# Tagged Questions

Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).

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### Counting the number of names for elements of a certain name.

I'm self studying the proof of consistency of MA on Jech's Set Theory (Theorem 16.13, p. 272). There is a step which I can't understand. To simplify the notation, I will try to "extract" the relevant ...
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### Measure of an antichain of the Random forcing

I'm dealing with the random forcing $\mathbb{P}=\mathbb{M}\mathbb{B}(X,\mu)$ defined in Kunen's 2013 book. Recall that the conditions of that forcing are the equivalence classes of positive measureble ...
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### Property ccc of the random real forcing

I'm dealing with the random forcing $\mathbb{P}=\mathbb{M}\mathbb{B}(X,\mu)$ defined in Kunen's 2013 book. Recall that the conditions of that forcing are the equivalence classes of mesurable sets ...
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### Forcing with a “shrinked” poset

Assume, in $V$, that we have some forcing $P$, a $P$-name $\tau$ and some sentence in the forcing language $\varphi(\tau)$ (it may include other names but we focus on $\tau$). Now we take some ...
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### A poset oriented proof for the intermediate model theorem.

The intermediate model theorem: If $M\subseteq N\subseteq M[G]$ are all models of $\sf ZFC$, with $G$ generic over $M$, then $N$ is a generic extension of $M$, and $M[G]$ is a generic extension of ...
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### Reference about $\sigma$-linked posets and related notions

In this link, the following list appears: Some chain conditions [of posets], listed from easiest to satisfy to hardest to satisfy: ccc powerfully ccc productively ccc $\sigma$-...
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### The Meta-Mathematics of Multiple Forcing

In forcing we have the forcing theorem (also called the truth and definability theorem). It guarantees that forcing works. What are the similar theorems for multiple forcing? To elaborate: Kunen, in ...
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### Maximal antichains in a forcing which adds surjections

Let $P$ be a separative partial order such that $\left| P \right| \leq \left| \alpha \right|$ and $$\Vdash_P\exists f(f\colon\omega\to\alpha\text{ is surjective}\land f\notin\check V).$$ I want to ...
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### Well-pruned $\kappa$-Suslin trees have the $\kappa$-Baire property

I'm having troubles seeing that a well-pruned $\kappa$-Suslin tree, with $\kappa$ uncountable and regular, has the $\kappa$-Baire property. I saw in Kunen's Set Theory that it can be seen by proving ...
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### Separative forcings add new sets

Let $\mathbb{M} = (M, \in)$ be a class which models ZFC. A forcing $\mathcal{P} = (P, \leq)$ on $M$ (where $P \in M$ and $\mathbb{M} \vDash P \in M$) is separative if for every $r \in P$ there are ...
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### Does ZF $\vdash$ Con(ZF) $\rightarrow$ Con(ZF+$\lnot$AC)?

Certainly ZFC $\vdash$ Con(ZF) $\rightarrow$ Con(ZF+$\lnot$AC), but the usual forcing argument to construct a model of ZF+$\lnot$AC seems to require choice to find a generic filter.
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### Is there a forcing extension $M[G]$ of $M$ that adds a new $\omega$-sized subset to $\omega_2$ without adding any new subsets of $\omega$?

I should add that the forcing extension must preserve the cardinals $\aleph_1$ and $\aleph_2$. Note that such a forcing extension cannot add any new $\omega$-sized subsets to $\omega_1$, and also ...
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### Can all Power Sets be Limit Cardinals?

Is it possible to create a model of ZFC, so that the cardinality of each power set is a limit cardinal (as opposed to GCH where they are always successor cardinals)? Take for example the following ...
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### Cardinality of power sets decides all of cardinal arithmetic?

Assuming ZFC, is it possible to have two models which agree on the cardinality of all the power sets, but disagree on the cardinality of some other cardinal exponentiation (meaning that they agree on ...
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### What is the intuitive meaning of the notion of absolutness in set theory?

I would love to know what the notion of absolutness in set theoy mean intuitively. I wonder if anyone can provide me a heuristics about it. I know the formal definition and that it corresponds to ...
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### Forcing names, parameters in definitions, and the Iterative Conception of Set

So, I've been trying to learn as much as I can about forcing. I know that a model provides its own (trivial) forcing extension. What I'm curious about is whether there is a way to think of the ...
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### Do the ZF-provable forcing principles differ from the ZFC-provable forcing principles?

In "The Modal Logic of Forcing", Joel David Hamkins and Benedikt Löwe show that the ZFC-provable forcing principles are exactly those of the modal logic S4.2 (interpreting $\Diamond \phi$ as asserting ...
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### Applications of forcing to Topology

I'm insterested on set theory and general topology: particulary on forcing and compactness. I'm searching for a book which study the interaction between both topics or even the interaction between Set-...
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### Keeping unwanted generic sets out of limit stage of iterated forcing

My question is about keeping unwanted generic sets from appearing at the first limit stage of an iterated forcing. The usual motivation for this is preserving some property $P$ of ZFC models: $P$ ...