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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random real forcing, independent real
Assume all independent reals that are added by random real forcing. Take enumeration of each independent real. Is the family of all enumerations dominating?
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How to make countably closed forcing "nice" without choice
When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice:
If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In ...
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Comparing Mathias forcing notions relative to various filters
Let $\mathcal F$ be a (non-principle, non trivial, ...) filter on $\omega$. The Mathias Forcing relative to $\mathcal F$ is the forcing notion $\mathbb M(\mathcal F)$ consisting of pairs $(s, X)$ with ...
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Ramsey-like property with order condition
I wonder if there are regular cardinals $\lambda$ and $\kappa$ such that $\kappa < \lambda \leq 2^\kappa$ and such that, consistently, the following holds:
Let $c: [\lambda]^2 \to \kappa$ be such ...
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How "much" does (Grigorieff) forcing destroy an ultrafilter?
Introduction. I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction ...
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Minimum transitive models and V=L
Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?
You may assume that ZFC has transitive models. ...
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Is there a more modern account of the main results of "Adding Dependent Choice" by D. Pincus?
I would like to read Pincus' article Adding dependent choice, where he proves, among other things, the consistency of the theory $\mathsf{ZF+DC+O+\neg AC}$, where $\mathsf{DC}$ stands for Dependent ...
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Comparing generic versions of $\mathbb{R}$
This question was previously asked and bountied at MSE, unsuccessfully.
I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and ...
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Highly improper forcings
The following question comes from a typo in an old notebook of mine (I changed what I was calling my forcing notion partway through writing the definition of properness):
Say that a forcing $\mathbb{P}...
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Forcing Martin's Axiom without cardinal arithmetic
We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which ...
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Cofinal well-founded subset in mod finite order
The mod finite order on ${}^\omega \omega$ is defined as $f \leq^\ast g$ if and only if $f(n) \leq g(n)$ except for finitely many $n \in \omega$. My question is: can we always extract a cofinal well-...
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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&...
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Iterating Neeman's forcing
In the paper, "Forcing with sequences of models of two types," (MR3201836), Neeman claims that, using a supercompact and a weakly compact above, one can force with his pure side conditions ...
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$2^{|V|}$ class cardinalities without global choice
Is it consistent with Morse-Kelley set theory without global choice (but with choice for sets) that there are $2^{|V|}$ proper classes of different cardinalities?
Alternative question: Is it ...
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Examples of the use of forcing to build up models of stronger theories?
I'm very new to the subject of forcing. I always got the impression that with forcing we begin with say a model $M$ of a theory $\sf T+I$ and produce another model $M[G]$ that is also a model of $\sf ...
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Is there a proof of independence of AC from Z that is done in Z?
The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing and constructibility, and these don't beg any consistency strength more than that of $\sf ZF$.
Is there a known similar proof ...
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Unbounded subset of $\omega$ in $V[G]$ has an unbounded subset in $V$?
This question is similar to a question I asked last year, but I'm not asking for the same thing.
Let $S$ be a set of ordinals, and consider the Levy collapse $\operatorname{Col}(\omega,S)$. Let $G$ ...
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Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$?
Is it consistent that there exists an inaccessible cardinal $\lambda$ and a forcing extension $V[G]$ so that $$V[G]\models\text{There is some non-trivial elementary embedding $j:V_{\lambda}^{V[G]}\...
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Forcing the existence of a weakly inaccessible cardinal in some strong set theory
Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $...
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Any connection between extension of algebraic structure and forcing of set theory?
Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
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Can we use forcing to adjoin this set to a model of ZF+j+$\alpha$?
Let $M$ be a countable transitive model of $\sf ZF + j +\alpha$, where $j:V_{\alpha+1} \to V_\alpha$ is an external [not used in separation and replacement] bijection such that for any $S \in V_{\...
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Destroying the P-filter-property
It is known that if a forcing notion is proper, then every P-filter will generate a P-filter in the generic extension (see, e.g., Shelah, Proper and Improper Forcing, VI.5)
On the other hand, if we ...
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Relationship between fragments of the axiom of choice and the dependent choice principles
The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...
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Preservation of stationary sets by Mitchell forcing quotients
It is well-known that Mitchell's forcing $\mathbb M$ for the tree property at $\omega_2$, which turns a weakly compact $\kappa$ into $\omega_2$ while adding many reals, is a projection of adding $\...
