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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8
votes
1answer
113 views

Does ZF + BPI alone prove the equivalence between “Baire theorem for compact Hausdorff spaces” and “Rasiowa-Sikorski Lemma for Forcing Posets”?

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $\mathbb{P}$ (i.e. $\mathbb{P}$ is a reflexive transitive relation) and for any countable family of dense subsets of $\mathbb{...
6
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0answers
253 views

“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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1answer
292 views

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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1answer
148 views

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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0answers
113 views

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 ...
43
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4answers
3k views

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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0answers
201 views

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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0answers
121 views

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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2answers
1k views

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. ...
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1answer
98 views

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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1answer
183 views

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 ...
3
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0answers
113 views

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 ...
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0answers
128 views

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 "$\mathrm{PFA}$ ...
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0answers
159 views

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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1answer
152 views

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 ...
10
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0answers
225 views

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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1answer
155 views

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 ...
6
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2answers
586 views

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 ...
12
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3answers
725 views

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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1answer
211 views

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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0answers
274 views

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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1answer
280 views

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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1answer
390 views

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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0answers
297 views

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
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1answer
252 views

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 $\rm ZFC$...
7
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2answers
740 views

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
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1answer
550 views

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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0answers
293 views

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....
6
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136 views

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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0answers
172 views

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 $\...
9
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0answers
229 views

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 ...
10
votes
1answer
326 views

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 $\sf ZFC$. ...
11
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2answers
421 views

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. ...
7
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0answers
207 views

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
1answer
407 views

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
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2answers
689 views

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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0answers
172 views

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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1answer
727 views

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 ...
5
votes
1answer
344 views

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 ...
62
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5answers
5k views

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
votes
1answer
245 views

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 ...
19
votes
1answer
650 views

How badly can the GCH fail globally?

It's known that we 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
2answers
407 views

$\omega_1$-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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0answers
166 views

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 $P(X_0)/\mathrm{NS} \cong P(X_1)/\mathrm{NS}$?
11
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0answers
426 views

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
1answer
266 views

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
1answer
464 views

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 ...
10
votes
1answer
583 views

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
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1answer
117 views

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
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0answers
206 views

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 ...

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