**22**

votes

**2**answers

2k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...

**13**

votes

**8**answers

3k views

### Forcing as a tool to prove theorems

It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to ...

**6**

votes

**3**answers

863 views

### Tractability of forcing-invariant statements under large cardinals

It is usual to mention theorems of the kind:
Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi ...

**14**

votes

**3**answers

606 views

### Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...

**9**

votes

**1**answer

554 views

### Resembling the Levy Collapse

Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V_\kappa$ and $\Vdash_{\mathbb{P}} \kappa = \aleph_1$, where $\mathbb{P}$ ...

**15**

votes

**9**answers

2k views

### Examples of ZFC theorems proved via forcing

This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it ...

**9**

votes

**2**answers

888 views

### Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?

Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? ...

**15**

votes

**3**answers

838 views

### Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails ...

**9**

votes

**1**answer

596 views

### Definability of ground model

I have seen mentioned that by a result of Laver, the ground model is definable in any set forcing extension (using parameters).
Does the same hold for class forcing? If it does, in order to establish ...

**7**

votes

**1**answer

194 views

### Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...

**6**

votes

**0**answers

429 views

### A proposed axiom of Laver (updated)

A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom:
(L) Some elementary embedding ...

**8**

votes

**1**answer

535 views

### Forcing Diamond

It is well known that adding a subset of a regular cardinal $\kappa$ with partial functions of size $< \kappa$ forces $\Diamond_\kappa$. One can also see that if $S \in V$ is a stationary subset ...

**7**

votes

**1**answer

326 views

### Consistency results using nonstandard models

Are there any consistency results in set theory (or in mathematics) that can be proved using nonstandard models of ZFC but not using transitive models of ZFC?

**6**

votes

**1**answer

205 views

### On intermediate transitive models for ZFC between M an M[G]

Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). Let $G$ be $B(P)$-generic over $M$, the ...

**4**

votes

**2**answers

441 views

### Mutually generics

Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of ...

**30**

votes

**1**answer

803 views

### Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...

**11**

votes

**1**answer

1k views

### Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...

**17**

votes

**2**answers

2k views

### Two versions of “absolutely ccc”

I have recently been slogging my way through Shelah's "Large continuum, oracles". Essentially from the start there has been a question needling me which I cannot seem to answer.
In the paper, ...

**14**

votes

**2**answers

615 views

### Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...

**12**

votes

**4**answers

1k views

### Forcing over models without the axiom of choice

In the vast majority of papers forcing is always developed over ZFC.
Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain ...

**9**

votes

**1**answer

649 views

### Reals added after Cohen forcing

Let $V_1$ be a generic extension of $V\models GCH$ obtained by adding $\aleph_{\omega}-$many Cohen reals. Then we have the following:
1- In $V_1$ there are $\aleph_{\omega+1}-$many reals,
2- In ...

**23**

votes

**2**answers

573 views

### When does the choice of the generic matter?

It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...

**15**

votes

**2**answers

814 views

### Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following:
Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$
In the paper "The ...

**14**

votes

**1**answer

493 views

### Is every class that does not add sets necessarily added by forcing?

We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. ...

**12**

votes

**3**answers

853 views

### A limit to Shoenfield Absoluteness

Shoenfield's Absoluteness Theorem states that if $\phi$ is any $\Sigma^1_2$ sentence of second-order arithmetic, then $\phi$ is absolute between any two models of $ZF$ which share the same ordinals. ...

**9**

votes

**2**answers

585 views

### Failure of diamond at large cardinals

What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact.
Remark. The problem of forcing ...

**7**

votes

**1**answer

636 views

### Iterated forcing and CH

I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert ...

**17**

votes

**1**answer

705 views

### Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...

**12**

votes

**4**answers

935 views

### Cantor-Bernstein for notions of forcing

For forcing notions $\mathbb{P}$ and $\mathbb{Q}$ let us write $\mathbb{P}\triangleleft\mathbb{Q}$ if forcing with $\mathbb{Q}$ always adds a $\mathbb{P}$-generic filter over $V$. In other words, ...

**11**

votes

**1**answer

275 views

### Which forcings preserve (some) determinacy?

The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...

**8**

votes

**1**answer

339 views

### Is there any forcing free proof for hard independence results?

We are forced to use forcing for almost all "hard" independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. The question simply is:
Primary Question: Is there any "forcing free" ...

**5**

votes

**1**answer

252 views

### Bad subforcings of nice forcing notions

Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$
...

**10**

votes

**3**answers

347 views

### Who proved “sets in every generic are already in the ground model?”

Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if ...

**9**

votes

**0**answers

416 views

### Existence of a regular subposet which collapses everything except the top cardinal

Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < ...

**9**

votes

**1**answer

488 views

### Can we change the Lebesgue measure by forcing?

Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...

**8**

votes

**1**answer

689 views

### Probabilities independent of ZFC?

Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...

**7**

votes

**1**answer

266 views

### A special c.c.c forcing notion and adding minimal generic reals

This question is related to my question "Forcing with c.c.c forcing notions, Cohen reals and Random reals".
A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds ...

**7**

votes

**1**answer

345 views

### Making all cardinals countable and its HOD

Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$?
Update. Are the ...

**3**

votes

**3**answers

390 views

### A Question on Special Forcings

The usual use of forcing begins with a "countable" and "transitive" ground model of $ZFC$ and reaches to a "countable" and "transitive" generic model of $ZFC$ with the "same ordinals". In a discussion ...

**13**

votes

**3**answers

542 views

### Products of Cohen forcings

Let $\lambda$ be an infinite cardinal. Does the full support product of $\lambda$ copies of $Add(\omega, 1)$ collapse $2^\lambda$ to $\aleph_0$?
For $\lambda = \omega$, it is known to be true (it is ...

**10**

votes

**2**answers

655 views

### Questions about Prikry forcing and Cohen forcing

I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**8**

votes

**2**answers

264 views

### Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...

**7**

votes

**6**answers

544 views

### Applications of forcing in model theory

What are the major applications of (set theoretic) forcing in model theory?

**6**

votes

**1**answer

341 views

### Can the Cohen forcing collapse cardinals?

Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...

**5**

votes

**1**answer

209 views

### $\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations:
$\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$;
$\mathfrak{b}$ is the bounding ...

**4**

votes

**1**answer

166 views

### How can I collapse all cardinals of ground model except one of them?

Let $\kappa$ be an uncountable cardinal of a c.t.m $M$ of $ZFC$.
Is there a generic extension of $M$ like $M[G]$ such that all uncountable cardinals of ground model collapse except $\kappa$?

**4**

votes

**3**answers

462 views

### Which notions of forcing add a cofinal branch to an $\omega_1$-tree?

I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree.
Question 1:
What kinds of forcings add cofinal branches to $\omega_1$-trees? What kinds of forcings cannot?
...

**4**

votes

**3**answers

456 views

### “name” for the ground model

I am a beginner of forcing, often I read from some articles something like "$p \Vdash \dot{G}$ is $P$-generic over $\check{M}$" (where $M$ is a countable transitive model, for instance).
Q1. I ...

**1**

vote

**2**answers

255 views

### Substructure Argument for Chain Conditions

Once, someone showed me a nice argument using elementary substructures for proving chain conditions about forcings. It was a basic example, maybe that Cohen forcing is ccc. I've been trying to look up ...

**14**

votes

**3**answers

484 views

### Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...