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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$\omega_1$-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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Facts on elementary submodels

In the paper of "Aspero, Larson, Moore - Forcing Axioms and the CH" three facts are stated as well-known. As i have not read them before, they are not that obvious to me. Maybe good references to ...
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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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35 views

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 ...
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extending automorphisms in complete boolean algebras

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$. Suppose $f : A \to A$ is an automorphism. Then $f$ can be extended to an automorphism of $B$. I can see this using the fact ...
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3answers
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Relation between topological denseness and denseness over poset

In the theory of forcing, the notion of dense set is important. Formally, a subset $D$ of a poset $P$ is dense if, for any $p\in P$ we can find some $q\in D$ with $q\le p$. Intuitively, denseness of ...
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On the paper “Forcing and the CH” by Aspero/Larson/Moore

Forcing Axioms and the CH by Aspero/Larson/Moore On page 11 of this paper I struggle with the beginning of the proof of Lemma 3.5. Coding: For $r \in 2^\omega$ and $A \in H(\aleph_1)$ let us say ...
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Iterated Forcing, to force $2^{\omega}=\kappa$ and $2^{\omega _1}=\lambda$

Hellow i'm stuck on some details in this iterated forcing exercise. Let $M$ be a countable transitive model of $ZFC+GCH$ and assume that $\kappa<\lambda$ are cardinals with $\aleph _0 ...
2
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1answer
85 views

(totally) (M,P)-generic forcing condition

We say a cardinal $\theta$ is sufficiently large for a forcing $Q$ if $\mathcal{P}(\mathcal{P}(Q)) \in H(\theta)$. And a set $M$ is a suitable model for $Q$ if $Q \in M$ and $M \prec H(\theta)$, $M$ ...
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$\mathbb{P}_{\kappa}$ forces $\text{non}(\mathcal{M})\leq \kappa$ and $\text{cov}(\mathcal{M})\leq \kappa$

Let $\mathbb{D}$ Hechler forcing. Let be $\kappa$ an uncountable regular cardinal. Consider the finite support iteration $(\langle \mathbb{P} \rangle _{\alpha < \kappa}, \langle \dot{\mathbb{Q}} ...
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4answers
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Why do we define an inner forcing relation?

Studying forcing I came across different definitions of the forcing relation $\Vdash$: the outer forcing relation $\Vdash^M$ where we define $p \Vdash^M \varphi(\tau_0,\dotsc,\tau_n)$ to hold if for ...
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2answers
35 views

Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
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1answer
78 views

Quotient of Cohen forcing

How do we know that the quotient of the Boolean algebra associated with Cohen forcing by a generic filter is either atomic or isomorphic to the Cohen forcing? I know that Cohen forcing is the unique ...
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2answers
55 views

A road-map through “Combinatorial Set theory: With gentle intro to independence proofs”

I'm going to study independence proofs form Halbeisen's book. It seems that some material is not needed to study independence proofs, so it seems that the book contains more material than my needs. ...
3
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86 views

$\operatorname{Fn}(\lambda,2,\lambda)$ collapses $\lambda^+$ to $\operatorname{cf}\lambda$ if $\lambda$ is singular?

It is an exercise problem in Kunen (VII G5). I shall show that $\operatorname{Fn}(\lambda, 2, \lambda)$ adds a map from $\theta = \operatorname{cf}\lambda$ onto $\lambda^+$ for singular $\lambda$. ...
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Good resources for studying independence proofs

I've finished most of Enderton's set theory. And I intend to spend some time studying independence proofs. I'm more interested in independence of axiom of choice not CH. From I know so far, there ...
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70 views

Cohen forcing factoring

I start from $M$ a transitive countable model of $ZFC + \mathbb V= \mathbb L$ and I add a single Cohen generic $G$. Now if $A \in M[G]$ is also Cohen generic over $\mathbb L$ and $M[A] \ne M[G]$, can ...
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Can every ccc forcing of size $2^{\aleph_0}$ be $\sigma$-linked?

It is well known that $MA_{\aleph_1}$ implies that every ccc forcing of size $< 2^{\aleph_0}$ is $\sigma$-linked (in fact - a countable union of filters). On the other hand, a separative forcing ...
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1answer
49 views

Why $\forall{f\in \omega^{\omega}\cap V}$ $ \forall^{\infty}n f(n)\neq f_{G}(n)$

Eventually different forcing, $\mathbb{E}=\{\langle s,H \rangle:s \in \omega^{\omega}\wedge H\subseteq [\omega^{\omega}]^{<\omega}\}$. ordered by $(s',H')\leq (s,H)$ iff $s \subseteq s'$ and $H ...
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1answer
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$V[G]=V[f_{G}]$ if $f_{G}=\bigcup\{s:(s,H)\in G\}$.

