Questions tagged [forcing]
Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
842
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"Relative plausibility" of some infinitary theories
We work in $\mathsf{ZFC+V=L}$.
Define a plausible theory to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely ...
7
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1
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Construction of a model of $ZFC+\neg Con(ZFC)$
By Gödel's second incompleteness theorem, the following assertion is true in ZFC:
$$
Con(ZFC)\rightarrow Con(ZFC+\neg Con(ZFC))
$$
Considering the completeness theorem, this assertion is equivalent to ...
3
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1
answer
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Grigorieff forcing and destruction of ultrafilters
I was interested in the Grigorieff forcing (you can read the definition here: How "much" does (Grigorieff) forcing destroy an ultrafilter?)
I couldn't prove that it destroys ultrafilters, ...
6
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Rigid boolean inclusions?
A boolean algebra $B$ is rigid if it has no nontrivial automorphisms and atomless if it has no minimal nonzero elements. $A \subseteq B$ is a complete boolean inclusion if $B$ is complete and $A$ is a ...
48
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4
answers
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Sheaf-theoretic approach to forcing
Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general ...
10
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Martin's Maximum implies stationary/club Chang's conjecture?
Chang's Conjecture (CC) states: for any $f: [\omega_2]^{<\omega} \to \omega_1$, there exists a set $X\subset \omega_2$ of order type $\omega_1$ such that $|f''[X]^{<\omega}|\leq \aleph_0$.
...
2
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Definability of the ground model in its class-forcing extension
It is known that Laver's ground model definability theorem doesn’t hold for all class forcing notions. That is, if $M$ satisfies ZFC then $M$ is not necessarily definable in $M[G]$, a class forcing ...
14
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Condensed / pyknotic sets in terms of forcing over Boolean-valued models of set theory / multiverse concepts?
Here is one way of saying what a pyknotic set is. Fix an inaccessible cardinal $\kappa$, and let $Proj_\kappa$ be the category of $\kappa$-small, extremally disconnected compact Hausdorff spaces. ...
1
vote
1
answer
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Second order theories for class forcing
We know that if $M$ is a model of ZFC, then taking $\mathcal{C}$ to be the collection of all classes definable in $M$ with set parameters, and taking $\in$ to be the obvious extension of $M$'s epsilon ...
4
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1
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Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?
My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...
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Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same question for $Σ_n(I_\text{NS})$ (i.e. using the ...
6
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Reference for "$\mathrm{PFA}$ implies $L(\mathbb{R}) \cap \bigcup_{1 \leq k < \omega} \mathcal{P}(\mathbb{R}^k)$ is productive"
The preprint of the recent result of Aspero and Schindler, "Martin's Maximum$^{++}$ implies Woodin's Axiom $(*)$", mentions productive pointclasses, and states that "$\mathrm{PFA}$ ...
3
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Independence through forcing vs generic collapses
Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
7
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Are generic filters that produce the same forcing extension related by a ground-model automorphism?
Suppose $M$ is a countable transitive model of some fragment of $\mathbf{ZFC}$, $\mathbb{P}\in M$ is a forcing notion and $G, H$ are $\mathbb{P}$-generic such that $M[G]=M[H]$. Does it then follow ...
11
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Can we bound $2^{\aleph_\omega}$ without pcf theory?
One of the famous applications of pcf theory is that if $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$. I'm curious whether any weaker result with the same ...
4
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1
answer
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Obtaining elements of a generic extension from a Boolean-valued model of ZFC
Let $\mathcal{M}$ be a countable transitive standard-model of ZFC.
Let $B \in \mathcal{M}$ be a boolean algebra that is complete in $\mathcal{M}$.
Further, let $\mathcal{M}^{(B)}$ be the corresponding ...
7
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2
answers
678
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Independence result where probabilistic intuition predicts the wrong answer?
In some of his writings, Paul Cohen gave an informal, motivational discussion about the word generic (as it is used in forcing). While very suggestive, the discussion leaves the meaning of the word ...
13
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3
answers
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What did Paul Cohen mean by saying that generic sets of natural numbers have "no asymptotic density?"
In Paul Cohen's original 1963 paper on forcing, The independence of the Continuum Hypothesis, published in PNAS, he gives his general proof sketch of how he intends to create a model of ZFC that doesn'...
6
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1
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Restricted notions of set-theoretic geology
We say that $W$ is a ground of the universe $V$ if $W$ is a model of ZFC and there is a poset $P\in W$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($\text{GA}$) ...
5
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Restricted notions of set-theoretic geology [duplicate]
We say that $W$ is a ground of $V$ if $W$ is a model of ZFC and there is a poset $P$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($\mathrm{GA}$) asserts that $V$ ...
8
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1
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Intuition behind Boolean-valued models of set theory
$\DeclareMathOperator\Card{Card}$The book Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise by Hoffmann provides an intuition behind boolean valued models of set theory which I will ...
3
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1
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Forcing, a technical detail
In the snippet below from Shelah's book P&I Forcing,
in the definition 5.2(2)
I do not follow why in this sentence [naturally extended to include $N\prec (H(\mu^\dagger),\epsilon),\mu\in N$] $N$ ...
6
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0
answers
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Temporary destruction of measures in intermediate models
It is a well-known theorem that if $\kappa$ is measurable, then there is a generic extension in which $\kappa$ is no longer weakly compact, but we can force its weak compactness back and recover the ...
8
votes
1
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Properness for small forcings
It is easy to see a forcing of size $\aleph_1$ is proper if and only if is semiproper. I was wondering when such an equivalency holds between semi-proper and stationary-preserving forcings in $\rm ZFC$...
