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Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
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Generic filters of inverse limits
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. …
6
votes
1
answer
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On the definition of $\alpha$-proper poset
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 $(*)$ …
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Is the forcing relation defined for mathematical formulas?
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). …
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What is the meaning of restricting a Boolean value to a subalgebra?
However, I have no insight at all about the meaning of $u\mapsto u:e$ with regard to forcing. …
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1
answer
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The GCH in a reverse Easton support iteration
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 …
6
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1
answer
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A question about the first Cohen model
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. …
3
votes
1
answer
416
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Representation of meager sets in Cohen extensions
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 $ …
6
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Iteration of random reals
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 …