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Consistency of a strange (choice-wise) set of reals, pt. 2
This is a follow-up on this question. Consider a set $X\subseteq \mathbb{R}$ such that
$X$ is not separable wrt its subspace topology
Every countable family of non-empty pairwise disjoint subsets of $...
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Consistency of a strange (choice-wise) set of reals
Consider a set $X\subseteq \mathbb{R}$ such that
$X$ is not separable wrt its subspace topology
For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$
In a ...
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For which class of forcings does the "name dichotomy" hold?
Let $\mathbb P$ be a forcing that does not collapse $\omega_1$, $\theta$ sufficiently large and regular and $X\prec H_\theta$ a countable elementary substructure with $\mathbb P\in X$ as well as $p\in ...
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How badly can the GCH fail globally?
It's known that we can have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms.
My question is whether we can have global ...
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Why can we assume a ctm of ZFC exists in forcing
Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...
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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<\...
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Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?
Roughly speaking, say that a logic $\mathcal{L}$ is self-equivalence-defining (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ ...
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Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?
Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
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Silver-like forcing preserves p-points (reference request)
A Silver forcing "below $2^n$" is defined e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It ...
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The set of ground model reals arbitrarily close to a new real in the forcing extension
Consider a forcing notion $\mathbb{P}$, a condition $p\in\mathbb{P}$, a $\mathbb{P}$-name $\dot{r}$ and a formula (with ground model parameters) $\varphi(x)$ such that $$p \Vdash \dot{r} \in \omega^\...
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A property of Levy collapse forcing
Consider the following nice property for a forcing notion $\mathbb{P}$ (in a transitive model $M$ of $\mathtt{ZFC}$):
Let $G_1,G_2$ $\mathbb{P}$-generic over $M$ and $M[G_1]$ respectively. Then, if $...
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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 $z$...
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The (non-)absoluteness of second-order elementary equivalence
Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...
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Does this proof by Shelah use any "hidden assumptions"?
Recall that the approachability ideal for $\kappa$, denoted $I[\kappa]$, consists of all sets $A\subseteq\kappa$ such that there is a sequence $\overline{a}=(a_{\alpha})_{\alpha\in\kappa}$ of bounded ...
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Necessary and sufficient conditions for a forcing to add a Cohen real
Are there some necessary and sufficient conditions for a forcing notion to add a Cohen real in the generic extension?
In other words, given a non-atomic forcing p.o. $\mathbb{P}$, what are the ...
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c.c.c forcing notions and adding minimal generic reals
Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any non-...
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How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?
Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{...
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$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$
Consider the sentence $\mathtt{PSP}_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set".
A priori this sentence is weaker than the ...
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When will the real numbers be Borel?
In set theory Borel sets are important, but we don't actually care about the sets. We can about the Borel codes. Namely, the algorithm to generate a given Borel set starting with the basic open sets (...
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An uncountable structure with unusual "relatively-computable shadow"
Below, all structures are infinite and in a finite language. Given a structure $\mathcal{A}$ with domain $\omega$, we conflate $\mathcal{A}$ with some reasonable encoding of its atomic diagram for ...
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Are there interesting examples of theorems proved using ‘height’ extensions?
It's well known that forcing is more than a tool for proving independence: We can prove theorems and formulate axioms in theories like $\mathsf{ZFC}$ by moving to forcing extensions (e.g. $\mathfrak{p}...
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Forcings that preserve $\mathtt{PSP}$
By $\mathtt{PSP}$ I mean the statement "every subset of the reals has the perfect set property, i.e. either is countable or it contains an homeomorphic copy of the Cantor space $2^\omega$".
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Generalized models of set theory
The forcing method can be viewed as building a Boolean-valued model of set theory. Some generalizations include Heyting algebra/sheaf/lattice-valued model. However, it seems these generalizations are ...
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Philosophy of forcing and ctm
I asked a similar question on SE before and received an answer. Not completely convinced, I decided to ask it here with some modifications. Note: I understand how forcing works and how it proves ...
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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 ...
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Does Laver forcing add cofinal branches to $\omega_1$-trees?
Given an $\omega_1$-tree $T$ in the ground model, can Laver forcing add a cofinal branch to $T$? Assume GCH in the ground model.
Definitions:
An $\omega_1$-tree is a well-founded tree of height $\...
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