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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The notations $ M[X] $ and $ V[A] $ and intermediate models

I am confused about Jech's notations $ M[X] $ and $ V[A] $. Let us begin with exercise 13.34 (see [Jec02, p. 199]). Let $ \mathbf{M} $ be a transitive model of $ \mathsf{ZF} $ containing all ...
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24 views

Complete embeddings and intermediate models

Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Theorem. Suppose that $ \mathbb{P} := (P, {\leq_P}, ...
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62 views

cov(meager) strictly between $\aleph_1$ and $2^{\aleph_0}$

Is is consistent that $\aleph_1 < \text{cov(meager)} < 2^{\aleph_0}$? I can only seem to find references for results that assert it is consistent that it (or other cardinal characteristics) is ...
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54 views

Proof of PFA from Supercompact

In Jech's proof (in chapter 31) that the consistency of a supercompact cardinal implies the consistency of PFA, he needs the following fact: Let $\mathbb{P}_\kappa$ be the countable support forcing ...
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2answers
50 views

Example of a non-proper product of two proper forcing notions

I am looking for an example of a proper forcing notion $ \mathbb{P} $ such that $ \mathbb{P} \times \mathbb{P} $ is not proper. Maybe someone knows an obvious example or can give a reference to such ...
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101 views

Does the Math-tea Argument Have Any Relevance to the Method of Forcing?

The Math-tea Argument (i.e. the argument that, for example, there must be real numbers that we cannot describe or define, because there are only countably many definitions, but uncountably many reals) ...
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76 views

An exercise in Kunen

The following exercise appears in Kunen; In $M$, let $\mathbb{P}=Fn(\kappa,\lambda)$ (i.e. the finite partial functions), where $\aleph_0\leq\kappa<\lambda$. Then $\lambda$ is countable in $M[G]$, ...
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1answer
93 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
2
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1answer
71 views

Some Questions Regarding Pointwise Definable Models of ZFC

In their paper "Pointwise Definable Models of Set Theory" Hamkins, Linetsky, and Reitz prove the following theorem: "Every countable model of ZFC has a pointwise definable class forcing extension." ...
2
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1answer
39 views

Adding subsets of regular cardinals (Jech p. 226)

On p. 226 of his ${\it Set}$ ${\it Theory}$, Jech considers adding $\lambda$ many subsets of $\kappa$ to a ground model $M$. He outlines a suitable partial order on the assumption that $M$ satisfies ...
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1answer
70 views

On the number of countable models of complete theories of models of ZFC

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
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70 views

Martin's Axiom and products of c.c.c. spaces

A topological space $X$ has the c.c.c. if every family of pairwise disjoint non-empty open subsets of $X$ is countable. Consider the following statement: $(P)$ The product of two c.c.c. spaces is ...
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2answers
82 views

Introduction to Proper Forcing Reference

What is a good introduction to proper forcing? I am aware of Shelah Proper and Improper Forcing, but I heard this book may be somewhat challenging to read. There is also Devlin's The Yorkshiremen's ...
2
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1answer
63 views

Equivalent (?) definitions of Axiom A

(Why) Are the following definitions of Axiom A equivalent? Soft question: Which one is more common, natural, or usually easier to verify? What was Baumgartner's original definition? ( B ...
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1answer
88 views

A question regarding Worldly Cardinals and L

For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals? The motivation ...
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1answer
41 views

Existence of Certain Names in Iterated Forcing

Suppose $\mathbb{P}$ is a forcing. Let $\dot{\mathbb{Q}}$, $\dot{<_\mathbb{Q}}$, and $\dot{1}_\mathbb{Q}$ be a name such that $1_\mathbb{P} \Vdash_\mathbb{P} ``\langle \dot{\mathbb{Q}}, ...
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96 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
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1answer
94 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
2
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2answers
101 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
2
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1answer
95 views

What does it mean to say that a forcing “collapses cardinals”?

I hear the following terminology a lot: "So-and-so forcing collapses cardinals." Does this just mean that certain cardinals in the ground model are no longer cardinals in the forcing extension? If ...
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33 views

Strategic closure of quotients

Is there an example of a poset $P$ that is a regular suborder of $Q$ such that $Q$ is $\omega_2$-strategically closed, but the quotient forcing $Q/P$ fails to be $\omega_1$-strategically closed? To ...
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82 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
4
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2answers
51 views

countably closed forcing cannot add a branch to a $\aleph_2$-tree if $\neg\mathsf{CH}$

I'm reading this survey. In it the author states the following result (fact 5.3) which is attributed to Silver: If $2^{\aleph_0}>\aleph_1$, countably closed forcing cannot add a new branch to ...
3
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1answer
61 views

Truth values in Boolean valued models

In Devlin's "The Joy of Sets" the author introduces the Boolean valued model $V^{{\mathcal B}}$ based on a given Boolean algebra ${\mathcal B}$ and describes how to assign a truth value ...
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1answer
33 views

Do Different Generic Filters Give Different Generic Extensions?

