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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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 $\mathbb{P}$...
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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 ...
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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 $\mathsf{SCH}$ ...
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Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal
Is anything known about the consistency strength of the following statement?
$\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap \...
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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 ...
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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 $V$...
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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 ...
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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 ...
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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!
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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 ...
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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. $|c"[H]^2|<\alpha$.
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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, $\operatorname{Col}(\lambda,\kappa)$ ...
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Forcing in Ackermann's Set Theory
How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 (...
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Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$...
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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 ...
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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 $...
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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 ...
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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-...
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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 ...
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History of forcing over admissible sets
In his paper "Forcing in admissible sets", Ershov writes
In unpublished lectures given at Novosibirsk State University in 1976-1977 on the theory of admissible sets, the author
showed that it is ...
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Does $\mathsf{MA}^+(\sigma\text{-closed})$ imply there are no Kurepa Trees?
The question in the title is somewhat self contained but let me make some definitions and remarks to clarify.
Recall that $\mathsf{MA}^+(\sigma\text{-closed})$ is the statement that if $\mathbb P$ is ...
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Class Forcing and Genericity: Predense sets vs Dense classes
In short my question is: why do we use definable classes in the definition of genericity for class forcing, instead of predense sets.
To elaborate, in Sy's book and indeed other sources on the subject ...
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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, ...
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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 ...
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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 $...
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How are Koepke's ordinal computability and E-recursion related?
In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result:
A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is ...
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A better way to explain forcing?
Let me begin by formulating a concrete (if not 100% precise) question, and then I'll explain what my real agenda is.
Two key facts about forcing are (1) the definability of forcing; i.e., the ...
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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 $V\vDash\mathsf{ZFC}$, ...
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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: ...
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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 $\...
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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 ...
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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(\...
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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 ...
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Background for classic forcing
When I learning forcing theory, I am surprised by various foring definition,I want to know the original purpose, why define partial order in such way and some background material which can help me ...
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Why does an $\varepsilon$-incomplete $Q$ remain so in the forcing extension of any $\varepsilon$-complete $P$?
This is Shelah's simpler notion of 'completeness' for not adding reals via forcing. Here $P$ and $Q$ are forcing notions.
Let $S_{\aleph_0}(\kappa)$ be the set of all countable subsets of a cardinal $\...
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Arithmetically-hyperimmune-free degrees are comeager
I think proposition XIII.1.22 of Odifreddi is false but I wanted to check I wasn't being dumb. Here's the claim.
Definition: The set$^1$ $A$ is arithmetically-hyperimmune-free if every function $f$ ...
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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 ...
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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 ...
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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 ...
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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}'\...
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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 ...
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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 $\...
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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 ...
CommunityBot
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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 ...
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