# 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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### 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 ...

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### Can we recover an inner model of CH after forgetting some generic information?

Suppose $\kappa$ is an inaccessible cardinal. Let $G \times H$ be $\mathrm{Col}(\omega_1,{<}\kappa) \times \mathrm{Add}(\omega,\kappa)$-generic over $V$. Let $X \subseteq \kappa$ be $\mathrm{Add}(...

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### Inner models with all sets generic

Question: Under large cardinal axioms, what is the intersection of all inner models $M$ of ZFC such that every set in $V$ is set-generic over $M$?
Every set belongs to a generic extension of HOD, and ...

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### Strong chains in $[\omega_2]^{\omega_2}$ mod finite of length $\omega_3$

Probing a bit the difference between $[\omega_1]^{\omega_1}$ and $[\omega_2]^{\omega_2}$ modulo the finite sets:
Question
Can there exist a family $\langle X_\alpha:\alpha<\omega_3\rangle$ ...

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### Strong Chains in $^{\omega_1}\omega_1$ of length $\omega_3$

In a previous question, I asked about the impact of strong chains in $^{\omega_1}\omega_1$ (e.g., sequences of functions $\langle f_\alpha:\alpha<\kappa\rangle$ in $^{\omega_1}\omega_1$ that are ...

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### Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$

Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property?
Because conditional $Σ^2_2$ absoluteness under $◊$ ...

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### NCF, P-points, weak P-points, and cardinalities

The post is a bit long, but all the questions are similar or concern the same topic.
Let $\omega^*=\beta\omega\setminus\omega$. A well-known topological definition of a P-point (on $\omega$) is as ...

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### Are there forcing notions adding $\kappa$ random, sacks, prikry, or Mathias reals?

This is my first question here, so if I am doing things incorrectly, please let me know. Now on to the question:
The forcing notion $Fn(\kappa,2)$, which constists of partial functions from $\kappa$ ...

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### On the role of $\diamondsuit$

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...

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### $\Sigma^2_1$ and the Continuum Hypothesis

This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...

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### Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?

A function $f:\omega\to\omega$ is called
$\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$;
$\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite.
Let us ...

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### Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$

Let $(P,\le)$ be a poset. For a point $x\in P$ let
$${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $...

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### Amoeba forcing adds a null set covering all old null sets

I am currently reading a paper by Goldstern, Kellner and Shelah, in which they, pretty nonchalantly, state "Amoeba forcing will add a null set covering all old null sets", without proving this fact or ...

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### Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets

We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$
Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...

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### A question on simple $P_{\aleph_2}$-points

This question is motivated by discussion surrounding this MO question.
An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\...

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### The existence of $T$-ultrafilters in ZFC

Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on $T$-ultrafilters hoping that somebody on MO can solve it.
First I recall the necessary ...

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### A new cardinal characteristic (related to partitions)?

In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic.
By a ...

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### Compactifications with remainder $[0,\omega_1]$ and convergent sequences

Is the following statement consistent?
$(\star)$ Let $K$ be a separable compact Hausdorff space containing the space $[0,\omega_1]$ so that the complement $K\setminus[0,\omega_1]$ is discrete. Then ...

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### Suslin representation of sets and limits to Shoenfield's Absoluteness

For any natural number $k \in \omega$ and any set $X$ the set $$T \subseteq \bigcup_{m \in \omega} (\omega^m)^k \times X^m $$ is a tree on $\omega^k \times X$ iff
$$(t_o, \ldots, t_k) \in T \: \...

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### Models of ZF intermediate between a model of ZFC and a generic extension

Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with
$$M\subseteq N \subseteq M[G]$$
and that $N=M(x)$ for some $x\in ...

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### How bad a proper forcing of size $\aleph_1$ can be?

This question concerns proper forcings of size $\aleph_1$. In the context of
$\rm ZFC+\neg CH$, I couldn't find any counter example to the following property. Suppose $\mathbb P$ is a proper forcing ...

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### Forcing square introduces diamond

Let $\mathbb S_\kappa$ be the standard forcing for $\square_\kappa$ by initial segments. This is $(\kappa+1)$-strategically closed.
Observation: Let $T \subseteq \kappa^+$ be stationary. If $T$ ...

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### Can $H_{\omega_1}$ and $H_{\omega_2}$ be in bi-interpretation synonymy?

This question concerns the possibility of the bi-interpretation synonymy of the structure
$\langle
H_{\omega_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle ...

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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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### A definable gender problem related to Sacks reals

Let $L$ be the ground model, and $a\in2^\omega$ be a Sacks-generic real over $L$. Note that any real $x\in S=(2^\omega\cap L[a])\setminus L$ is still Sacks-generic over $L$. Now assume that $\mathsf E$...

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### Preservation of chain condition under strategically closed forcing

It is well-known that $\kappa$-closed forcing preserves $\kappa$-c.c. posets. The same argument works for $\kappa$-strategically closed forcing. Here is the definition:
A poset $\mathbb P$ is $\...

