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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Countable closure of quotient forcing

Let us say that a partial order is "countably closed with infima" if every descending $\omega$-sequence has an infimum. Suppose $P$ and $Q$ are posets that are countably closed with infima, ...
Monroe Eskew's user avatar
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10 votes
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When are two forcing posets "the same"?

Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a ...
new account's user avatar
7 votes
1 answer
316 views

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
9 votes
1 answer
586 views

Small Violations of Choice: Can we force AC without collapsing the cardinalities of ordinals?

We say that a model $M$ of $\mathsf{ZF}$ satisfies Small Violations of Choice ($\mathsf{SVC}$) if all (any) of the following apply: There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, ...
Calliope Ryan-Smith's user avatar
4 votes
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Some questions on a paper of Gerald Sacks

I've been reading Sacks' Countable admissible ordinals and hyperdegrees as I'm interested in Theorem 5.3 of the paper: Let $M$ be a countable standard model of $\mathsf{ZF}$ and $V=L$. Suppose $\...
Lorenzo's user avatar
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Uniformization of almost disjoint families

Suppose $\mathcal{F} \subseteq \mathcal{P} (\omega) $ is an almost disjoint family and $\aleph_0 < \vert \mathcal{F} \vert = \kappa < 2^{\aleph_0} $. Is it consistent that for some such cardinal ...
Matteo Casarosa's user avatar
2 votes
0 answers
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When is the ground model $[\kappa]^\lambda$ cofinal in $[\kappa]^\lambda$ in a forcing extension?

Suppose that $\lambda\leq\kappa$ are infinite cardinals. Say that a notion of forcing $\mathbb{P}$ is $[\kappa]^\lambda$-bounding if, whenever $G\subseteq\mathbb{P}$ is $V$-generic, $$V[G]\vDash(\...
Calliope Ryan-Smith's user avatar
4 votes
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A Sacks-like model in which $2^{\aleph_0} = \aleph_3$

This question is a follow-up of this one. In the previous question I asked if it was known the consistency of $\operatorname{cof}(\mathcal{L})=\aleph_1 + 2^{\aleph_0}=\aleph_3$, and the answer in that ...
Lorenzo's user avatar
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9 votes
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Extending Namba forcing to arbitrary lengths

Namba forcing is stationary-preserving and forces $cf(\omega_2^{\mathbf{V}}) = \omega$. Ronald Jensen used $\mathcal{L}$-forcing to iterate Namba posets in order to solve the extended Namba problem: ...
Zoorado's user avatar
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8 votes
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Forcing a unique $\Delta_3^1$ generic real

I know Jensen developed a forcing notions in $L$ that adds a unique, minimal and $\Delta_3^1$ $L$-generic real. In his paper Definable sets of minimal degree he says that Solovay had already shown the ...
Lorenzo's user avatar
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2 votes
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Arithmetically-hyperimmune-free degrees are comeager

I think proposition XIII.1.22 of Odifreddi is false but I wanted to check I wasn't being dumb. Here's the claim. Definition: The set$^1$ $A$ is arithmetically-hyperimmune-free if every function $f$ ...
Peter Gerdes's user avatar
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4 votes
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Given an elementary embedding $j: M\to N$ and a strong master condition $q\in N$, how are we able to construct a generic over $M$ from $M$?

Suppose we have an elementary embedding $j: M\to N$, a forcing notion $\mathbb{P}\in M$, and a strong master condition $q\in j(\mathbb{P})$. A strong master condition for $j$ and $\mathbb{P}$ is a ...
Connor W's user avatar
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Are the completeness Games $G_{\lambda+1}(P)$ and $G_{\lambda^+}(P)$ equivalent for INC?

The completeness game $G_{\gamma}(P)$ for a partial order $P$ has players COM and INC play alternatingly and descendingly elements of $P$ with player INC playing first and player COM playing at limit ...
Hannes Jakob's user avatar
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Where can I find information about this concept of 'dual ideals'?

