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Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $ …

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\m …

I am trying to understand the proof that the GCH can first fail at a weakly compact cardinal. We assume the GCH and that there exists a weakly compact cardinal $\kappa$, and we construct a reverse Eas …

There, $\mathbb P_\lambda$ is the $\lambda$-th step of an iterated forcing construction which is the inverse limit of the previous steps, and $G$ is a $V$-generic filter. …

I am reading Uri Abraham's chapter on Proper Forcing in the Handbook of Set Theory and I have a quite trivial question on the definition of $\alpha$-proper forcing. … In Proper and Improper Forcing, Shelah requires $i\in M_i$ in the definition of $\alpha$-tower, Abraham does not include this, but I have needed an even stronger hypothesis in order to prove that $(*)$ …

My first question is that I suspect that this is also valid for the definition of the forcing relation $p\Vdash \phi$, i.e.: Question 1: Can the forcing relation $p\Vdash \phi$ be defined for mathematical … This question is related to theorem III 2.11 of Shelah's book Proper and Improper Forcing (second edition). …

However, I have no insight at all about the meaning of $u\mapsto u:e$ with regard to forcing. …

I see that all the axioms have a clear translation in standard terms of forcing but the last one, which asserts more or less what I am trying to prove. …