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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$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ 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
29 views

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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1answer
42 views

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
2
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1answer
49 views

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
66 views

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
53 views

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 ...
2
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40 views

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 ...
2
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1answer
61 views

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
45 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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3answers
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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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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? I know that the sign $p \Vdash \phi(x_1,...,x_n)$ somehow suppose to tell me that for any generic filter which contains ...
6
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1answer
93 views

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
50 views

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 ...
3
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1answer
68 views

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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0answers
64 views

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
82 views

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 $B$ ...
2
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1answer
43 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
48 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|$.?
5
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1answer
73 views

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
26 views

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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65 views

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 ...
0
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1answer
40 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
59 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 $h:A\rightarrow{B}$ ...
4
votes
0answers
90 views

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
votes
1answer
74 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
votes
1answer
44 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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38 views

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
54 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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0answers
118 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 $AC$ ...
2
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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 ...
2
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0answers
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
44 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
93 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 ...
3
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1answer
64 views

About $\kappa$-Knaster and $\kappa$-linked

For an infinite cardinal $\kappa$ and a partial order $\mathbb{P}$, we say: (1) $\mathbb{P}$ has the $\kappa$-linked is a union of $\kappa$-many linked subsets. (2) $\mathbb{P}$ has the ...
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1answer
30 views

No generic is definable in a perfect notion of forcing of a model of Peano Arithmetic

I would like to prove Lemma 6.1.2.2 from The Structure of Models of Peano Arithmetic by Kossak and Schmerl. Let $\mathcal{M}$ be a countable model of Peano Arithmetic and $\mathbb{P}=\langle P, \le ...
3
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0answers
38 views

When a $\mathbb{P}$ - generic filter is $\kappa$ - complete?

By definition a $\mathbb{P}$ - generic filter $G$ over a ground model $M$ is $\aleph_0$ - complete because for any finite set of conditions in $G$ there is a condition $p\in G$ such that $p$ is ...
0
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0answers
37 views

Connection beween closure property in forcing and preservation of $H_{\kappa}$

I would like to know about the relation between closure property of forcing notions and preservation of hierarchy of hereditary small sets, $\langle H_\lambda \mid \lambda\in \mathrm{Card}\rangle$, at ...
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1answer
43 views

Are objects built by a generic filter which is not in the ground model necessarily out of the ground model?

Let $G$ be a $\mathbb{P}$-generic filter over a ground model $M$ of ZFC and $G\notin M$. Are all objects built by this generic filter necessarily out of the ground model $M$? Particularly is limit of ...
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Questions of Hechler forcing

Shows that Hechler forcing adds Cohen real. A suggestion please. Can you tell me reference Hechler forcing.
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2answers
153 views

A question about $\aleph_1$-dense sets and the basis problem for uncountable linear orderings

I have a question which I have been unable to find a reference for and which I explain as follows: Recall that a set of reals $X$ is $\kappa$-dense if between any two real numbers there are exactly ...
8
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1answer
137 views

Examples of Forcing in Model Theory

My question is exactly my title: What are some examples of (set theoretic) forcing in model theory? I have been studying (combinatorial) set theory and model theory (independently of one another) for ...
3
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1answer
61 views

Can somebody explain (and ideally reference) this strange use/version of the Pressing Down Lemma?

In Stevo Todorcevic's "A dichotomy for P-ideals of countable sets" (link, page 261 at the bottom [page 11 in the pdf]), the following confusing situation comes up: (Context: $\mathcal I$ is a P-Ideal ...
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1answer
44 views

A Question Regarding Representing $\mathscr P$($\omega$) as a Digraph and CH

It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy ...
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2answers
77 views

Adding Substructures by Forcing

Consider a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $M$. Let $V$ be a model of ZFC (or ZF) the general question is that what would happen to classes, $Sub(M):=\{N~;~N~\text{is ...
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1answer
102 views

If $\mathbb{P}$ is a separative poset that doesn't add $\theta$-sequences then every intersection of $\theta$ dense open sets is dense in $\mathbb{P}$

I am looking for a hint (not a solution) to exercise IV.7.28 of Kunen's Set Theory book (2013). Recall that a poset $\mathbb{P}$ is separative if for every $p,q\in \mathbb{P}$, $p\nleq q$ implies ...
2
votes
2answers
67 views

A forcing that is $\omega_1$-closed and $\omega_2$-c.c.

I am reading an article (on second order characterizability) which at some point in a proof states that by forcing with $\mathbb P=\{f:\alpha\to\{0,1\},\alpha\in\omega_1\}$ we do not add subsets to ...
2
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1answer
83 views

A question from Kunen's book: chapter VII (H9), about diamond principle

Assume $(\mathbb{P}$ is c.c.c.$)^M$ and $\Diamond$ holds in $M[G]$. Show that $\Diamond$ holds in $M$. Hint: It is sufficient to verify $\Diamond^-$ in $M$. Should I try to create a ...
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1answer
60 views

PDE Transport Equation(?) with Decay and Forcing Term

So I am kind of lost on how to solve this PDE IVP. $${\mathrm du \over \mathrm dt}+2{\mathrm du \over \mathrm dx}+4u=x$$ where $t>0$ and $x$ is in $R$, with the initial condition ...
5
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0answers
50 views

Type-definable Forcing or forcing in a non-first order setting

Roughly speaking, in set forcing the forcing notion is a set from ground model's perspective and in class forcing its a definable subset of the ground model given by solutions of some formula with ...