Eventually different forcing, $\mathbb{E}$, consists of pairs $(s,F)$, where $s \in \omega^{<\omega}$ and $F$ is a finite set of reals. $(s,F)\leq (t,G)$ iff $t \subseteq s$ and $G \subseteq F$ ...
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0answers
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If $\alpha \leq \delta$ and cf$(\lambda)\geq\kappa$, then $\mathbb{P}_{\lambda}\simeq \text{limdir}_{\alpha<\lambda}\mathbb{P}_{\alpha}$.

let $\kappa$ be an infinite cardinal, $\delta$ an ordinal and $\mathcal{I}=\{C\subseteq \delta :|C|<\kappa \}$. Consider an $\mathcal{I}$-support iteration $(\langle \mathbb{P}_{\alpha \leq ...
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What conditions must be checked for that $c$ is Cohen over $V$.

$\textbf{Hechler forcing} $ Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ ...
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Exercise on forcing

I got this homework in my forcing class: Let $G\subseteq P$ be generic over M. Show that there is a cardinal of M, $\lambda$ such for every set of ordinals $X\in M\left[G\right]$ there is a set ...
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2answers
131 views

What are some simple example of “forcing” in set theory?

Can someone illustrate the idea of "forcing" in set theory through some simple examples? The article on forcing on wikipedia goes straight to axiom of choice and continuum hypothesis, I wonder if ...
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62 views

Not meager in $V^{\mathbb{C}_I}$

Assume $A\subseteq 2^\omega $ is not meager in any non-empty open set, in The ground model $V $. Then is not meager in any non-empty open set, in $ V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen ...
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1answer
45 views

P-names are comprehensible in the ground model

If my understanding of $\mathbb{P}$-names $\tau$ is correct, they must be comprehensible by someone living in the ground model $\mathcal{M}$. However, the value $\tau[G]$ of $\tau$ need not be ...
5
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1answer
80 views

Find a dense embedding from specific forcing poset to any countable forcing poset

I tried to prove this in the Kunen's set theory: Let $P$ be a countable non-atomic partial order. Show that there is a dense embedding from $T = \{p\in\operatorname{Fn}(\omega,\omega) ...
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Definition of $\mathbb{P}$-name with index number

I've just started studying forcing. Currently, I am struggling to understand what is a $\mathbb{P}$-name in the first chapter in the Shelah's book page 6 ...
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In P. Cohen's models (or others) may we have $\neg\mathsf{AC}+\mathsf{CH}$? May we have $\neg \mathsf{AC} + \neg \mathsf{CH}$?

I know we have the consistency of $\mathsf{ZF} + \mathsf{AC} + \mathsf{GCH}$, $\mathsf{ZF} + \neg \mathsf{AC}$, and $\mathsf{ZF} + \mathsf{AC} + \neg \mathsf{GCH}$. What about $\mathsf{ZF} + \neg ...
5
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show that there is a $\mathbb P$-name $\sigma$ such that $M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$

I want to show that there is a $\mathbb P$-name $\sigma$ such that for every $G$ a generic filter we will have $$M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$$ while $\phi (x)$ is a ...
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$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ has property $\mu$-linked and $\mu$-centered.

Let $\mathbb{P}$ be a poset and $\mathbb{\dot{Q}}$ be a $\mathbb{P}$-name of a poset. If $\phi(\mathbb{P})$ and $\Vdash_{\mathbb{P}} \phi(\mathbb{\dot{Q}})$, then $\phi(\mathbb{P}\ast ...
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1answer
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forcing over a dense set.

I want to show that given a countable transitive model $M$ a notion of forcing $\mathbb P$ and a generic filter $G$, then forcing over a dense set of $\mathbb P$, $D$ is just as forcing over $\mathbb ...
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1answer
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The Amoeba poset has property Knaster.?

The Amoeba poset has property Knaster.? Amoeba poset $\mathbb{P}=\{p \subseteq \mathbb{R}: p$ $\text{is open} \wedge \mu(p)< \epsilon\}$ for $\epsilon>0$. Any suggestion. Thanks
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Why this set is finite $E_n=\{l \in \omega: \exists{j<n}[r_j \Vdash \check{l} \leq \dot{h}(n)]\}$?

Lemma IV (4.9) (Kunen book). Let $M$ be a countable transitive model of $ZFC$ and fix $\mathbb{Q}\in M$ such that $(|\mathbb{Q}| \leq \aleph_{0})^{M}$. Let $G$ be $\mathbb{Q}$-generic over $M$. Then ...
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1answer
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What axioms are needed in proofs of the independence of the continuum hypothesis?

