nontrivial ground model W V in which σ is true.
The same argument works with the various other versions of the maximality principle, such as MP Γ (X), where only forcing notions in a class Γ are considered and statements with parameters from X. The same argument as in theorem 12 shows that MP Γ (X) implies the Γ-ground model reflection principle with parameters from X.
A similar argument can be made from the inner-model hypothesis IMH, which is the scheme of assertions made for each sentence σ, that if there is an outer model with an inner model of σ, then there is already an inner model of σ without first moving to the outer model. This principle also can be described in modal vocabulary as the scheme of assertions σ =⇒ σ, where the up-modality refers here to possibility in outer models and the down-modality refers to possibility in inner models. See [Fri06] for details about the IMH; the axiom is naturally formalized in a multiverse context of possible outer models, although Antos, Barton and Friedman [ABF] show that the axiom is expressible in the second-order language of set theory in models of GBC + Σ 1 1 -comprehension, without direct reference to outer models. In formulating the IMH, one may equivalently insist on proper inner models.
Theorem 13. The inner-model hypothesis implies the lightface inner-model reflection principle.
Proof. If σ is true in V , then in any nontrivial extension of V , there is a proper inner model in which σ holds, namely V itself. So if the inner-model hypothesis holds, then there must already be such an inner model of V , and so the lightface inner-model reflection principle holds.
Let us now consider a downward-directed version of the maximality principle, which can be viewed itself as a kind of reflection principle, studied in [HL13]. Namely, let us say that the ground-model maximality principle holds, if and only if any statement σ that holds in some ground model and all grounds of that ground, is true in V . This is expressible as σ =⇒ σ, where and are the modal operators of "true in some ground model" and "true in all ground models," respectively. 1 By considering ¬σ and the contrapositive, the ground-model maximality principle is easily seen to be equivalent to the following assertion: if a sentence σ is true, then every ground model has itself a ground in which σ is true. This formulation of the maximality principle reveals it to be a particularly strong reflection principle, when combined with the assertion that indeed there are nontrivial grounds.
Theorem 14. If V δ ≺ V , then V has a ground model W that satisfies the groundmodel maximality principle for assertions allowing parameters of rank less than δ in W .
Proof. The proof uses the recent result of Toshimichi Usuba [Usu17], showing that the strong downward-directed grounds hypothesis (DDG) holds; that is, for any set I, there is a ground W contained in r∈I W r , where W r denotes the r th ground as in the statement of the ground-model enumeration theorem.
1 In light of the downward orientation of this axiom, however, the 'maximality' terminology may be distracting, as any deeper ground, for example, will also satisfy the ground-model maximality principle. What is being maximized here is not the model, but the collection of truths that are downward-necessary. The principle is related to S5 for grounds, as in [HL13], since the axiom ϕ → ϕ is the defining axiom of S5 over S4; but the ground-model maximality principle is not identical to the validity of S5, as the principle is (trivially) true under the ground axiom, whereas the modal logic of grounds in this situation strictly exceeds S5.
Assume V δ ≺ V . By the strong DDG, there is a ground W with W ⊆ W r for all r ∈ V δ . Suppose that W |= ϕ(a) for some a ∈ V δ ∩W . So W has a ground model U , such that U |= ϕ(a); that is, U satisfies ϕ(a) and so does every ground of U . Since U is a ground of W , which is a ground of V , it follows that U is a ground of V and consequently U = W r for some r by the ground-model enumeration theorem. The least rank of an r whose corresponding ground W r has the properties of U that we have mentioned is definable in V using a as a parameter, and consequently there is such an r already in V δ . In this case, W ⊆ U = W r by the assumption on W . Since W ⊆ U ⊆ V , where W is a ground of V , it follows from the intermediatemodel theorem (see [FHR15,fact 11] or [Jec03,corollary 15.43]) that W is also a ground of U , where ϕ(a) holds, and so ϕ(a) is true in W , as desired.
The previous argument, using the strong DDG, provides a more direct method than [HL13,theorem 8] of providing a model with the ground-model maximality principle.
Corollary 15. If V δ ≺ V and V has no bedrock, that is, no minimal ground, then there is a ground model W satisfying the ground-model maximality principle and also the ground-model reflection principle, for assertions using parameters of rank less than δ.
Proof. Assume V δ ≺ V and V has no bedrock. Let W be the ground model identified in theorem 14, which satisfies the ground-model maximality principle for assertions with parameters of rank less than δ in W . Since there is no minimal ground, it follows that W also has no minimal ground. By the ground-model maximality principle, every statement ϕ(a) true in W with a ∈ V δ ∩ W is true densely often in the grounds of W , that is, true in some deeper ground of any given ground of W . Since there are such proper grounds of W , it follows that any such statement ϕ(a) is true in some proper ground of W , and so W satisfies the ground-model reflection principle for these assertions.

