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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About $\mathbb{P}$-Knaster and $\mathbb{P}$-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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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 ...
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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 ...
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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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In preparation forcing and large cardinal textbooks

Everybody in set theory refers to texts like Kunen, Jech and possibly Halbeisen's books as elementary references for forcing. Also Drake and Kanamori's books are well-known references for large ...
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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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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 ...
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115 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 ...
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53 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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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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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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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 ...
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64 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 ...
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68 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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47 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 ...
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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 ...
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230 views

Is there any category theoretic proof for independence of Continuum Hypothesis?

Both of set theory and category theory could be a foundation for mathematics. Many set theoretic arguments could be translated to a category theoretic argument and vice versa. Question: Is there ...
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31 views

Generic in Boolean-Valued-Models

Let $M$ be a transitive $\in$-interpretation of a extension $T$ of $ZF$ in $ZF$,and let $B$ such that $$T\vdash B\in M\wedge B\text{ is a complete Boolean algebra}$$ Then, using the fact that any set ...
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36 views

A filter $G$ on $P$ is generic iff whenever $W$ is a partition of $P$, $G \cap W \not = \emptyset$

I'm finally struggling a little bit with the notion of forcing, so I decided to do some of the exercises in Jech's book (2nd Edition) in order to sharpen a little bit my intuitions about the whole ...
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Introducing a new element to make a new model of set theory

Say we have a model of set theory $V$ and a partially ordered structure $\mathbb{P}$, and I want to talk about a $V$-generic filter $G$. A $V$-generic filter is a filter such that for every $D\in V$ ...
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57 views

How to make a large cardinal unique?

A general form of questions regarding large cardinals is the following: Let $A(x)$ be the formula asserting "$x$ is a large cardinal of type $A$" then is the following true? ...
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105 views

Consistency of restricted forms of Martin's Axiom with the negation of the Continuum Hypothesis

Consider $\mathsf{MA}(S)$, the forcing axiom for all ccc posets which preserve a Souslin tree $S$. is $\lnot \mathsf{CH}$ consistent with $\mathsf{ZFC}+\mathsf{MA}(S)$? Does there exist a model for ...
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How to force p<b?

Two cardinal characteristics (cardinals between $\aleph_1$ and $\mathfrak{c}$ are: $\mathfrak{b}$, the least size of an unbounded family in $\omega^{\omega}$ ordered under eventual domination ...
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79 views

Absoluteness of $\Sigma_2$ sentences in forcing

Let $M$ be a model of ZFC and $M\models \varphi$ such that $\varphi$ is a $\Sigma_2$ sentence in the language of set theory. Let $M[G]$ some forcing extension of $M$. Is $M[G] \models \varphi$? What ...
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Why adding a club of $\aleph_1$ collapses $\aleph_1$ to $\aleph_0$?

Let $\{S_n \mid n < \omega\}$ be a partition of $\aleph_1$ into countably many disjoint stationary subsets. Why adding a club of $\aleph_1$ to each $\aleph_1 \setminus S_n$ collapses $\aleph_1$ to ...
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37 views

Forcing $M[G] \models CH$

I have seen the proof of transitioning from a model in which CH holds to a model in which CH fails. However, how do we force the other direction? More explicitly, suppose that $M$ is a countable, ...
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If models of set theory can be construed as categories, can notions of forcing be construed as functors?

Consider the following passage from Blass and Scedrov's paper "Complete topoi representing models of set theory"(Annals of Pure and Applied Logic, vol. 57 (1992),PP. 1-26) where 'set theory' in this ...
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127 views

Miller's Construction, Partition Principle and Failure of Axiom of Choice

Partition Principle ($PP$) is the following statement: For all sets $a$, $b$ there is an injection $f:a\rightarrow b$ iff there is a surjection $g:b\rightarrow a$ It is known that $ZF\vdash ...
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Trouble understanding Jech's version of Easton's theorem

On page 232 of Jech's Set Theory (3rd Edition, 2003), we have the following statement of Easton's theorem. Theorem 15.18 (Easton). Let $M$ be a transitive model of ZFC and assume that the ...
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Cardinality of Countable Support Iteration of Proper Forcing

Suppose $\mathsf{CH}$ holds. Suppose $\mathbb{P}$ is a forcing such that $\mathsf{ZFC} \models \mathbb{P} \text{ is proper and } |\mathbb{P}| = 2^{\aleph_0}$. For instance, Sacks forcing is such an ...
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Proper Forcing and Sequence of Names for Reals.

I read something that seems to suggest the following is true: If $\mathbb{P}$ is a proper forcing, $|\mathbb{P}| = \aleph_1$, and $\mathsf{CH}$ holds, then there exists a sequence $\{(p_\xi, ...
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Why is ZF favoured over NBG

So why is ZF favoured over NBG? Is it historical, but I've read after Gödel's monograph was published, NBG was more prominent. Or is the reason that NBG gets the cold shoulder to do with forcing and ...
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Redundancy in definiton for forcing poset

The following definition appears in Kunen (2nd edition): For any sets $I,J$ and cardinal $\lambda$: $\text{Fn}_{\lambda}(I,J)$ is the set of all $p\in{[I\times{J}]}^{<\lambda}$ such that $p$ is ...
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What is the definition of the Feferman-Levy model?

Any (reference to) definition of Feferman-Levy model in set theory? I cannot find any... Though I know what is Levy collapse.
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137 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
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Existence of a generic ultrafilter over constructible universe

I am referring to the Jech's "Set Theory" book. He wants to show that, if $(P,<)$ is the notion of forcing defined as: $p \in P$ iff $p$ is a finite sequence of ordinals less than $\omega_{1}^{L}$ ...
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67 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
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Exercise on preservation of cardinals

I'm trying to solve exercise 12.1 in Prof. Monk's lectures on set theory which asks me to show that $Fn(\omega_1, 2, \omega_1)$ preserves cardinals larger than $\omega_2$. Now, the only method in the ...
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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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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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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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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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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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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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151 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 ...
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83 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." ...
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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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93 views

On the number of countable models of complete theories of models of ZFC [duplicate]

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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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 ...