Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M $ and $ \mathbb{Q} := (Q, {\leq_Q}, \mathbb{1}_Q) \in M $ are forcing notions.

## Reminder

**Definition** ([Kun80, VII.7.1]). A mapping $ i \colon P \to Q $ is a *complete embedding* of $ \mathbb{P} $ into $ \mathbb{Q} $ iff

(i) $ \forall r, s \in P \ (r \leq_P s \implies i(r) \leq_Q i(s)) $,

(ii) $ \forall r, s \in P \ (r \perp_P s \implies i(r) \perp_Q i(s)) $, and

(iii) $ \forall q \in Q \ \exists p \in P \ \forall r \in P \ (r \leq_P p \implies i(r) \parallel_Q q) $.

A condition $ p \in P $ as in (iii) is called a *reduction* of $ q $ to $ \mathbb{P} $.

*Remark.* Note that in clause (ii) equivalence holds because of (i).

**Definition** ([Kun80, VII.7.7]). A mapping $ i \colon P \to Q $ is a *dense embedding* of $ \mathbb{P} $ into $ \mathbb{Q} $ iff

(i) $ \forall r, s \in P \ (r \leq_P s \implies i(r) \leq_Q i(s)) $,

(ii) $ \forall r, s \in P \ (r \perp_P s \implies i(r) \perp_Q i(s)) $, and

(iii) $ i[P] $ is dense in $ \mathbb{Q} $.

*Remark.* Every dense embedding is a complete embedding.

**Theorem** ([Kun80, II.3.3]). There exists a dense embedding of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{P}) \setminus \{ \mathbb{0} \} $.

**Lemma** ([Kun80, VII.Ex.C2]). If $ \mathbb{P} $ and $ \mathbb{Q} $ are separative and $ i \colon P \to Q $ is a complete embedding of $ \mathbb{P} $ into $ \mathbb{Q} $, then $ i $ is one-to-one, $ i(\mathbb{1}_P) = \mathbb{1}_Q $, and $ (r \leq_P s \iff i(r) \leq_Q i(s)) $ holds for all $ r, s \in P $.

Now, consider the following statements:

*(C1)* For each $ \mathbb{Q} $-generic $ H $, there exists a $ G \in M[H] $ such that $ G $ is $ \mathbb{P} $-generic over $ M $. (Then $ M[G] \subseteq M[H] $.)

*(C2)* For each $ \mathbb{P} $-generic $ G $, there exists a $ \mathbb{Q} $-generic $ H $ such that $ G \in M[H] $. (Then $ M[G] \subseteq M[H] $.)

*(C3)* There exists a complete embedding $ i \in M $ of $ \mathbb{P} $ into $ \mathbb{Q} $.

*(C4)* There exists a complete embedding $ i \in M $ of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{Q}) \setminus \{ \mathbb{0} \} $.

*(D1)* For each $ \mathbb{Q} $-generic $ H $, there exists a $ G \in M[H] $ such that $ G $ is $ \mathbb{P} $-generic over $ M $ and $ H \in M[G] $. (Then $ M[G] = M[H] $.)

*(D2)* For each $ \mathbb{P} $-generic $ G $, there exists a $ \mathbb{Q} $-generic $ H $ such that $ G \in M[H] $ and $ H \in M[G] $. (Then $ M[G] = M[H] $.)

*(D3)* There exists a dense embedding $ i \in M $ of $ \mathbb{P} $ into $ \mathbb{Q} $.

*(D4)* There exists a dense embedding $ i \in M $ of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{Q}) \setminus \{ \mathbb{0} \} $.

*(Pr)* If $ \mathbb{Q} $ is proper, then $ \mathbb{P} $ is also proper.

## Question

What implications between the above statements are provable in $ \mathsf{ZFC} $? Which are not?

If it is helpful, you may assume that $ \mathbb{P} $ and $ \mathbb{Q} $ are separative partial orders (in the strict sense).

## Main problem

Suppose that *(C1)* holds. What additional assumptions do we need to show *(Pr)*?

(Note that *(C2)* implies *(Pr)*. So what additional assumptions does one need to show *(C2)* from *(C1)*?)

## Bibliography

[Kun80] Kenneth Kunen: *Set Theory: An Introduction to Independence Proofs.* North Holland, 1980