**6**

votes

**1**answer

671 views

### Is there a monster behind the trees?

First Fix the following notation:
$\forall \kappa\in Card~~~Tp(\kappa):="\kappa~has~tree~property"$
The large cardinals as "monsters of heaven" live everywhere in the land of ...

**3**

votes

**1**answer

294 views

### Countable support iterated forcing of length $\alpha$ which forces $|\alpha| > \aleph_2$

Let $cf(\alpha) > \omega$, and $P_\alpha := \langle P_{\beta}, \dot{Q}_{\beta} : \beta < \alpha \rangle$ be a countable support iterated forcing construction (so for each $\beta$, $P_{\beta} = ...

**5**

votes

**1**answer

261 views

### The Ground Axiom for special statements of set theory

The Ground Axiom (GA), introduced by Hamkins and Reitz, asserts that the universe is not a nontrivial set forcing extension of any inner model, and it is known that GA is consistent relative to ZFC. ...

**7**

votes

**1**answer

342 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 ...

**8**

votes

**0**answers

154 views

### Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$

In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other ...

**6**

votes

**1**answer

268 views

### If there is a non-constructible real, is there an $L$-generic real?

If we assume that $\Bbb R^V\neq\Bbb R^L$, can we deduce that there is some $x\in\Bbb R^V$ which is $L$-generic?
Of course if $V$ is a generic extension of $L$ this is true, but if $V=L[0^\#]$ this is ...

**7**

votes

**2**answers

261 views

### How to make countably closed forcing “nice” without choice

When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice:
If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In ...

**7**

votes

**2**answers

499 views

### Does forcing generally go one way?

Question Is there any forcing free proof for hard independence results? talks about the use of forcing for independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. For that case ...

**8**

votes

**1**answer

336 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" ...

**13**

votes

**3**answers

471 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 ...

**14**

votes

**2**answers

799 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 ...

**8**

votes

**1**answer

243 views

### On free limits of iterations

Shelah isolates the notion of "$\aleph_1$-free iteration" in the first two sections of Chapter IX from Proper and Improper Forcing, and he proves there that properness is preserved by this sort of ...

**4**

votes

**1**answer

254 views

### Large cardinals and mild extensions

It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are:
Question ...

**12**

votes

**0**answers

433 views

### Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...

**3**

votes

**2**answers

192 views

### Preservation of measurable cardinals in mild extensions

I would like some help with the proof of the preservation of measurable cardinals in mild extensions, as I am a bit new with forcing.
By mild extensions, I mean the generic extension produced from a ...

**9**

votes

**2**answers

576 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 ...

**4**

votes

**1**answer

384 views

### set theory forcing

Suppose $M$ satisfies the $CH$ and that we force over $M$ with $\mathbb{P}=Fn(I,2)$ where $(\omega_{2} \leq |I|))^{M}$, that is, with the finite partial functions from $I$ to $2$. If $f \in M[G] \cap ...

**9**

votes

**0**answers

415 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 < ...

**3**

votes

**1**answer

247 views

### Trivial forcings which are not very trivial

Suppose that $M$ is a model of $\sf ZFC$, and we add some generic set $G$. Then it is not hard to see that for every $x\in M[G]$ it holds $M\subseteq M[x]\subseteq M[G]$.
Given $x\in M[G]$ such that ...

**6**

votes

**3**answers

330 views

### Class forcing: Pelletier vs Friedman

[Apologies in advance for a fluffy question]
I'm reading this old paper by Pelletier, where he gives a Boolean-valued model version of class forcing, assuming that the Boolean algebra in question can ...

**4**

votes

**1**answer

139 views

### Amalgamation of two ccc algebras may collapse the continuum

The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but unfortunately I ...

**7**

votes

**2**answers

427 views

### collapsing successor of singular

Let $\lambda$ be a singular cardinal. Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$?
(Note that even without ...

**5**

votes

**1**answer

251 views

### Question about Shelah's version of “Shooting a club” found in PIF

Suppose $S \subset \omega_{1}$ is stationary co-stationary. Then there is a forcing notion $P_{S}$ which shoots a closed unbounded $C \subset S$ without collapsing cardinals (or
changing ...

**10**

votes

**1**answer

453 views

### Forcing mildly over a worldly cardinal.

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a ...

