# Questions tagged [forcing]

Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

743
questions

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### 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{...

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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**answer

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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, ...

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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 ...

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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 ...

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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 $|f''[X]^{<\omega}|\leq \aleph_0$.
...

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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 ...

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**2**answers

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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vote

**1**answer

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**

votes

**1**answer

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 ...

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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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### 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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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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### 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 ...

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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 ...

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**1**answer

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 ...

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**2**answers

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 ...

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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'...

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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 ($\text{GA}$) ...

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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 $V$ ...

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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 ...

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**1**answer

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 $N\prec (H(\mu^\dagger),\epsilon),\mu\in N$] $N$ ...

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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 ...

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**1**answer

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 $\rm ZFC$...

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votes

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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 ...

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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 $\...

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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....

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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....

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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 $\...

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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 ...

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**1**answer

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 $\sf ZFC$.
...

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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. ...

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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 $\varphi$ ...

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votes

**1**answer

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, ...

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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 $...

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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 ...

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**1**answer

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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 ...

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**1**answer

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 ...

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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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**1**answer

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 ...

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**1**answer

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### 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 ...

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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 ...

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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 $P(X_0)/\mathrm{NS} \cong P(X_1)/\mathrm{NS}$?

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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 ...

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**1**answer

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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 ...

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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 ...

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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 ...

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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\...

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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 ...