Questions tagged [forcing]

Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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Independence result where probabilistic intuition predicts the wrong answer?

In some of his writings, Paul Cohen gave an informal, motivational discussion about the word generic (as it is used in forcing). While very suggestive, the discussion leaves the meaning of the word ...
Timothy Chow's user avatar
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Characterizing "bounded" distributivity in terms of dense open sets

Prikry forcing is a well-known example of a forcing which adds an $\omega$-sequence to some cardinal $\kappa$, but does not add bounded subsets to $\kappa$. There are other examples of forcings which ...
Asaf Karagila's user avatar
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Countable Product of Class Forcing Notions

Is the following consistent? There are definable class forcing notions $\lbrace \mathbb{P}_{n}\rbrace_{n\in \omega}$ such that: The product of any finitly many of them preserves $\text{ZFC}$ and all ...
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Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
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Solovay's paper from AD+ that all sets are Ramsey

For my research I've been trying to locate Solovay's proof that AD+ implies that all sets are Ramsey (or completely Ramsey, depending on the terminology). I know that it uses Mathias forcing but I ...
Daniel Walker's user avatar
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Forcing axiom for a single poset

Let $FA_\kappa (\mathbb{P})$ be the claim that for every family $\mathscr{D}$ of dense sets in the poset $\mathbb{P}$ with $\vert \mathscr{D} \vert = \kappa $ there is a filter $G$ such that for all $...
Matteo Casarosa's user avatar
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Does Laver Forcing add an infinitely often equal real?

Given a model $V$ of set theory and an inner model $W \subseteq V$, a real $x \in V \cap \omega^\omega$ is infinitely often equal over $W$ if for each real $y \in \omega^\omega$ there are infinitely $...
Corey Bacal Switzer's user avatar
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On a theorem of Zhang Jinwen about models of arithmetic

In the paper ''A Nonstandard Model of Arithmetic Constructed by means of Forcing Method'', Zhang Jinwen states the following in his abstract: The first nonstandard model of arithmetic was given by ...
Mohammad Golshani's user avatar
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Tree Version of Hechler Forcing

In their recent (2009) paper Eventually Different Functions and Inaccessible Cardinals, Brendle and Löwe consider a 'tree version' of the Hechler forcing. This forcing $\mathbb{D}$ consists of ...
Justin Palumbo's user avatar
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Forcing out of L[U] when we have a precipitous ideal in V

The following theorem of Jech, Magidor, Mitchell and Prikry is well-known. Theorem. (1) If $\kappa$ is a regular cardinal that carries a precipitous ideal, then $\kappa$ is a measurable cardinal in an ...
Toby Meadows's user avatar
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Non-homogeneous forcing and HOD

Is there a separative forcing notion $\mathbb{P}$ such that: 1) For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to any homogeneous forcing notion, ...
Mohammad Golshani's user avatar
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Iteration of random reals

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\...
Carlos's user avatar
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How Random is Cohen?

Suppose that $V$ is a universe of $\sf ZFC$, and $c$ is a Cohen generic real over $V$. Is it possible that $c$ is also generic in other senses? I know that it can't be random or Sacks or whatnot ...
Asaf Karagila's user avatar
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Consistency results using nonstandard models

Are there any consistency results in set theory (or in mathematics) that can be proved using nonstandard models of ZFC but not using transitive models of ZFC?
Mohammad Golshani's user avatar
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subalgebra of a simple forcing

Let $\alpha > 0$ be any ordinal. Consider the forcing $Add(\omega,\alpha) * Add(\omega_1,1)$. Let $B$ be its boolean completion. Let $\dot{X}$ be the canonical $B$-name for the generic subset of ...
Monroe Eskew's user avatar
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Suslin algebras

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...
Monroe Eskew's user avatar
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Proof (or reference) about the cc-ness of termspace forcing

Recall that for $P$ a forcing order and $\dot{Q}$ a $P$-name for a forcing order, the termspace forcing $T(P,\dot{Q})$ consists of minimal-rank names for elements of $\dot{Q}$, ordered by $\dot{q}'\...
Hannes Jakob's user avatar
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Preservation of projective stationarity

