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The present question is a follow-up to this one. Assume GCH holds in $V$ and suppose $G \subseteq Add(\omega_1,\omega_2)$ is generic over $V$. For any $g \subseteq Add(\omega_1,\alpha)$ in $V[G]$ fo …

We can justify this with a general fact about forcing. Fact: Let $G$ be $\mathbb P$-generic, where $\mathbb P$ is separative. …

Another solution can be obtained by adding many reals and using a lemma of Spencer Unger that generalizes the lemma used by Silver. Lemma (Unger) Suppose $\mu \leq \kappa$ are regular, there is $\ …

Suppose $j : M \to N$ is an elementary embeddings between transitive models of ZFC. Everyone knows that if $G$ is $\mathbb{P}$-generic over $M$, $H$ is $j(\mathbb{P})$-generic over $N$, and $j[G] \su …

Jech states in Chapter 30 of his book that $\diamondsuit$ implies that there exists a Suslin tree whose boolean completion is a simple algebra. This means that it has no nontrivial subalgebras. Such …

You’re right; this kind of strong master condition cannot exist in the usual context where $j : M \to N$ is a class of $M$. The relevance is when we’re in an intermediate stage of lifting an embeddin …

Does forcing with $\mathbb{P}$ add an unbounded $A \subseteq \kappa$ such that for all unbounded $X \subseteq \kappa$ in the ground model, $X \cap A \not= \emptyset$ and $X \setminus A \not= \emptyset$ …

The factor forcing P/R is ccc and $\omega$-distributive, a Suslin algebra. A theorem (of Jech?) says that any Suslin algebra has size at most $2^{\omega_1}$. So there is a bound on the size of P. …

Theorem: Let $\kappa$ be a regular cardinal in $M$. Let $\mathbb{P} \in M$ be a separative partial order of size $\kappa$, and let $G$ be $\mathbb{P}$-generic over $M$. Then the following are equival …

Yes, look at Sacks forcing. Add a Sacks real $s$ to $L$. It's a fact (see Jech) that there are no proper intermediate models between the ground model and the Sacks extension. … If so, any forcing over $L$ gets a similar situation. In general, every intermediate submodel of ZFC of a forcing extension containing the ground model is a forcing extension. …

I think it is historically important to see that the earlier results involving generic ultrapowers, such as those of Solovay in the late 1960s, did not depend on the far-future realizations of Laver a …

Can you name a ccc forcing with the following properties? …

It is well known that adding a subset of a regular cardinal $\kappa$ with partial functions of size $< \kappa$ forces $\Diamond_\kappa$. One can also see that if $S \in V$ is a stationary subset of $ …

Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$? … (Note that even without the size requirement this implies a failure of the Jensen covering property, so such a forcing does not necessarily exist.) …

submodels and sufficiently generic conditions: "If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, then the stationarity of $S$ is preserved by $\kappa$-closed forcing …