**4**

votes

**1**answer

179 views

### Which $\omega_1$-trees are proper?

Consider a tree $(T, <_T)$ of height $\omega_1$, with countable levels. One can view $T$ as a forcing poset by calling a condition $s\in T$ stronger than $t\in T$ if $t <_T s$.
My question is: ...

**4**

votes

**1**answer

291 views

### Examples of stationary set preserving forcings that are not semiproper?

A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of course proper) notions ...

**4**

votes

**1**answer

164 views

### Hereditarily Countable Names and Proper Forcing

The 'hereditarily countable names' are as defined in Shelah's Proper and Improper Forcing, Chapter 3 Definition 4.1. Let $\mathbb{P}$ be a proper forcing notion and $\dot{Q}$ a $\mathbb{P}$-name such ...

**9**

votes

**2**answers

349 views

### The transcendence degree of $\mathbb R$ after adding a Cohen

Let $V\models\sf ZFC$, and let $V[r]$ be a generic extension obtained by adding one Cohen real, or equivalently $\omega$ Cohen reals.
It is clear that $\Bbb R^{V[r]}$ and $\Bbb R^V$ have the same ...

**8**

votes

**1**answer

532 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 ...

**4**

votes

**1**answer

269 views

### Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal

Is anything known about the consistency strength of the following statement?
$\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap ...

**1**

vote

**0**answers

162 views

### What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...

**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

**1**answer

393 views

### Random reals and strongly meager sets

Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see this question).
The notion of strong measure zero sets has its dual concept in the category branch -- ...

**4**

votes

**2**answers

331 views

### Set forcing and ultrapowers

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter):
(Woodin) Let $V[G]$ be a ...

**7**

votes

**2**answers

442 views

### subalgebra of a simple forcing

Let $\alpha > 0$ be any ordinal. Consider the forcing $Add(\omega,\alpha) * Add(\omega_1,1)$. Let $B$ be its boolean completion. Let $\dot{X}$ be the canonical $B$-name for the generic subset of ...

**9**

votes

**1**answer

547 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 $\mathbb{P}$ ...

**11**

votes

**1**answer

484 views

### Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals?

The question is whether, when you add a Cohen subset to a cardinal
$\kappa$, that cardinal becomes a characteristic of the resulting forcing extension $V[G]$. Or can there be strange instances in ...

**9**

votes

**1**answer

327 views

### Homogeneous Namba-like forcing

Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly ...

**9**

votes

**1**answer

349 views

### Forcing to “minimally” add new reals.

Suppose that I want to force to add a "single" new subset of $\omega$ and not much else. For example, consider the Cohen forcing consisting of finite partial functions from $\omega$ to 2. The forcing ...

**3**

votes

**1**answer

259 views

### The canonical forcing of the GCH and direct limits.

The motivation for this question is that I am working through an exercise to force the GCH (generalized continuum hypothesis) over a model of ZFC and obtain a model of ZFC where GCH holds.
The ...

**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? ...

**14**

votes

**1**answer

492 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 $M$. ...

**5**

votes

**1**answer

211 views

### Forcing over the poset of nonempty open subsets of a nice topological space

Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...

**5**

votes

**1**answer

260 views

### How to find a sub-forcing?

Suppose that $M$ is a model of ZFC, $P\in M$ is a notion of forcing and $G$ is a $P$-generic filter over $M$.
It is a well-known theorem that if $M\subseteq N\subseteq M[G]$, and $N$ is a model of ...

**6**

votes

**2**answers

393 views

### From the product lemma to to a result about powersets

Recall the product lemma from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one factor $\lambda^+$-cc ...

**3**

votes

**1**answer

270 views

### Can we weaken GCH in this class forcing?

I've just stumbled across the following theorem (here):
Theorem Let $P$ be a partial class order in $M$, a transitive model of ZF (resp. ZFC). Suppose for arbitrarily large cardinals $\kappa$ we have ...

**2**

votes

**0**answers

219 views

### Can we prove the completeness of FOL based on forcing?

I asked this question at http://math.stackexchange.com but I didn't get any answer there.
In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a ...

**12**

votes

**1**answer

632 views

### I'll admit it: I don't understand the definition of the Easton product.

I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the ...

**5**

votes

**2**answers

365 views

### Suslin algebras

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...

**6**

votes

**3**answers

371 views

### Why does the Solovay-Tennenbaum theorem work?

I have decided to learn iterations at long last, and I am reading through Jech's Set Theory for now. The standard example after explaining what is an iteration with finite support is the following ...