Questions tagged [forcing]
Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
843
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preservation of forcing rigidity in iterations
Say that a partial order $P$ is forcing-rigid in a model $V$ if whenever $G \subseteq P$ is generic over $V$, then in $V[G]$, $G$ is the only filter which is $P$-generic over $V$. This implies there ...
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Is the axiom $\Diamond\Box\varphi\to\Box\Diamond\varphi$ in c.c.c. forcing potentialism equivalent to the productivity of c.c.c. forcing?
This question arose in connection with a lecture series on
Potentialism
that I have just completed here in Hejnice in the Czech Republic at
the Winter School 2018 (see
Slides). Several of us discussed ...
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9
votes
1
answer
538
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What are the typical forcings to shoot a club through a stationary subset of $[\lambda]^\omega$
Let $\lambda\geq \omega_2$ be a regular cardinal and $S\subset[\lambda]^\omega$ be a stationary set. I'm looking for a property of $S$, say "shootable", such that there exists a forcing extension ...
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1
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Ways to add Aronszajn trees which are neither Souslin nor special
By an Aronszajn tree, I mean a tree of height $\omega_1$ with countable levels and no branch. Such a tree is Souslin if it has no uncountable antichains and special if it can be written as the ...
- 3,138
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1
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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}...
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1
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Iterated forcing and the super tree property at $\omega_2$
It is a theorem of Baumgartner and Laver that iterating Sacks forcings of weakly compact length gives rise to the tree property at $\omega_2$. Natural questions (at least for me) are: do we get ...
- 3,138
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Query about iterated collapse forcing
I have heard it said that it is possible to use Woodin's iterated collapse forcing to prove that the theory $\textsf{ZF}$+"there exists a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow ...
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3
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1
answer
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Set forcing over transitive models of NBG
What is known about set forcing over (transitive) models of NBG?
Where can I read about it?
More specifically: Given some set-sized complete Boolean algebra $\mathbb P$ (or simply a poset) it does ...
- 1,655
6
votes
2
answers
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Categoricity of the complex field in the generic extensions
Let $V[G]$ be a generic extension of $V$ by adding a new Cohen real (or generally a generic extension which adds new reals and do not blow up the power of the continuum). Working in $V[G],$ we can ...
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Prikry forcing and Cohen generic
Let $\kappa$ be a measurable cardinal and let $\mathcal{U}$ be a normal measure on $\kappa$. Let $\mathbb{P}$ be the standard Prikry forcing using $\mathcal{U}$. Let $\mathbb{Q} = \text{Add}(\kappa, 1)...
11
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Forcing, cuts, and Dedekind-finite cardinalities
Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...
CommunityBot
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1
answer
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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 $p$ ...
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Interesting examples of countable support iteration of ccc forcings
I am looking for examples in the literature of countable support iterations of ccc (particularly $\sigma$-centered) forcings, possibly with some emphasis on iterations that avoid adding Cohen reals.
...
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A forcing which can build weird models of $\neg$ADS
There is a class of forcing notions I've been playing around with recently. They have a couple nice properties, and all have the same theme, but I've found them difficult to analyze beyond the basics. ...
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5
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1
answer
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Can Woodin's fast function forcing kill Shelah cardinals?
Definition 1. An uncountable cardinal $\kappa$ is Shelah if for every function $f:\kappa\rightarrow \kappa$ there exists a transitive class $M$ and a non-trivial elementary embedding $j:V\rightarrow M$...
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2
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Does fast function forcing really have $\kappa$-Knaster property?
I ran into a claim concerning Woodin's fast function forcing in the following paper of Apter and Cummings which sounds no right to me:
A. Apter, J. Cummings, Blowing up the power set of the least ...
CommunityBot
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1
answer
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views
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 $...
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PCF conjecture and fixed points of the $\aleph$-function
Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and ...
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342
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The failure of GCH al $\aleph_\omega$ by nice forcing
There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:
1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.
2) Woodin's ...
8
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0
answers
220
views
When can we force two frames to be homeomorphic?
Recall that if $M,N$ are two structures of the same type, then
$M$ is $\mathcal{L}_{\infty,\omega}$ elementarily equivalent to $N$ precisely when $M$ and $N$ are isomorphic in some forcing extension. ...
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4
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2
answers
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The $\delta$-approximation property for ground models
Definition 1 (Hamkins).
Suppose $V \subseteq W$ are transitive models of $\mathrm{ZFC}$ and $\delta$ is a cardinal in $W$.
$(V,W)$ has the $\delta$-cover property iff for each
$A \in W$ with $A \...
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2
answers
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Can there be an almost-special not-fully-special Aronszajn tree?
Question. Can there be an Aronszajn tree $T$, such that no
c.c.c. forcing extension adds a cofinal branch to $T$, but there is an
$\omega_1$-preserving forcing extension adding a cofinal branch to
$T$?...
- 3,138
7
votes
1
answer
259
views
rigid collapse to $\aleph_1$
Suppose $\kappa$ is inaccessible (or more). Does there exist a $\kappa$-c.c. partial order $\mathbb P \subseteq V_\kappa$ that forces $\kappa = \aleph_1$, with the following property?-- Whenever $G \...
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3
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0
answers
93
views
What can the set of all intersections of a set and the class of all ground models look like?
Does there exist a model $V$ of $ZFC$ with the following property?
Suppose that $X$ is a set and $\mathcal{A}\subseteq P(X)$ is a collection of subsets such that $X\in\mathcal{A}$ and where $\mathcal{...
