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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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 ...
6
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1answer
81 views

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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105 views

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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57 views

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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77 views

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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1answer
56 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.
6
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74 views

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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3answers
80 views

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 ...
2
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1answer
61 views

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 ...
2
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3answers
65 views

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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2answers
112 views

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 ...
7
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1answer
168 views

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 ...
10
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1answer
174 views

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 ...
2
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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 ...
5
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1answer
65 views

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 ...
8
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1answer
114 views

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 ...
2
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1answer
85 views

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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52 views

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: $P$ ...
5
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1answer
54 views

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 ...
0
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1answer
51 views

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?
8
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1answer
144 views

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 ...
3
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1answer
69 views

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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1answer
107 views

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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0answers
42 views

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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1answer
59 views

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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1answer
54 views

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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20 views

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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84 views

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 ...
2
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1answer
68 views

$\sigma \colon (M[G];\in) \prec (N[H]; \in)$ 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 ...
4
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2answers
138 views

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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44 views

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 ...
2
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1answer
111 views

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 ...
2
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1answer
37 views

$\omega_1$-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 ...
6
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1answer
94 views

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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52 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 ...
4
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1answer
74 views

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 ...
2
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3answers
53 views

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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0answers
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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 ...
5
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1answer
101 views

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
107 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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1answer
40 views

$\mathbb{P}_{\kappa}$ 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
45 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
96 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
63 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
votes
1answer
110 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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180 views

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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1answer
83 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 ...
4
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1answer
106 views

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 ...