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Let $cf(\alpha) > \omega$, and $P_\alpha := \langle P_{\beta}, \dot{Q}_{\beta} : \beta < \alpha \rangle$ be a countable support iterated forcing construction (so for each $\beta$, $P_{\beta} = P_{\beta …

But $P_1$ forces CH and forcing with $P_{\alpha}$ is the same as forcing with $P_1$ then with $P_{\alpha}$ defined in $V[G_1]$, since $P_1$ is proper. Thus $P_{\alpha}$ forces $|\alpha| > \aleph_2$. …

For a forcing notion $Q$, let $\dot{S^Q}$ be the $Q$-name for the class of ordinals $\{\kappa : \kappa = \omega_1^{V}$ $or$ $\kappa$ $is$ $a$ $regular$ $uncountable$ $cardinal \}$ in $V^Q$. …

Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). …

This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. … Shelah claims every $RUCar^{V}$-semiproper forcing notion is proper. Is it really easily observable? …

I know that, in the presence of $CH$, Namba forcing does not add reals. But when $CH$ fails, is it consistent that it still does not add reals? …

Let $\lambda$ be so that $\varepsilon$, $Q$, $P \in H(\lambda)$. We shall show that for any $p \in P$, there is $p^{*} \in P$, $p \leq p^{*}$ with $p^{*} \Vdash$ "if $\lambda$ is a cardinal, then $Q$ …

What I am confused about is, if we force in $\mathbf{V}$ with an atomless forcing notion that does not add any countable sequence of elements of $\mathbf{V}$ (eg. any atomless countably-closed forcing) …

Given $\mathbb{B} = \langle B, \wedge, \vee, \neg, 0, 1 \rangle$ an atomless complete Boolean algebra that has a $< \mkern-4mu \kappa$-closed dense subset and is $\kappa^+$-c.c., we define a forcing notion … EDIT: I try to describe better what the conditions of the forcing entail, and made requirement 3. optional if it helps with answering the questions without that requirement. …

I am interested in the scenario where one iterates this ultrapower construction in a larger forcing extension. …

Namba forcing is stationary-preserving and forces $cf(\omega_2^{\mathbf{V}}) = \omega$. … Ronald Jensen used $\mathcal{L}$-forcing to iterate Namba posets in order to solve the extended Namba problem: for any strongly inaccessible $\kappa$, he constructed a stationary-preserving forcing notion …

This is Shelah's simpler notion of 'completeness' for not adding reals via forcing. Here $P$ and $Q$ are forcing notions. … My reference for this is Shelah's 'Proper and Improper Forcing', Chapter 5, Section 1. http://projecteuclid.org/DPubS? …

I am mainly concerned about revised countable support iteration of $\dot{S}$-semiproper forcing notions, for a specific class of $\dot{S}$. … Also, there are a number of preservation theorems under various iterations investigated by Shelah in the 1980s-1990s, most of which can be found in his book Proper and Improper Forcing. …

Then there is a forcing notion $P_{S}$ which shoots a closed unbounded $C \subset S$ without collapsing cardinals (or changing cofinalities). … Clearly $Q$ is a forcing notion of power $\aleph_{1}$, so it cannot collapse cardinals or regularity of cardinals except possibly $\aleph_{1}$ (all finite subsets of $S$ belong to $Q$). …

Let $\mathbb{P}$ be a forcing notion. …