- 27.8k
7
votes
1
answer
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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 ...
- 3,138
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votes
2
answers
332
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Laver property, non-meager reals and cardinal characteristics
Let $V$ be a model of set theory with CH. Recall the following definitions.
A forcing $\mathbb{P}\in V$ has the Laver property if for any $\mathbb{P}$-generic filter $G$ over $V$, functions $f\in\...
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4
answers
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Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?
Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? ...
9
votes
2
answers
527
views
Reals which must, can't or might be added by forcing
Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the ...
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5
votes
1
answer
199
views
Consistency of Strong reflection principle with the existence of a Suslin tree
In Woodin's book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal" Remark 2.55 (5), it states SRP by Todorcevic (defined below) is consistent with the existence of a Suslin tree (...
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1
answer
221
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Restricting extenders to a ground model
Let V=W[g], where g is P-generic over W for some poset P in W. Let F be a V-extender with critical point κ such that P ∈ VκW. If the support of F is sufficiently closed, say strength(F)=...
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2
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views
Examples of stationary set preserving forcings that are not semiproper?
A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of course proper) notions ...
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3
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Preserving $\omega_1$ is Inaccessible to the reals
$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$.
The question is if $\omega_1$ is inaccessible to the reals in $V$ and $\...
- 1,798
6
votes
2
answers
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views
Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness
It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...
- 1,798
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votes
3
answers
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views
When can we reach a real by forcing?
I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such ...
- 1,798
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2
answers
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Iterated forcing and CH
I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert \dot{Q}_\alpha\rvert\...
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1
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When is a direct limit of nice models of set theory is again a model of set theory?
It is often useful to allow taking direct limits in set theory. This happens all the time when taking about ultrapowers.
But let's not limit ourselves to ultrapowers. Suppose that we have a sequence ...
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Characterization of Cohen reals
The following is a well-know fact:
Theorem The real $r$ is Cohen over $V$ iff if it does not belong to any meager Borel
set coded in $V$.
Now suppose that $\kappa$ is an uncountable cardinal and ...
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2
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Paul Cohen on genesis of method of forcing and mathematical similarities
We have on record Paul Cohen's comments on being inspired by issues of formalizing algorithms in number theory (this needs to be verified as per comment) as well as related remarks on computability. ...
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0
answers
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What does the algebras of elementary embeddings look like in this forcing extension?
In this paper, Laver claims (on page 14 of the Arxiv version) that if there exists a rank-into-rank embedding, then in some upward Easton forcing extension, there are elementary embeddings $j,k:V_{\...
- 27.8k
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votes
1
answer
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views
moving up a consequence of PFA
The Proper Forcing Axiom (PFA) implies that every forcing which adds a subset of $\omega_1$ either adds a real or collapses $\omega_2$. Is it consistent that every forcing which adds a subset of $\...
8
votes
2
answers
448
views
Ultrafilters preserved by $\mathbb{P}$ but not by products?
Let $U\in V$ be an ultrafilter on $\omega$. We say $U$ is preserved under forcing with $\mathbb{P}$ if $\Vdash \forall x\subset \omega \ \exists Z\in U \ Z\subset x \vee Z\subset x^c$. In other words,...
- 13.9k
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1
answer
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Extending ground model ultrafilters
Work over a model of $\sf GCH$. Suppose that $\kappa$ is some regular cardinal, and $U$ is a uniform ultrafilter on $\kappa^+$.
Does $U$ have some canonical extension after forcing with $\...
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What are some kinds of models where DC holds?
There are a lot of ways to build a model where DC fails. However, all of them that I'm aware of involve adding at least a messy set of reals (or rather, taking a forcing extension and then passing to ...
- 72.3k
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0
answers
201
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Does Solovay's $\Sigma$-construction preserve "niceness"?
Suppose I have two forcing notions $\mathbb{P}$ and $\mathbb{Q}$, and a $\mathbb{Q}$-name $\nu$ for a real. Let $G$ be $\mathbb{P}$-generic, and suppose $r$ is a real in $M[G]$ such that for some $H$ ...
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2
votes
1
answer
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Class forcings and elementary embeddings
In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:
"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...
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2
answers
498
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Intermediate submodels without Boolean algebras
My question is motivated by the following well-known fact regarding intermediate submodels of generic extensions. I would like to know if it can be proven using posets without the need for Boolean ...
5
votes
2
answers
369
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Independent families of functions on $\omega$ of size continuum
In Hausdorff's article "Über zwei Sätze von G. Fichtenholz und L. Kantorovich''(1935) one can find the (simplified) proofs of the following two theorems:
1) There are continuum many essentially ...
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2
answers
222
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Miller real is not in the closure of sets under some conditions
Background
We can define Miller Forcing as the poset of nonempty perfect rational trees. That is, we define:
$p\subset 2^{<\omega}$ is a perfect tree iff it is closed downwards (for all $s, n$, ...
CommunityBot
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3
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235
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$(\kappa, \kappa, 2)$-saturated ideals?
Is it consistent to have a $(\kappa,\kappa,2)$-saturated ideal $I$ on $\kappa$ that is $\kappa$-complete and $\kappa$ is not weakly compact? Here $\kappa$ is inaccessible. An ideal is $(\kappa,\kappa, ...
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What is the precise relationship between forcing on a poset and the topos of double-negation sheaves on this poset?
I've seen various statements that the Boolean-valued models of ZFC occurring in model-theoretic forcing are "really" the topos of sheaves on an appropriate site, but never a fully precise statement. ...
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