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Later a more elementary proof was found, using no 'metamathematical' concepts as forcing or ultrapowers of the universe, similar to the history of the proofs of Silvers theorem. …

I am also highly interested in the dual situation, where we replace Cohen forcing with Random forcing in the above. Can we guarantee that there is a real $r$ which is Random over all $V^{P_n}$'s? …

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 course proper) notions of forcing are stationary set preserving. …

Is it possible to have an $\omega_2$-length sequence of ($\omega_1$-)Suslin trees such that if one builds the product of finitely many trees in that sequence, one ends up with a Suslin tree again? Th …

If we start with $L$ as our ground model then whenever $T$ is a Suslin tree, the forcing $\mathbb{P}_T$ which shoots a branch through $T$ will always introduce new Suslin trees. … Thus in $L$, $\mathbb{P}_T$ is an example of a forcing which has the ccc, adds Suslin trees and does not add a Cohen real. …

Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic filter for the forcing. … Note that the statement is true if $\mathbb{P}$ is just the trivial forcing, however I would be interested in the case when we collapse cardinals down to $\omega_1$. …

Scott quickly after the invention of forcing, used a forcing argument to show that the Continuum Hypothesis is independent of an axiomatization of Second Order Analysis. … Jech's book, which explains forcing as forcing with Boolean valued models of the universe $V$. …

It is well known that if $P$ is a proper notion of forcing and $\Vdash_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. … My intuition would tell me no, we could probably add in the first step a new stationary set, which gets killed after the second forcing. …

So: The following concept is due to Shelah and I have some issues with a claim using this notion: Suppose that $\nu$ is a limit ordinal and that $P_\nu$ is an iteration of forcing notions. …

The usual strategy to prove that the negation of $CH$ is consistent with $ZFC$ is the following: One shows (under the assumption $Con(ZFC)$ that for any finite fragment $T$ of $ZFC$ we have that $T+ …

There exists a stationary subset of $P_{\omega_1} (\omega_2)$ of size $\aleph_2$. This is a result of Baumgartner and you can find a proof for this here: Why is this set stationary? I don't dare to a …

However a (probably very easy) question came up: Assume that we force with a Boolean algebra of the form $P(A) / I$, where $A$ is an arbitrary set and $I$ is an ideal over $A$; i.e. our forcing conditions …

It is true that you can use Löwenheim Skolem to get a countable model $M$ of ZFC assuming $Con(ZFC)$. But to use Mostowski you need additionally the well foundedness of that model, which doesn't have …

Assume that $(P,\le)$ is a notion of forcing. …

I think that I might have found a solution to this rather dispensable question. I will sketch it: Consider the following characterization of properness: $P$ is proper iff for all $\lambda > 2^{|P|}$ …