Forcing axioms
Let us consider next the question of whether strong forcing axioms might settle the inner-model or ground-model reflection principles. Work of Caicedo and Veličković [CV06, corollary 2] shows that it is relatively consistent that the universe is a minimal model of the bounded proper forcing axiom BPFA, that is, that the universe satisfies BPFA, but has no proper inner model satisfying BPFA. In this situation, of course, the inner-model reflection principle must fail, and so it seems natural to inquire whether PFA or MM might outright refute the ground-model reflection principle. The next theorem shows, however, that this is not the case. Proof. We use the fact that both PFA and MM are necessarily indestructible by <ω 2 -directed closed forcing (see [Lar00,theorem 4.3]). Let P be the Easton-support class product used in the proof of theorem 3, but with nontrivial forcing only at stages ω 2 and above. This forcing is consequently ω 2 -directed closed, and therefore preserves PFA and MM, if these forcing axioms should hold in the ground.
The proof of theorem 3 then shows that the extension satisfies the ground-model reflection principle, as desired.
Meanwhile, if (after suitable preparatory forcing coding sets into the GCH pattern) one should use an Easton-support iteration, rather than a product, then again the forcing is ω 2 -directed closed, and the main result of [HRW08] shows that the extension satisfies the ground axiom: it has no nontrivial ground models by set forcing. Thus, the ground-model reflection principle fails in this extension.

Expressibility of Inner-Model Reflection
Lastly, let us consider the question of whether the inner-model reflection principle might be expressible in the first-order language of set theory, or whether, as we expect, it is a fundamentally second-order assertion. The ground-model reflection principle, as we have pointed out, is expressible as a scheme in the first-order language of set theory. But the same does not seem to be true for the inner-model reflection principle, in light of the quantification over inner models. How can we prove that indeed there is no first-order means of expressing the principle?
As a step towards this, let us first show that the existence of an inner model satisfying a given sentence σ is not necessarily first-order expressible.
Theorem 17. With a mild consistency assumption, there are models M, ∈, S 0 and M, ∈, S 1 of GBC set theory with the same first-order part M and a particular first-order sentence σ, such that S 1 has a proper inner model of M satisfying σ, but S 0 does not.
Proof. Let us assume V = L, plus the existence of a truth-predicate Tr L for truth in L. So Tr L is a non-definable class, which satisfies the Tarskian recursion for the definition of satisfaction for first-order truth. The consistency strength of this assumption is strictly less than Kelly-Morse set theory, which is itself strictly weaker than ZFC plus the existence of an inaccessible cardinal, since KM implies the existence of such a truth predicate (see [GHH + ] for further discussion of truth predicates and the strength of this hypothesis in the hierarchy between GBC and KM). So it is a mild consistency assumption.
In L, let P be the Easton-support product that adds a Cohen subset via Add(δ, 1) for every regular cardinal δ. Suppose G ⊆ P is L-generic, and consider L[G]. Let S 0 consist of the classes that are definable in the structure L[G], ∈, G , and let S 1 be the classes definable in L[G], ∈, G, Tr L , where we also add the truth predicate. (A similar argument is made in [HR].) Note that in light of the definability of the forcing relation for this forcing, it follows that S 1 includes a truth predicate for the extension L[G]. Both models L[G], ∈, S 0 and L[G], ∈, S 1 satisfy GBC. Inside the latter model, let T be a class of regular cardinals that codes the information of the truth predicate Tr L in some canonical and sufficiently absolute manner, such as by including ℵ L α+1 in T exactly when α codes a formula-parameter pair ϕ[ a] that is declared true by Tr L . Let G T be the restriction of the generic filter G to include the Cohen sets only on the cardinals in T . It is not difficult to see that in L[G T ], the cardinals that have L-generic Cohen subsets of a regular cardinal δ are precisely the cardinals in T . Therefore, the model L[G T ] satisfies the assertion that "the class of regular cardinals δ for which there is an L-generic Cohen subset of δ codes a truth-predicate for truth in L." So S 1 has an inner model, namely L[G T ], that satisfies this statement. But S 0 can have no such inner model, since there can be no truth predicate for L definable in L[G], ∈, G , as in this case we could use the definable forcing relation and thereby define a truth predicate for the full structure L[G], ∈, G itself, contrary to Tarski's theorem.
In order to show that the inner-model reflection principle is not first-order expressible, however, one would need much more than this. It would suffice to exhibit a positive instance of the following: Question 18. Are there two models of GBC with the same first-order part, such that one of them is a model of the inner-model reflection principle and the other is not?
In particular, for this to happen we would at the very least need to strengthen theorem 17 by producing a model M, ∈, S |= GBC and a sentence σ that is true in M and also true in some inner model W M in S, but which is not true in any proper inner model of M that is first-order definable in M allowing set parameters. We are unsure how to arrange even this much.