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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An iteration of proper forcing without proper iterands

Famously, a countable support iteration with proper iterands is again proper. This is mostly stated as follows: Suppose $(\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha})_{\alpha<\lambda}$ is a ...
Hannes Jakob's user avatar
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7 votes
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A reference for forcing projections

The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
Miha Habič's user avatar
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11 votes
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Is ground model $\Sigma_n$ truth $\Sigma_n$-definable in every forcing extension?

It is well-known (and quite useful) that for any formula $\phi$, forcing poset $\mathbb{P}$, $p\in\mathbb{P}$, and $\mathbb{P}$-name $\dot{a}$, $p\Vdash_{\mathbb{P}}\phi(\dot{a})$ is definable as a ...
Ben Goodman's user avatar
9 votes
0 answers
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Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$

I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
new account's user avatar
2 votes
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Namba forcing, one Cardinal up

The classical Namba forcing collapses $\omega_2$ to have cofinality $\omega$ while preserving the cardinal $\omega_1$. Higher analogs were constructed to (under some additional hypotheses) give a ...
Hannes Jakob's user avatar
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Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?

Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...
Hanul Jeon's user avatar
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14 votes
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Changing the cofinality of a regular cardinal without collapsing any cardinals?

I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals: Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals? Is ...
user2925716's user avatar
7 votes
1 answer
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Can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$?

Since $\operatorname{cf}(\aleph_\omega)=\omega$, $\aleph_\omega<\aleph_\omega^\omega$. However, can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$? I am especially interested in models for ...
Calliope Ryan-Smith's user avatar
6 votes
1 answer
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Preservation of cardinals implies preservation of cofinalities when $V=L$?

Kunen mentions the result stated in the title. It would be much appreciated if someone can give a reference for this. Thank you!
LYS's user avatar
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Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$

The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$. ...
Jiachen Yuan's user avatar
6 votes
1 answer
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A variation on pinned equivalence relations

Recall (see e.g. Zapletal, Pinned equivalence relations) that a Borel equivalence relation $E$ on $\omega^\omega$ is pinned iff for every forcing $\mathbb{P}$ and every $\mathbb{P}$-name $\nu$ we have ...
Noah Schweber's user avatar
5 votes
0 answers
205 views

Questions about very fat sets

If $\kappa$ is a regular uncountable cardinal, we call a set $S\subseteq\kappa$ fat if for every $\alpha<\kappa$ and every club $C\subseteq\kappa$, there is a closed subset of $S\cap C$ of ...
Hannes Jakob's user avatar
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3 votes
1 answer
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Can a non-free Whitehead group embed as a discrete subgroup of a normed space?

Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
Tomasz Kania's user avatar
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7 votes
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On the existence of a real which is not set-generic over $L$

Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$. I know that Jensen's ...
Lorenzo's user avatar
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8 votes
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Precipitous ideal and inner model

Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$. If we don't have the inner model condition, then we only know that $\...
Reflecting_Ordinal's user avatar
5 votes
0 answers
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When does an iteration not add functions $\eta\to V$ at the final stage?

I am interested in better understanding the following property: Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...
Calliope Ryan-Smith's user avatar
4 votes
2 answers
187 views

Does $\mathsf{ZF}$ prove $\operatorname{Col}(\lambda,\kappa)$ preserves cardinals below $\lambda$?

Let $\lambda<\kappa$ be cardinals and consider the forcing $\operatorname{Col}(\lambda,\kappa)$ adding a generic surjection $\lambda\to\kappa$. More formally, $\operatorname{Col}(\lambda,\kappa)$ ...
Hanul Jeon's user avatar
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15 votes
3 answers
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May two Cohen reals collapse cardinals?

My question is the following: Let $M$ be a c.t.m. of $\mathsf{ZFC}$. Are there two reals $r_0,r_1 \in \mathbb{R}$ such that $r_i$ is Cohen over $M$ for $i=0,1$ and such that $\omega_1^M$ is countable ...
Lorenzo's user avatar
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6 votes
1 answer
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A strange product forcing

Given two forcing notions $\mathbb{P}_0,\mathbb{P}_1$, we know that the product $\mathbb{P_0}\times\mathbb{P_1}$ will produce the following diagram of models ordered by inclusion: where $M$ is the ...
Lorenzo's user avatar
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7 votes
1 answer
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Consistency strength of some problems about singular cardinals

What is the consistency strength of singularizing a regular cardinal with forcing? Is it exactly a measurable cardinal? Of course the consistency strength of "$V\subseteq W$, $\kappa$ regular in $...
n901's user avatar
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9 votes
1 answer
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Natural set-theoretic principles implying the Ground Axiom

The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By Reitz, it is first-order expressible and easy to force over any given ZFC model with class-...
Monroe Eskew's user avatar
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5 votes
1 answer
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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
3 answers
720 views

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
329 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
611 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
0 answers
173 views

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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5 votes
1 answer
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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
100 views

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
9 votes
1 answer
337 views

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
1 answer
313 views

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
1 answer
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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
1 answer
117 views

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
  • 113
5 votes
1 answer
191 views

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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4 votes
0 answers
204 views

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
105 views

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
0 answers
72 views

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
7 votes
1 answer
249 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
0 answers
111 views

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
573 views

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
  • 18.1k
5 votes
0 answers
102 views

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
221 views

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
943 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
6 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
  • 18.1k
7 votes
0 answers
236 views

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
  • 37.9k
5 votes
1 answer
204 views

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
241 views

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
7 votes
0 answers
225 views

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
421 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
  • 2,134
76 votes
3 answers
18k views

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
  • 1,023
7 votes
1 answer
333 views

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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