The $\delta$-approximation property for ground models

Question

Definition 1 (Hamkins). Suppose $V \subseteq W$ are transitive models of $\mathrm{ZFC}$ and $\delta$ is a cardinal in $W$.

1. $(V,W)$ has the $\delta$-cover property iff for each $A \in W$ with $A \subseteq V$ and $\operatorname{card}^{V}(A) < \delta$ there is some $B \supseteq A$ with $B \in V$ and $\operatorname{card}^{V}(B) < \delta$.

2. $(V,W)$ has the $\delta$-approximation property iff for each $B \in W$ with $B \subseteq V$:

   If $B \cap C \in V$ for all $C \in V$ with $\operatorname{card}^{V}(C) < \delta$, then $B \in V$.

The standard covering argument shows:

Lemma 2. Let $\delta$ be a regular cardinal, let $\mathbb{P}$ be a forcing with the $\delta$-c.c. and let $g$ be $\mathbb{P}$-generic. Then $(V,V[g])$ satisfies the $\delta$-cover property.

With regards to the approximation property, we have:

Theorem 3 (Hamkins). Let $\delta$ be a cardinal, let $\mathbb{P}$ be a nontrivial forcing of size $\le \delta$ and let $\dot{\mathbb{Q}}$ be such that $$ \Vdash_{\mathbb{P}} \dot{\mathbb{Q}} \text{ is } \le \delta \text{ strategically closed.} $$ Then any forcing extension via $\mathbb{P} * \dot{\mathbb{Q}}$ satisfies the $\delta^{+}$ approximation (and the $\delta^{+}$-cover) property.

Corollary 4. Let $\delta$ be a cardinal, let $\mathbb{P}$ be a forcing of size $\le \delta$ and let $g$ be $\mathbb{P}$-generic. Then $(V,V[g])$ satisfies the $\delta^{+}$-cover property.

Question 5. Suppose $\delta$ is a cardinal and $\mathbb{P}$ is a $\delta$-closed forcing. Let $g$ be $\mathbb{P}$-generic. Does $(V,V[g])$ satisfy the $\delta^{+}$-approximation property?

If the answer is positive: What if $\mathbb{P}$ is only $\kappa$-strategically closed?

In light of Hamkins' theorem the question essentially reduces to:

Question 6. In Hamkins' theorem is it crucial that $\mathbb{P}$ be nontrivial?

Looking at its proof, it seems to me that the answer to this question should be 'no' but Hamkins' phrasing of his theorem makes me wonder whether I missed something. In any case, the main question I'm interested in is:

Question 7. Are there other known properties of a forcing $\mathbb{P}$ to guarantee that its forcing extensions satisfy the $\delta$-approximation property for small $\delta$ compared to $\operatorname{card}(\mathbb{P})$?

This question is intentionally broad. If it turns out that there is an overwhelming wealth of known results, I'd be happy to phrase it more carefully.

Lastly, I'm also interested in strong negative answers, should they be known. But I'm not entirely sure how to phrase some such question.

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References.

Hamkins. Extensions with the approximation and cover properties have no new large cardinals

Answer 1

Regarding question 7: an often-used property of a poset which implies the $\delta$-approximation property (but which doesn't imply $\delta$-cc) is for the poset to be "strongly proper" with respect to a stationary set of models of size $<\delta$. This notion was introduced by Mitchell in the early 2000s, and there is a lot of recent literature on it. An even weaker property than strong properness, which still implies $\delta$-approximation property, is ``Y-properness", as discussed in Chodounsky-Zapletal "Why Y-cc?".

Some examples of strongly proper posets (for $\delta = \omega_1$) are: Baumgartner's finite condition forcing to add a club to $\omega_1$; Todorcevic's finite $\in$-collapse; various "side condition" forcings of Mitchell, Friedman, Krueger, Neeman, and others.

Answer 2

To answer question 6, yes, it is essential that $\newcommand\P{\mathbb{P}}\P$ is nontrivial. For a counterexample, take $\mathbb{Q}=\text{Add}(\delta^+,1)$, which is $\leq\delta$-closed, but this forcing adds a fresh set to $\delta^+$, that is, a new set all of whose initial segments are in the ground model. Such a set violates the $\delta^+$-approximation property, since the set is new, but all its approximations are in the ground model.

One thing to say in answer to question 7 is that an improved version of theorem 3 requires only that $\P$ is absolutely $\delta$-c.c., rather than than it is small, and this idea was apparantly implicit in early work of Mitchell. See Spencer Unger's paper, Fragility and indestructibility II.