A set $S \subseteq [\kappa]^\omega$ is called projective stationary if for every stationary $A \subseteq \omega_1$, and every algebra $F : \kappa^{<\omega}\to\kappa$, there is $z\in S$ such that $z$...
Monroe Eskew's user avatar
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Dropping "generic" from the definition of forcing

Back when I was first learning about forcing and trying to understand the need to consider generic filters, I came up with the following question. Suppose we have a countable transitive model $M$. ...
Timothy Chow's user avatar
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Forcing the negation of CH without adding Cohen reals over L

Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?
Ohad Drucker's user avatar
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Forcing Extension of Countable Linearly Iterable Structures

Let $V$ satisfy there exists a measurable cardinal. Let $\kappa$ be a measurable cardinal and $U$ be the normal measure on $\kappa$ witnessing this. Let $\mathbb{P}$ be a forcing of size less than $\...
William's user avatar
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Generic filters of inverse limits

Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper Fragile measurability, Journal of Symbolic Logic 59 (1994) 262-282. There, $\mathbb P_\...
Carlos's user avatar
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Can the Cohen forcing collapse cardinals?

Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...
Yair Hayut's user avatar
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GCH+ Kurepa Families

I have a couple of questions about known theorems for GCH+Kurepa families. Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ ...
Ioannis Souldatos's user avatar
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Symmetric extensions and class forcing

Suppose $V\models ZFC$ and $P\in V$ is a poset of forcing conditions. It is a basic theorem in forcing that $V[G]\models ZFC$ for any generic extension by a $V$-generic filter $G$. It is also known ...
Asaf Karagila's user avatar
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How hard is it to get "absolutely" no amorphous sets?

A beautiful and surprising (to me at least) result around the axiom of choice is that, while full $\mathsf{AC}$ is preserved by forcing, a model of $\mathsf{ZF}$ + "There are no amorphous sets&...
Noah Schweber's user avatar
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Class forcing as set forcing followed by truncation

My question arises from thinking about how we can obtain class forcing extensions from truncations of set forcing extensions in the presence of a (strongly) inaccessible cardinal in the following ...
Alexander Block's user avatar
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1 answer
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Pseudo-Prikry sequences vs Prikry sequences

Definition: Let $V\subseteq W$ be two transitive models of $ZFC$. A pseudo-Prikry sequence, $s$, at a cardinal $\kappa$ for $(V, W)$ is an $\omega$-sequence, cofinal at $\kappa$ such that for every ...
Yair Hayut's user avatar
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Characterization of $\kappa$-closed forcings

For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like. For a partial order $\mathbb{P}$, the ...
Monroe Eskew's user avatar
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Reals added after Cohen forcing II

This is related to my previous question "Reals added after Cohen forcing" Reals added after Cohen forcing The fact is that what I had in mind by $\lambda-$many Cohen reals was different from what was ...
Mohammad Golshani's user avatar
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1 answer
673 views

complete embeddings of boolean algebras and preservation of stationarity

Define a complete embedding of Boolean algebra as an homomorphism of Boolean algebras which preserves also the sup and inf operations. Notice that if $\mathbb{B}$ and $\mathbb{D}$ are complete boolean ...
matteo viale's user avatar
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Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets

We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$ Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...
Dominic van der Zypen's user avatar
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Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give

Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing ...
Mohammad Golshani's user avatar
7 votes
1 answer
226 views

Restrictions of the Stationary Tower forcing providing various critical points

Following Larson's "The Stationary Tower", let $\mathbb{P}_{<\delta}$ be the full stationary tower on $\delta$, and for a stationary $S\subset \mathcal{P}_\delta(V_\delta)$, $\mathbb{P}_{<\delta}...
Ur Ya'ar's user avatar
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Silver forcing and Cichoń's diagram

Recall that the Silver forcing $\mathbb{P}$ is defined as the set of all partial functions $p\in 2^{\le\omega}$ such that $\omega\setminus dom(p)$ is infinite. As usual, $p\le_\mathbb{P}q$ if $p$ ...
Damian Sobota's user avatar
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2 answers
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Set theoretic forcing, large cardinals and probabilistic methods