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### When does “sufficient genericity” actually suffice?

Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is $\mathbb{P}$-enforceable if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for ...

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### About product of Baire spaces and forcing

Let $\mathbb{P}=\langle P, \leq \rangle$ be a p.o.
Two elements $p$ and $q$ of it are called
compatible if there is an $r \in \mathbb{P}$ such that $r\leq p$ and $r \leq q$; otherwise they
are called ...

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### Saturation of non-stationary ideal on $\omega_2$?

It is known that $NS_{\omega_2}$ cannot be saturated (namely there cannot be $\aleph_3$ many stationary subsets of $\omega_2$ any two of which have non-stationary intersection). However, it may be the ...

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### On the relationship between Martin's Axiom, the countable chain condition and the Knaster property

This is a repost of a question that went unanswered on MSE
We say that a poset $P$ has the Knaster property (or is Knaster) if every uncountable subset of $P$ contains an uncountable subset of ...

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### Forcing and new ordinals

$\textbf{Question}$: In expositions to forcing, why do we insist on not adding new ordinals to a countable transitive model $M$ of ZFC?
For example, after ruling out transitive proper class models ...

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### Why do ordinal collapsing functions use regular cardinals?

Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...

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### Is the category of atomless Boolean algebras with complete embeddings closed under coproducts?

Consider the category whose objects are atomless Boolean algebras (not necessarily complete) and whose arrows are complete embeddings.
Does a coproduct exist in this category for any two atomless ...

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### Solovay’s model

Solovay proved that if $\kappa$ is inaccessible, then if we adjoin a generic $G \subseteq \mathrm{Col}(\omega,{<}\kappa)$, then in the extension, every set of reals in $L(\mathbb R)$ is Baire- and ...

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### How to increase unbounding and dominating numbers for $(\kappa^\lambda,\leq^*)$

Let $\kappa$ be regular and $\lambda\geq\kappa$. For $f, g\in\kappa^\lambda$ say that $f\le^* g$ if the set $\{\gamma<\lambda:f(\gamma)>g(\gamma)\}$ has
size less than $\kappa$. Set
$\mathfrak{...

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### Generic properties of dominating/etc. reals with non-Cohen working parts

The first and foremost forcing that adds a real is the Cohen forcing which approximates a new real number by giving finite approximations to its characteristics function.
But very quickly after that, ...

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### Effect of adding one Hechler real versus adding two on the meager ideal

In "The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)" by Haim Judah and Saharon Shelah
JSL Vol. 55, No. 3 (Sep., 1990), pp. 909-927 ([...

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### Forcing absoluteness in the setting of second-order arithmetic

There are some results about (set-size) forcing absoluteness for first-order properties in $L(\mathbb{R})$ and for $\mathbf{\Pi}^1_{\infty}$ properties when one works over $\mathsf{ZFC}$. My question ...

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### ZFC applications of Shelah's creature forcing

Shelah's creature forcing is a very powerful method, with wide range of applications. The method also has some applications in ZFC, let's quote a few of them that I am aware of:
(1) In A partition ...

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### ladder system uniformization at successors of singulars

Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...

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### A simple cardinal characteristic associated with $\omega^\omega$

We can define a very simple cardinal characteristic in the following way. Recall the relation $\leq^*$ on $\omega^\omega$ defined by $x\leq^* y$ iff $x(i)\leq y(i)$ for all but finitely many $i$. For $...

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### Examples of independent $\Sigma_4^1$ statements

As in the title, I'm looking for examples of $\Sigma^1_4$ (preferably complete) sentences which are independent from ZFC in both ways, namely given a model $V$ we can extend it to $V'$ where such a ...

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### What examples of existence forcing proofs are there?

Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing.
There are only a handful of ...

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### Generic saturation of inner models

Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses ...

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### Proof that Namba Forcing preserves stationary subsets of $\omega_1$

I am looking for a proof (or reference) of the fact that Namba Forcing preserves stationary subsets of $\omega_1$. This fact is stated and used throughout the literature(whenever you google it you'll ...

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### Constructing a model of $\mathrm{DCF}_0$ via forcing

As is mentioned in the introduction of this paper of Spodzieja there is a lack of 'natural' examples of differentially closed fields. The immediate naive guesses, namely the field of germs of ...

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### Making the precaliber number bigger than all Knaster numbers

Write $\mathfrak m_k$ for the Martin's axiom number for $k$-Knaster, i.e., for the smallest size of a family of dense subsets of some $k$-Knaster poset for which there is no generic filter. (A poset ...

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### Radin forcing preserving large cardinals

I'm wondering if there are any known result for the maximum large cardinal strength
which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can ...

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### Radin generics from iterated ultrapowers

Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}_w$. Specifically,...

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### Semi-rigid boolean algebras

A boolean algebra is rigid if it has no nontrivial automorphisms. Call it semi-rigid if none of its nontrivial automorphisms has any fixed points other than 0 and 1.* The four-element algebra $\{0, b, ...