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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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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51 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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51 views

Question about 2-step iterations [closed]

Let $\mathbb{P}$ be poset. If $\mathbb{\dot{Q}}=\dot{G}$, the $\mathbb{P}$-name of the $\mathbb{P}$-generic filter, as a suborder of $\mathbb{P}$ then $\Vdash_{\mathbb{P}} \mathbb{\dot{Q}} ...
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1answer
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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 $H ...
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$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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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 ...
4
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1answer
53 views

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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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 ...
4
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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
39 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
74 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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2answers
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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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1answer
74 views

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 ...
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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 ...
4
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1answer
78 views

$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ 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 ...
2
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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 ...
2
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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 ...
2
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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 ...
2
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1answer
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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 ...
3
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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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1answer
47 views

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 ...
9
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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 ...
7
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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 ...
3
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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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1answer
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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 $B$ ...
3
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1answer
65 views

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 ...
2
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1answer
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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, $\sigma$ ...
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Questions of $\mathbb{P}$-name for a set and functions

Let $\mathbb{P}$ be poset. Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function ...
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1answer
43 views

Generalization of name and nice names

Let $\mathbb{P}$ be poset. Let $D, B$ be sets. We say that a $\mathbb{P}$-name $\dot{z}$ is a nice name for a function from $D$ into $B$ if there is $\left<{A_{d},h_{d}}\right>_{d \in D}$ such ...
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1answer
64 views

Names and nice names

Let $\mathbb{P}$ be poset. Let $B$ be a set. We say that a $\mathbb{P}$-name is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function $h:A\rightarrow{B}$ ...
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Non-Forcing and Independence

Do there exists sentences which are independent of ZFC, cannot be shown to be independent through some method of forcing, and do not increase the consistency strength of ZFC (e.g. so Large Cardinal ...
3
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1answer
80 views

Question of $\Diamond$ in Generic Extension

Let $M$ is a transitive model of $ZFC$ and $G$ is filter which a countable transitive model. Assume $( \mathbb{P}$ is c.c.c and $|\mathbb{P}|\leq \omega_{1})^{M}$ and $\Diamond$ holds in $M$. I want ...
2
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1answer
46 views

How to find centered subsets in the forcing Hechler

Let Hechler forcing $\mathbb{D}$. Define $\mathbb{D}=\omega^{<\omega }\times{}\omega^{\omega }$( not cardinal arithmetic) ordered as $(t,g)$ iff $s \subseteq t$ and $f \leq g$ (that is, ...
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Class of forcings with an approximation property to subsets of \omega_1

Is there a class of forcing notions which has the following property? For every $A \subseteq \omega_1 \cap V[G]$, there exists $A' \subseteq \omega_1 \cap V$ with $|A \triangle A'| \leq \omega$? That ...
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1answer
64 views

I am confused about poset $\sigma$-centered.

Assume that $2 \leq |J| \leq \aleph_{0} $. Let $\mathbb{P}=\operatorname{Fn}(I,J)$ $\mathbb{P}=\operatorname{Fn}(I,J)$ is $\sigma$-centered iff $|I| \leq \mathcal{c}$ where ...
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130 views

Choice-less Set Theory for Dummies

In almost every graduate set theory text there are some parts about equivalences of $AC$, its consequences, some axioms like $AD$ which imply $\neg AC$, some well-known axiomatic systems which $AC$ ...
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Questions of complete embedding and dense embedding

Let $\kappa$ a regular uncountable cardinal and $\mathbb{P}$ and $\mathbb{Q}$ poset. $\mathbb{P}$ has the $\kappa$-Knaster property iff for every $A \subseteq \mathbb{P}$ of size $\kappa$ there is $B ...
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70 views

Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
2
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1answer
48 views

Is this presentation of the Random real forcing separative and $\sigma$-linked?

Random real forcing is the poset formed by the closed subsets of $[0,1]$ that are non-null (with respect to the Lebesgue measure), ordered by $\subseteq$. Is the Random real forcing $\sigma$-linked? ...
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CH is preserved under a $Fn(\kappa,\lambda)$ forcing? (Kunen IV.7.10)

The following is Exercise IV.7.10 in the 2013 edition of Kunen's "Set Theory": Let $M$ be a countable, transitive model for ZFC. In $M$, let $\aleph_0 \le \kappa < \lambda$ be cardinals and ...
2
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1answer
109 views

Questions of $n$-linked in poset

Let $2\leq n \leq \omega$ and $F$ be a set of size $\leq n$ . Let $\mathbb{P}$ be a poset and $Q \subseteq \mathbb{P}$ an $n$-linked subset. Questions: if $\dot{a} $ is a name for a menber of ...