Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use [tag:differential-equations].

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Enlarging the continuum with $\sigma$-centered forcing

How large can we force the continuum to be if we force with a $\sigma$-centered forcing notion? References to texts discussing the subject would be much appreciated. [A forcing notion $P$ is called ...
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Kunen exercise III.8.21

Let $f: \omega_1\to \mathbb{R}$ be one-one. Let $g:[\omega_1]^2\to 2$ be such that for any $\alpha<\beta<\omega_1$, $g(\{\alpha, \beta\})$ is $0$ when $f(\alpha)<f(\beta)$, and $1$ otherwise. ...
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GCH is preserved when forcing with $Fn(\lambda,\kappa)$.

Given a countable transitive model $M$ where $GCH$ holds it is an exercise from Kunen's book to show that GCH also holds in $M[G]$ when $G$ is a $P-$generic filter over $M$, and $P=Fn(\lambda,\kappa)$ ...
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$\kappa$-closed forcing preserves stationary sets.

Let's take an uncountable cardinal $\kappa$ which is regular inside the ground model $M$ and $\mathbb{P}\in M$ a forcing notion which is $\kappa-$closed in $M$. I'm trying to prove that every ...
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Countable Transitive model where $\exists A\subset \omega_1\;(L[A]\vDash\, \neg CH)$

It is well known that for every subset $A\subset \omega_1$ if $V=L[A]$ then $L[A]\vDash GCH$. In particular $L\vDash \exists A\subset \omega_1\,(L[A]\vDash\, GCH)$. Nonetheless, it is also consistent ...
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Preserving weakly inaccesible cardinals in generic extensions

I'm trying to see that every weakly inaccessible cardinal $\kappa$ in a ctm $M$ remains weakly inaccesible if we force with a forcing which preserves cofinalities (thus, preserves cardinalities). It's ...
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Do the notations for relative constructible universe and for forcing extention coincide?

We have the notations $L[A]$ for the inner-model constructible relative to some $A$, and the notation $M[G]$ for a generic extention of the model $M$. Do they coincide? That is, if we look at the ...
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Forcing exercise from Kunen's book

I'm new in the study of the forcing method and I having some troubbles to solve some of the exercise from Kunen's book (edition 2013): specifically, problem IV 2.46 from page 271. It says the ...
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Understanding sets added by a forcing notion

Consider a coloring $c:[\kappa]^2 \to 2$ ($\kappa$ a regular uncountable cardinal, can be assumed to be $\omega_1$ for simplicity) s.t. the following holds: For every $A \subset ...
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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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Given two forcing extensions, is there a common extension?

Working in ZFC. Say $\mathbb V$ is the ground model, and $\mathbb V[G_1]$ and $\mathbb V[G_2]$ are forcing extensions. Is there a forcing extension $\mathbb V[H]$ containing $\mathbb V[G_1]$ and ...
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Do the relative consistency results involving the axiom of choice use completeness of FOL?

The proof of the Completeness Theorem for first order logic uses the axiom choice and is a central result in model theory. Chang and Keisler claim that model theory has important applications to set ...
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Equivalence of forcing notions from dense embedding between them

In general I want to prove that $\mathbb{P}=\left(P,\leq\right)$ and $\mathbb{Q}=\left(Q,\leq\right)$ are forcing notions and there is a dense embedding $h:P\longrightarrow Q$ , then ...
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Does forcing preserve the least undefinable ordinal from a model of ZFC?

Let $M$ be a transitive model of ZFC. For convenience let assume that $M$ is countable. Now let us consider the least undefinable ordinal $\vartheta_M$ which is not definable from elements in $M$ and ...
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In what sense does forcing increase the width of a set-theoretic hierarchy?

I've often read that, whereas top extensions increase the height of the cumulative hierarchy, forcing extensions increase its width (see, e.g., the answer to this question where's it's said that ...
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Are two generic filters in a common generic extension?

Let $M$ be a countable transitive set. Suppose $\mathbb{P}$ is a forcing in $M$. Let $G$ and $H$ be two generic filters for $\mathbb{P}$ over $M$. My questions are: Is there a forcing $\mathbb{Q}$ ...
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What is forcing isomorphism? [closed]

This question is from Kunen's set theory book. My questions are: What is the definition of isomorphism between forcing notions? When do we say that two forcing notions $\mathbb{P},\mathbb{Q}$ ...
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Shelah's materialize vs. realize;note: tags are badly chosen due to the lack of them

Can someone please explain to me in some detail the exact difference between materialize and realize for a Galois type $p$? Esp. is realize a special case for materialize? Why is it so? What is the ...
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Minimal Possible Ordinal for a Cardinal

For a given $\aleph_\alpha$, what is the minimal ordinal possible such that $|\beta| = \aleph_\alpha$? More precisely, assume we have a model $N$ of ZFC. $N$ could be an extension of many different ...
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59 views

If $M[G] \subseteq M[H]$ are forcing extensions, why is $M[H]$ a generic extension of $M[G]$?

I know that, wlog, we may assume $G = H \cap A$ for some complete subalgebra $A$ of the complete Boolean algebra $B$ over which $H$ is $M$-generic.
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Question about collapsing cardinals

Suppose, in $M$, $\kappa$ regular, $\lambda>\kappa$ regular. Is there a generic extension of $M$ in which $\kappa^+ = \lambda$ and in which cardinals $\leq \kappa$ and $\geq \lambda$ are preserved? ...
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Is there is a real $r$ and a countable transitive model $M$ such that $r$ is not in any forcing extension of $M$?