Let $\mathbb{P}$ be a forcing. If $G \subseteq \mathbb{P}$ and $H \subseteq \mathbb{P}$ are two $\mathbb{P}$-generic filters over $V$ and $G \neq H$, does this imply that $M[G] \neq M[H]$. If $G$ ...
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163 views

About proving that the Continuum Hypothesis is independent of ZFC

In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the ...
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73 views

Choice in a Particular HOD Type Model

Let $V \models ZFC$. Let $P$ be a forcing and $G \subseteq P$ be generic over $V$. Let $x \in V[G]$. Let $M$ be the class of set that are hereditarily definable (in $V[G]$) using as parameters $x$ ...
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20 views

Intermediate Extensions and Coding Sets by Ordinals [duplicate]

Lemma 15.43 in Jech's "Set Theory" states that if $V \subseteq M \subseteq V[G]$ where $G \subseteq \mathbb{P}$ is some $V$-generic filter and $M$ is a transitive models of ZFC, then $M = V[D \cap G]$ ...
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1answer
101 views

A question regarding the status of CH in the Gitik model

Consider models of ZF+"Every uncountable cardinal is singular" (eg. Moti Gitik: "All uncountable cardinals can be singular", Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be ...
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1answer
41 views

Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
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57 views

Quotient Forcing in Iterations

I am trying to understand a proof of a lemma used to prove a preservation theorem for $^\omega \omega$-bounding for countable support iterations. In that quotient forcing is used to get a certain ...
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1answer
37 views

Defining forcing relation in base transitive model $M$

In page 177 of Set Theory for the Working Mathmatician, on chapter forcing it says: Theorem 9.2.7 For every formula $\varphi(x_1,..., x_n)$ of set theory there exists another formula ...
2
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1answer
75 views

Viewing forcing as a result about countable transitive models

I think of forcing in the following context: Truth and Definability Lemmas. So forcing is a schema in the meta theory. Now in his Set Theory book (the first edition), Kunen claims that setting up the ...
2
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1answer
71 views

Can set-theoretic forcing exist without law of excluded middle?

Of course the law of excluded middle is accepted by almost every mathematician except a few constructivists, but then I was wondering if set-theoretic forcing can exist without law of excluded middle. ...
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68 views

P-generic filter [closed]

Let $M$ be a countable transitive model of $ZF$ and let $P\in M$ be a partial order then how can we see If $P$ is non atomic partial order and $G$ is a P-generic filter over $M$, then $G\notin M$. ...
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2answers
28 views

Boolean-valued model and the use of generic ultrafilter in ZFC

So I asked the question about generic filter; but I was also reading http://math.mit.edu/~tchow/forcing.pdf which is a forcing (in ZFC) guide for dummies. Then I was struck with the part where it ...
3
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2answers
61 views

How generic ultrafilter is used in forcing

So I just learned what ultrafilter is and generic filter is. As all maths are for beginners, it just looks like pure concepts, and I don't see how they are going to be applied in forcing. I am looking ...
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236 views

Does the Laver real determine the generic filter?

Let us concern the Laver forcing $ \mathbb{L} $. Let $ G $ be $ \mathbb{L} $-generic over a c.t.m. $ M $ for ZFC. Let $$ x_G := \bigcup \{ \operatorname{stem}(p) : p \in G \} $$ be the Laver real ...
3
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1answer
56 views

Absoluteness of $\mathbb{P}$-names

In Kunen's book he says that $\mathbb{P}$-names are absolute for a transitive models of ZFC using a theorem to the effect that functions defined by recursion are absolute, i.e; Let $R$,$A$,$G$ be ...
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1answer
94 views

Inaccessible cardinals

Let $M[G]$ be the full Solovay model, and let HOD be the model of hereditarily ordinal definable sets in $M[G]$. Is it possible for HOD not to have an inaccessible cardinal? Does HOD satisfy GCH?
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43 views

Describing two-step iteration in terms of complete Boolean algebras.

Suppose $B$ is a complete Boolean and let $\dot{C}\in V^B$ be such that $$\|\dot{C}\text{is a complete Boolean algebra}\|_B=1.$$ Let us consider all $\dot{c}\in V^B$ such that ...
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48 views

Full Theory of Structure of Set Theory and Generic Extensions

Let $V$ be a model of $ZFC$. Let $M \in V$ be (a possibly not transitive) countable model of large fragments of $ZFC$ (for example countable substructures of large $H_\Theta$). If $M$ models enough of ...
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170 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...
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1answer
103 views

A Question Regarding Forcing in Gödel's Constructible Universe in Infinitary Logics

In his answer to the MathOverflow question Gödel's Constructible Universe in Infinitary Logics, Prof. Hamkins gives a very interesting answer and proof to user46667's second question: (2) What is ...
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120 views

Levy collapse gone bad

Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following? (1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, ...
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173 views

About generically Knaster property

We say that a poset $\mathbb{P}$ is absolutely Knaster if, for every $ccc$ poset $\mathbb{Q}$, $1 \Vdash_{\mathbb{Q}} \text{``$\mathbb{P}$ is Kanster''}$. In general, we say that a poset $\mathbb{P}$ ...
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1answer
65 views

All models of $\mathsf{ZFC}$ between $V$ and $V[G]$ are generic extensions of $V$

I'm reading the proof of lemma 15.43 of Jech's Set Theory: Let $G$ be generic on a complete Boolean algebra $B$. If $M$ is a model of $\mathsf{ZFC}$ such that $V\subset M\subset V[G]$, then there ...
8
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2answers
146 views

$\kappa$-c.c. vs. $\kappa$-Knaster

For an infinite cardinal $\kappa$ and a partial order $\mathbb{P}$, we say: (a) $\mathbb{P}$ has the $\kappa$ chain condition ($\kappa$-c.c.) iff there is no subset of $\mathbb{P}$ of size $\kappa$ ...
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1answer
124 views

Consistency strength of Turing measurability

This is probably well-known to recursion theorists, but as google didn't help me, I'll ask it here. Convention: All sets of reals in the following discussion are assumed to be closed under Turing ...
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1answer
67 views

Difference between $V^P$ and $V[G]$

This may be a basic question. I am studying forcing at Kunen's book. However, in several other papers that I am reading, they use that something is true in $V^P$ instead of $V[G]$. I know that if ...