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If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and th …

Clearly you cannot add elements to sets with forcing. When does $X$ remain Borel? Or just generally, when does $\Bbb R$ itself stay Borel in its generic extension? …

To someone who needs to use forcing as a blackbox, understanding the forcing relation is probably slightly more important, but the specific construction of $\Bbb P$-names is perhaps not as important. … Without once mentioning formulas and the language of forcing, or even the forcing relation, in technical terms. Yes, this is still lacking, and yes this is really just aimed at the casual reader. …

Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing. … There are only a handful of exceptions, where we prove that a forcing notion that does the trick exists, but we use abstract argument instead of specifying the forcing. …

Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent: For every $x\in\Bbb R^V$ there is some $P_x\in W$ such that for some $G\subseteq P_x$ whi …

In the vast majority of papers forcing is always developed over ZFC. … I am not looking for ways to develop forcing extensions of ZF without the axiom of choice; rather I am looking for theorems such as c.c.c forcing does not collapse cardinals and similar theorems extended …

The usual presentation of forcing uses models, because the uses of forcing are to prove consistency results. … The approach via models also clarifies how forcing works, what are the names, and what the forcing relation truly means. …

If you look closely, you will see that this is not just presented in the "old method of forcing", but it is also from an era where the distinction between generic and symmetric extension wasn't fully explained … Many of the complicated choice-related constructions of the time (Morris' model, Pincus' work, Sageev's work, and others) were written before a very clear understanding of forcing, iterations, and symmetric …

But equivalently, we can say that the forcing defined with the non-empty open sets is a ccc forcing. … This lends itself to many other definitions that can be translated from forcing terms to topological spaces. …

We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb P=\left\{(p,A)\mid p\in[\kappa]^{<\omega}, A\in\mathcal U, \max p<\min A\right\}.$$ We also define the order, $(q,B)$ is stronger … have some $x\in[\kappa]^\omega$ such that for some $\cal U$ in $M$ which is a normal measure on $\kappa$, $x\subseteq^*A$ for all $A\in\cal U$, then $x=x_G$ for some generic filter $G$ over the Prikry forcing …

In $\sf ZFC$, any forcing of the form $\operatorname{Add}(\kappa,1)$, for any $\kappa$, will destroy the properness of some forcing. Indeed, much more of that is true. Theorem. …

Interestingly, assuming CH this is the same as the standard collapsing forcing. … Is this so-called "Prikry–Silver collapse" provably equivalent to the standard collapsing forcing? …

Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion. …

In a comment to Miha's question in Forcing PFA with ccc forcing, I suggested that if such situation is even possible, it might be achieved by screwing with PFA by some ccc forcing (e.g. adding a Cohen … real), and then "fixing all the problems" via a ccc forcing. …

There are two types of "working with forcing": We can develop the theory of forcing, e.g. iterations, where working with canonical forcing notions is somewhat preferable, so dealing with complete Boolean … Finally, a word about the foundations of forcing. When one learns about forcing, it is often confusing. …