It is a theorem that if $M[G]$ is a generic extension $M$, then for every model $N$ of ZFC with $M \subset N \subset M[G]$, $\ N$ is some generic extension of $M$ (and is, in fact, $M[G\cap D]$ for ...
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Definition of the forcing relation in class-forcing

Jech really glosses over class forcing. I cannot find a good reference online. I have two questions about it. 1) Jech says, "As for the forcing relation in general, we cannot generally define ...
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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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1answer
66 views

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 ...
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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: $P$ ...
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Cardinals and generic extensions

Let us consider a model of ZFC $M$, a forcing $\mathbb{P}\in M$. If $G$ is a $\mathbb{P}$-generic filter over $M$ we can construct the generic extension $M[G]$ of $M$. Given any cardinal number ...
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Consistency of the Subcomplete Forcing Axiom (relative to a supercompact cardinal)

In the introduction to his Singapore lecture Jensen mentions that the Subcomplete Forcing Axiom is consistent relative to a supercompact cardinal. Can anyone refer me to a proof of this claim?
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Transitive models and CH

Suppose $M, N$ are two countable transitive models of ZFC which have same ordinals, cofinalities and reals (but not necessarily same sets of reals!). Suppose $M$ models the continuum hypothesis. Can ...
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Elementarily equivalent forcing extension?

Is it possible to take a forcing extension which is elementarily equivalent to the ground model? Here I'm assuming the extension is proper, that is, it adds a new set. It's clear it can't be an ...
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Reflection Principle vs. Löwenheim-Skolem-Theorem

From my undestanding a standard method of deducing relative consistency results is the following: By a combination of the Levy reflection principle, Skolem-Hulls and Mostowski Collapse we show: If ...
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Forcing, $ p \Vdash q \in \dot{G} \Rightarrow p \leq q $

I was wondering if a poset is separative if $ p \Vdash q \in \dot{G} ~~ \Rightarrow p \leq q$ I think it's clear that $ p, q \in G $ and hence are compatible but I am not seeing why ( if it's true ...
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Some questions regarding a theorem of Paul J. Cohen

In his paper "Automorphisms of Set Theory", Paul Cohen proved the following theorem: "There exist models of $ZF$ admitting automorphisms of order two. More exactly, If $M$ is any countable model ...
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Cohen forcing question

Suppose $M$ is a countable transitive model of ZFC and $(x, y, z)$ is Cohen generic point in $\mathbb{R}^3$ over $M$: This means that for every open dense set $U \subseteq \mathbb{R}^3$ in $M$, $(x, ...
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Question of Hechler forcing and $\mathbb{LOC}$ notion Localization forcing

Let $\mathbb{D}$ notion Hechler forcing and $\mu$ uncountable regular cardinal. All subalgebra of $\mathbb{D}$ of size $<$ $\mu$ is $\mu$-centered. Let $\mathbb{LOC}$ notion Localitation forcing. ...
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Why does Namba forcing preserve $\omega_1$?

I have seen proofs that under CH, Namba forcing does not add reals, and thus preserves $\omega_1$. How do you prove in ZFC alone that it preserves $\omega_1$? I have also seen the stronger claim ...
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$\sigma \colon (M[G];\in) \prec (N[H]; \in)$ implies $\sigma \restriction M \colon M \prec N$?

Let $M,N$ be transitive models of sufficiently many $\operatorname{ZFC}$ axioms and let $$ \sigma \colon M[G] \prec N[H] $$ be an elementary embedding, where $G$ is $\mathbb P$-generic over $M$ for ...
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Using a forcing extension $V[G]$ to determine properties of $V$.

It is well known that you can use forcing to change the truth value of various sentences ($CH$, $\Diamond$, et cetera). However, often when performing such a construction over a model $V$, the action ...
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The generic function added by the localization forcing for $h : \omega \to \omega$ determines the generic extension

Let $h:\omega\to \omega$ is a monotone increasing function that goes to infinity. For $n<\omega$ let $\mathcal{S}_{n}(\omega,h)=\{s:n \to{\omega}^{<\omega}: \forall i< \omega(|s(i)|\leq ...
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Concequences of the result of Woodin's problem

In his book "The Axiom of Determinacy, Forcing Axioms, and the Non-Stationary Ideal" Woodin formulated tho following problem [No 22]: Are there two $\Pi_2$-sentences $\psi_1$ and $\psi_2$ in the ...
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1answer
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Good introduction to “forcing” and “inner models”?

I've occasionally come across the use of forcing, e.g., JDH's exploration of the modal logical of forcing. I know it is a massively important proof technique for, e.g., independence proofs. I've also ...
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$\omega_1$-closedness and fullness for $\searrow$ $\omega$-sequences

Let $\pi$ is a $\Bbb{P}$-name for a partial order, i.e. there is a name $\pi'$ and $\pi''$ such that $$1\Vdash_\Bbb{P} \pi '' \in \pi\land (\text{$\pi'$ is a partial order of $\pi$ with largest ...
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Sets Forced to be Equal in All Extensions

My question is: Let $\mathbb{P}$ be a forcing and $\tau \in V^\mathbb{P}$ is a name. Suppose that $$1_{\mathbb{P} \times \mathbb{P}} \Vdash_{\mathbb{P} \times \mathbb{P}} \tau_\text{left} = ...
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Cohen Forcing in Set Theory - Proof that Forcing is Equivalent to intersection of Dense Sets

Cohen's book "Set Theory and the Continuum Hypothesis" on Page 126/127 (see below) shows that the existence of a completed new set a' is equivalent to its intersection with all dense subsets in M. I ...