This question is motivated by the work of Ajtai "The complexity of the pigeonhole principle" and similar works. In this paper, the author proves that $PHP_n$, the pigeonhole principle for $...
Mohammad Golshani's user avatar
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1 answer
286 views

Extending Sacks forcing

Sacks forcing allows us to build a model $V[G]$, such that there is no "intermediate model" between $V$ and $V[G]$, meaning if $V \subseteq W \subseteq V[G]$ is a model of ZFC then either $W = V$ or $...
Alon Navon's user avatar
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Intermediate submodels which do not satisfy AC

The following is known: Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of $...
Mohammad Golshani's user avatar
7 votes
1 answer
408 views

A special c.c.c forcing notion and adding minimal generic reals

This question is related to my question "Forcing with c.c.c forcing notions, Cohen reals and Random reals". A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds ...
Mohammad Golshani's user avatar
7 votes
1 answer
295 views

Consistency of Rado's conjecture with not CH

Rado's conjecture (one of many equivalent formulations) states: any non-special tree has a non-special subtree of cardinality $\aleph_1$. "Special" means a tree can be decomposed into countably many ...
Jing Zhang's user avatar
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How many closed measure zero sets are needed to cover the real line, really?

This is a refinement of an earlier question. This question assumes familiarity with combinatorial cardinal characteristics of the continuum. For the reader's convenience, I reproduce below the ...
Boaz Tsaban's user avatar
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On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement: For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with $|\mathfrak{B}|=\kappa$...
Rahman. M's user avatar
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1 answer
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On the definition of $\alpha$-proper poset

I am reading Uri Abraham's chapter on Proper Forcing in the Handbook of Set Theory and I have a quite trivial question on the definition of $\alpha$-proper forcing. Since there are many equivalent ...
Carlos's user avatar
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Understanding descending intersections of generic extensions

Let $B_{0}\supseteq B_{1}\supseteq\dots\supseteq B_{\alpha}\supseteq\dots\,\,\left(\alpha<\kappa\right)$ be a descending sequence of complete Boolean algebras, $B_{\kappa}:=\bigcap_{\alpha<\...
Ur Ya'ar's user avatar
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1 answer
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Construction of a model of $ZFC+\neg Con(ZFC)$

By Gödel's second incompleteness theorem, the following assertion is true in ZFC: $$ Con(ZFC)\rightarrow Con(ZFC+\neg Con(ZFC)) $$ Considering the completeness theorem, this assertion is equivalent to ...
Ka Ho's user avatar
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What is forcing indescribability?

Suppose $m, n\in\omega$ and $\kappa$ is a cardinal. Then $\kappa$ is $\Pi^m_n$-indescribable if every $\Pi^m_n$-sentence true about $\kappa$ is true about some $\lambda<\kappa$; formally, if for ...
Noah Schweber's user avatar
7 votes
1 answer
270 views

Modifying a Cohen generic

Let $M$ be a transitive model of ZFC (set- or class-sized) and let $\kappa\in M$ be a regular cardinal (in $V$). Let $G$ be $M$-generic for $\operatorname{Add}(\kappa,1)$. Now suppose that there is an ...
Miha Habič's user avatar
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Consistency of Weak Diamond with a Weak Version of Martin's Axiom

If $S \subset \omega_1$ is stationary, then the weak diamond principle $\Phi(S)$ states that for any $F: 2^{<\omega_1} \to 2$, there is a $g: \omega_1 \to 2$ such that for all $f: \omega_1 \to 2$, ...
Danielle Ulrich's user avatar
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1 answer
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Embeddings of forcing notions - preserve properness?

Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M $ ...
Justus87's user avatar
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1 answer
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Is there a monster behind the trees?

‎First Fix the following notation:‎ ‎ $‎‎\forall ‎\kappa‎\in Card~~~Tp(‎\kappa‎):="‎\kappa‎~has~tree~property"$‎ ‎‎‎‎ The large cardinals as "monsters of heaven" live everywhere in the land of ...
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