Let $\lambda$ be a singular cardinal. Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$? (Not …

Suppose $S \subset \omega_{1}$ is stationary co-stationary. Then there is a forcing notion $P_{S}$ which shoots a closed unbounded $C \subset S$ without collapsing cardinals (or …

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$ …

Consider a tree $(T, <_T)$ of height $\omega_1$, with countable levels. One can view $T$ as a forcing poset by calling a condition $s\in T$ stronger than $t\in T$ if $t <_T …

A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of cou …

Let $V\models\sf ZFC$, and let $V[r]$ be a generic extension obtained by adding one Cohen real, or equivalently $\omega$ Cohen reals. It is clear that $\Bbb R^{V[r]}$ and $\Bbb R^ …

The 'hereditarily countable names' are as defined in Shelah's Proper and Improper Forcing, Chapter 3 Definition 4.1. Let $\mathbb{P}$ be a proper forcing notion and $\dot{Q}$ a $\m …

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 …

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Krei …

Is anything known about the consistency strength of the following statement? $\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ su …

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by Hamkins, Kirkmayer and Perlmutter): (Woodin) Let $V[G]$ …

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the om …

Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see this question). The notion of strong measure zero sets has its dual concept in the …

Let $\alpha > 0$ be any ordinal. Consider the forcing $Add(\omega,\alpha) * Add(\omega_1,1)$. Let $B$ be its boolean completion. Let $\dot{X}$ be the canonical $B$-name for the …

Recall the product lemma from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one fa …