I have come across the following notion of (what I am calling) dual ideals, and I am looking for any work in which this notion has been considered, and particularly anything about transferring ...
Calliope Ryan-Smith's user avatar
2 votes
1 answer
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Closure properties of elementary embeddings resulting from generic iterations

In the context of generic embeddings, we fix a set $Z$, an ideal $I$ on $Z$, and $G$ a generic ultrafilter on $\mathcal{P}(Z) / I$. In $V[G]$, we can define the generic ultrapower $Ult(V, G)$ and an ...
Zoorado's user avatar
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2 votes
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Question related to number of distinct forcing extensions of a countable model

A bit of context: the below question is motivated by roughly the following scenario: we have some countable model $\mathcal{M}$, and want to “count” the number of functions/sets $f$ such that $\...
Oliver Korten's user avatar
6 votes
1 answer
235 views

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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3 votes
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Hereditarily Lindelöf spaces with density continuum

Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof ...
GAW's user avatar
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12 votes
2 answers
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Amoeba collapse

Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V_\kappa$ is a partial order and $p \in P$. We define the ...
Monroe Eskew's user avatar
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5 votes
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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 ...
Corey Bacal Switzer's user avatar
4 votes
1 answer
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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 ...
Matteo Casarosa's user avatar
17 votes
3 answers
910 views

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. ...
Dmytro Taranovsky's user avatar
17 votes
5 answers
1k views

Strategic vs. tactical closure

The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when ...
Monroe Eskew's user avatar
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6 votes
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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 ...
Asaf Karagila's user avatar
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5 votes
1 answer
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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-...
Matteo Casarosa's user avatar
5 votes
1 answer
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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}...
Noah Schweber's user avatar
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Is this equivalent to (some version of) Hechler forcing?

Let $\omega^{<\omega}$ be the set of finite strings of naturals, and let $\omega^{<\omega}_{\not=\emptyset}$ be the set of nonempty finite strings of naturals. Consider the following forcing ...
Noah Schweber's user avatar
12 votes
1 answer
409 views

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 ...
Lorenzo's user avatar
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73 votes
3 answers
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Czelakowski's claimed proof of the Twin Prime Conjecture

It seems like the article "The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I)" by Janusz Czelakowski ...
Glycerius's user avatar
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7 votes
1 answer
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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
5 votes
0 answers
121 views

$2^{|V|}$ 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 ...
Dmytro Taranovsky's user avatar
3 votes
0 answers
100 views

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 ...
Zuhair Al-Johar's user avatar
6 votes
1 answer
319 views

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 ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
99 views

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 $G$ ...
Clement Yung's user avatar
2 votes
0 answers
95 views

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?
XL _At_Here_There's user avatar
2 votes
0 answers
60 views

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_{\...
Zuhair Al-Johar's user avatar
5 votes
1 answer
139 views

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 $\...
Monroe Eskew's user avatar
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4 votes
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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 $...
Lorenzo's user avatar
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7 votes
2 answers
690 views

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 ...
Lorenzo's user avatar
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7 votes
1 answer
199 views

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 ...
Andreas Lietz's user avatar
7 votes
1 answer
180 views

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
  • 327
10 votes
3 answers
830 views

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 ...
Guest's user avatar
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4 votes
1 answer
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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 ...
Taras Banakh's user avatar
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3 votes
0 answers
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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^\...
Lorenzo's user avatar
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2 votes
0 answers
134 views

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 $...
Lorenzo's user avatar
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7 votes
1 answer
208 views

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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5 votes
0 answers
189 views

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 ...
Hannes Jakob's user avatar
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4 votes
0 answers
125 views

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 ...
Lorenzo's user avatar
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6 votes
1 answer
174 views

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{...
James Hanson's user avatar
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3 votes
1 answer
113 views

$\mathtt{PSP}$ 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 ...
Lorenzo's user avatar
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