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Yes, forcing can add fundamentally new knots, not equivalent to any ground model knot. … By absoluteness, this remains true in all forcing extensions. So you cannot add fundamentally new tame knots. …

By combining the forcing into an iteration, we may view $V[G][H]$ as a single-step forcing extension of $V$, and so it has some essential size over $V$. … S4.3, for a wide collection of forcing classes $\Gamma$. …

Solovay proved shortly after Cohen's result on the independence of CH that in any model of set theory $V$, if $\kappa^\omega=\kappa$, then there is a forcing extension in which $2^\omega=\kappa$. … The forcing is simply $\text{Add}(\omega,\kappa)$, the forcing to add $\kappa$ many Cohen reals. …

And similarly with many other forcing arguments. … In many forcing arguments, it is not enough just to build a partial order consisting of tiny pieces of the desired object, since one also wants to know that the forcing preserves other features. …

The situation isn't that any given theorem off the shelf might have a forcing proof, but rather that it is a fascinating situation when one can use forcing to prove a theorem purely about the ground model … The classical proof uses ergodic theory, but there is a quick forcing argument. Suppose it did reduce; this fact is absolute to the forcing $V[c]$ adding a Cohen real. …

Like forcing potentialism, this would be a case of width potentialism and height actualism, since outer models increase only the width of the universe and not the height. … This can be made true in an outer model (by forcing) and once true, remains true in all further outer models; and the statements can be controlled independently. …

Consider the context of c.c.c. forcing over the set-theoretic universe $V$. … So there is a c.c.c. forcing notion $\mathbb{P}$ forcing $\Box\varphi(a)$. …

Since the forcing will also have size $\kappa$, in the direct limit case, it follows that the forcing is isomorphic to $\text{Coll}(\omega,\kappa)$, and so $\sigma$ holds in the model $V[G]$, so Alice … Using the Foreman-Woodin theorem that it is relatively consistent that GCH fails everywhere, we can perform additional forcing by first adding a generic class of cardinals, and then forcing certain instances …

how one can take the common set-theorist's talk of "forcing over $V$" at face value. … Strictly speaking, one doesn't need that the entire model is countable, but rather only that the model $M$ has only countably many dense subsets of the forcing notion $P$ being used for the forcing. …

In this sense, the measure of a measurable set cannot be affected by forcing. … The set has full measure in the constructible universe, of course, but it is easily made to have measure zero in a forcing extension. …

The answer is that the FPP is not absolute, and indeed, even the unit interval loses the FPP in a forcing extension. … The unit interval famously has the FPP, but I claim that in any forcing extension having a new real, such as the forcing extension $V[c]$ obtained by adding a Cohen real, which preserves cardinals, the …

desired explanation of the illusion of forcing extensions of $V$. … That is, Koellner argues that the details of the proof of the naturalist account of forcing is how one explains away the illusion of forcing. So that would seem to be a coherent view. …

With forcing, we can just go all the way to a fully generic real, and this simplifies things. …

For the larger large cardinals, this is generally proved by forcing, and there are several natural ways to force $V=\HOD$. … You can find further uses of the CCA in Set-theoretic geology, which also contains full details of this kind of forcing and variations. …

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits automatic mutual genericity, if whenever $G,H\subseteq\Q$ are distinct $V$-generic filters (existing, say, in some forcing extension of … In any forcing extension $V[G]$ by such forcing, $G$ must be the only $V$-generic filter; so this kind of forcing leads to unique generics. Thus, it is a rigidity concept. …