8
votes
2
answers
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Non-set-theoretic consequences of forcing axioms
This article by Quanta Magazine states:
... forcing axioms ... are workhorses that regular mathematicians “can actually go out and use in the field, so to speak,” ...
What are some examples of uses ...
11
votes
1
answer
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Approximating a real in the ground model
Let $\mathbb{P}$ be a proper notion of forcing, having the Sacks property. Suppose that $\dot{D}$ is a $\mathbb{P}$-name for an infinite subset of $\omega$. I'm looking for a set which approximates $\...
10
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What kind of objects can code a universe?
Jensen proved that given $V\models\sf ZFC+GCH$, there is a class generic real $r$, such that $V[r]=L[r]$, and no cardinals are collapsed.
We know that this can be modified such that $r$ is minimal, i....
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Preserving supercompactness in intermediate forcing extensions
Let $\kappa$ be supercompact in $V$. Let $\mathbb{P}$ be one of the standard forcing notions (or an iteration of such), and for simplicity assume that $\mathbb{P}$ is ${<}\kappa$-directed closed (e....
4
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PFA for cardinal preserving forcing notions and the CH
Let $FA_{\aleph_1}$(cardinal preserving proper forcings) be the forcing axiom: if $\mathbb{P}$ is a cardinal preserving proper forcing notion and $(D_\xi)_{\xi<\omega_1}$ are dense subsets of $\...
13
votes
1
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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 ...
11
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1
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If all transitive models have the same height, are they all "simple"?
Suppose that $\alpha$ is the unique ordinal for which $L_\alpha$ is a model of $\sf ZFC$. In other words, there is no transitive model of $\sf ZFC$ in which there is a transitive model of $\sf ZFC$.
...
12
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2
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548
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Forcing notions adding minimal reals
I am looking for a comprehensive list of known forcing notions which add a minimal real into the ground model. I know some of them like the Sacks forcing, or the Judah-Shelah's example of a c.c.c. ...
14
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1
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Is there an infinitary sentence which is absolutely not second-order expressible?
This is a "forcing-absolute" followup to this question, whose answer was largely unsatisfying. The question is:
Suppose $V=L$. Is there an $\mathcal{L}_{\infty,\omega}$-sentence $\varphi$ ...
6
votes
1
answer
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Set-theoretic geology III: inside the core
Thanks to Jonas, Asaf, and Gabe I understand a little more of grounds and the mantle (or mantles, because it looks like there may be more than one).
But, set-theoretic geology, or so it seems to me, ...
15
votes
2
answers
730
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Good forcings with bad squares
Consistently with $\mathsf{ZFC}$ there is a forcing which preserves cardinals but whose square does not always preserve cardinals - that is, some $\mathbb{P}$ such that for every $\mathbb{P}$-generic $...
6
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0
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Collapse successor of singular while preseving supercompactness
Suppose $\kappa$ is a supercompact cardinal. Is it possible to find a forcing which collapses $\kappa^{+\omega+1}$ to $\kappa^{+\omega}$ (all those $\kappa^{+n}$'s are preserved) while the ...
11
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1
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Set-theoretic geology: controlled erosion?
I have to say that after the two last posts by Timothy Chow on Forcing I got so intrigued that I am trying to rethink the little I know about this formidable chapter of mathematics.
I have also to add ...
6
votes
1
answer
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Forcing, constructibility, and random functions
This question is in some ways an offshoot of my recent question about trying to explain forcing to someone (such as Scott Aaronson, whose questions have prompted my questions) encountering it for the ...
72
votes
6
answers
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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 ...
5
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1
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Consistency strength of lifting through a lot of collapsing
What is the consistency strength of the following situation?
$j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$.
$\mathbb P$ is a forcing that ...
22
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1
answer
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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 ...
12
votes
2
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$\omega_1$-approximation property for Sacks iteration— contradiction in literature?
The following is a folklore result.
Suppose $P$ is a countable support iteration of nontrivial forcings, $\langle P_\alpha, \dot{Q}_\alpha : \alpha < \omega_1 \rangle$. Then there is a complete ...
10
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0
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Isomorphisms mod nonstationary
Suppose $G \subseteq \mathrm{Add}(\omega_1)$ is generic over $V$. Let $X_i = \{ \alpha : G(\alpha) = i \}$. Is it true that $P(X_0)/\mathrm{NS} \cong P(X_1)/\mathrm{NS}$?
12
votes
0
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Can a generic $\mathbb{R}$ have a new cardinality?
This question was asked and bountied at MSE, without success.
My main question is whether, starting with a model of determinacy, a "generic $\mathbb{R}$" could be different in cardinality ...
12
votes
1
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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 ...
12
votes
1
answer
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Specific notions of forcing from the point of view of category theory
I'm trying to learn about the topos of sheaves and the double negation topology to try to go through the independence of CH from a categorical perspective. I'm curious in general about what the ...
11
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1
answer
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Which is the more popular approach to forcing in the literature?
There are some interesting questions and answers on the site discussing the different approaches to forcing in set theory, and I understand that the two most important are the ones using countable ...
5
votes
1
answer
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Maximality principle in symmetric extensions
Let $M$ be a ctm and $P\in M$ a forcing order.
In regular forcing extensions, we have the following well-known Principle:
$$p\Vdash_{M,P}\exists x\phi[x]\;\Longrightarrow\;\exists\sigma\in M^P\;p\...
9
votes
0
answers
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Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
17
votes
1
answer
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Topos extensions
In set theory, starting from a model $V$ of $ZFC$, a forcing notion $\mathbb{P}$, and a generic filter $G \subset \mathbb{P}$ over $V$, we can find a generic extension which is a model of $ZFC$ and is ...