My understanding is that the proofs that CH and not-CH are consistent with ZFC are both about ZFC and in ZFC. Is it possible to do these proofs about ZFC but in a weaker axiomatic system? (It is also ...
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Forcing reference

Who first proved that, over ZF, the statement (1) The reals are well-orderable is strictly stronger than the statement (2) Every real-indexed family of nonempty sets of reals has a choice ...
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Forcing $M[G] \models \text{cov(meager)}\geq \kappa$

Let $M$ a countable transitive model $ZFC$. In $M$, let $\mathbb{P}=Fn(\kappa,2)$, where $\kappa$ is any cardinal. Let $G$ be $\mathbb{P}$-generic over $M$. Then $M[G] \models \text{cov(meager)}\geq ...
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1answer
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Why can fix $W_0 \subseteq \kappa$ and select $W$ such that $W_0\subseteq W \subseteq \kappa$?

Lemma V$.2.19$ (book Kunen) In $M$, let $\mathbb{P}=Fn(\kappa,\omega)$, where $\kappa$ is any cardinal. Let $K$ be $\mathbb{P}$-generic over $M$. Then $M[K]\models \mathfrak{d} \geq \kappa$ Proof: ...
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What is the difference between $\pi \in \sigma$ to $\pi \in dom(\sigma)$

I am reading the proof of 4.20 here below and I don't understand: What is the different between $\pi \in \sigma$ to $\pi \in dom(\sigma)$? I am also not sure what is $dom(\sigma)$. Is there anything ...
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What is a $P$-name in forcing theory

I am having troubles understanding what is a $P$-name is forcing theory and what is the purpose of this term in the forcing tecnique. Is there any simple way to explain this term. If there was I ...
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1answer
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Can anyone explain what is the intuition behind the following definition of $p \Vdash^* \phi $?

Can anyone explain what is the intuition behind the following definition? Definition 4.25 Let $\Bbb P$ be a poset. Let $\phi(x_1,\ldots,x_n)$ be a formula, $p\in\Bbb P$, and let ...
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how to collapse $\omega_2$ to a smaller cardinal

Let $M$ be a model of ZFC and take the forcing notion $P(\omega,\omega_2)$ where: $P(\omega,\omega_2)=\{p|p \space is \space a \space function \space and \space \exists n \space s.t. (dom(p)=n) \space ...
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1answer
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Function $f\in M[G]$, $f:\kappa\to M$ is in the ground model implies $\kappa^+$-Baire

Let $M$ be countable transitive model of ZFC, $P\in M$ be poset and $\kappa$ be a cardinal in $M$. In addition, for every $P$-generic filter $G$ over $M$, if a function from $\kappa$ to $M$ is in ...
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Simple question about of $\Vdash \varphi$ [closed]

Let $\mathbb{P}$ a poset. The following are equivalent. $(1)$ $p\Vdash \varphi$. $(2)$ $\forall r\leq p(r\Vdash \varphi)$. $(3)$ $\{r: r\Vdash \varphi\}$ is dense below $p$. I am confused when ...
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Simple question of poset and names.

If $\mathbb{P}$ be a poset, $\dot{Q}$ a $\mathbb{P}$-name and $\mu$ an infinite cardinal such that $\Vdash 0<|\dot{Q}|\leq\mu$. $(a)$ Exist names $\langle \dot{q}_\alpha\rangle_{\alpha<\mu}$ ...
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1answer
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How to define a nice name?

Let $\mathbb{P}$ be a poset and $B,D$ be sets. Let $p \in \mathbb{P}$ and $\sigma$ be a $\mathbb{P}$-name such that $p \Vdash \sigma \in B$. Then there exist a nice name $\tau$ for an object in $B$ ...
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1answer
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How to show that the Cohen forcing adding arbitrary many reals adds no dominating real

Let $\lambda$ be any infinite cardinal and let $Fn(\lambda, 2)$ be the set of finite partial functions from $\lambda$ into $2$. This is a forcing notion adding $\lambda$ many Cohen reals. It is a ...
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1answer
56 views

A question about of $\mathbb{P}$-name

Let $ \mathbb{P}$ be poset and $\dot{Q}$ a $\mathbb{P}$-name. Question: we can find some cardinal $\mu$ in the ground model such that $ \Vdash |\dot{Q}| \leq |\mu|$.?
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1answer
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Why we use $\mathrm{Fn}(\kappa\times\lambda,2,\lambda)$ to force $2^\lambda\ge\kappa$ instead of $\mathrm{Fn}(\kappa\times \lambda,2)$

I am reading Kunen's Set Theory and I learn that, $\operatorname{Fn}(\kappa\times\omega,2)$ forces $2^{\aleph_0}\ge|\kappa|$, where $$\operatorname{Fn}(I,J,\lambda)=\{p:p\text{ ...
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0answers
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Simple question about $\mathbb{P}$-name [duplicate]

If $p \in \mathbb{P}$ and $\tau$ be a $\mathbb{P}$-name such that $p \Vdash \tau \in B$. Then there exist a nice name $\dot{b}$ for an object in $B$ such that $\Vdash \tau =\dot{b}$. Also, $\sigma$ ...