**4**

votes

**2**answers

356 views

### Forcing for Arbitrary First Order Theories

Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...

**3**

votes

**2**answers

296 views

### Antichains and the Knaster Property

This may be a naive question, but I'll pose it.
Is there an example of a notion of forcing $\mathbb{P}$ that has the $\kappa$-c.c. which is not also $\kappa$-Knaster Property also is not "...

**5**

votes

**1**answer

174 views

### Intermediate submodels which do not satisfy AC

The following is known:
Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of $...

**8**

votes

**1**answer

352 views

### Consistency of P1 on Kunen

It's the first time I'm posting here so I don't know if I really should put this question here... I tried to post it on math.stackexchange, but a friend told me I would get better results by posting ...

**4**

votes

**1**answer

200 views

### About subposet of Levy collapse

Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that ...

**11**

votes

**1**answer

319 views

### Class Forcing and Genericity: Predense sets vs Dense classes

In short my question is: why do we use definable classes in the definition of genericity for class forcing, instead of predense sets.
To elaborate, in Sy's book and indeed other sources on the ...

**9**

votes

**1**answer

347 views

### Suslin trees in ccc ideals

A countably complete ideal $I$ on a set $Z$ ideal is c.c.c. when there is no uncountable family of pairwise disjoint $I$-positive subsets of $Z$. If such an ideal exists, then there exists a weakly ...

**1**

vote

**1**answer

184 views

### Different Methods for Forcing Total Failure of Generalized Continuum Hypothesis

It seems there are few papers on total failure of Generalized Continuum Hypothesis.
As an example Merimovich in 2007 constructed a model of ZFC such that GCH fails everywhere assuming consistency of ...

**11**

votes

**3**answers

379 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 ...

**7**

votes

**3**answers

826 views

### PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$.
Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...

**5**

votes

**1**answer

154 views

### Shelah's proof that proper forcing preserves P-points

In Proper and Improper forcing, VI.5; Claim 5.1 part 1 is the following:
If $F$ is a P-point in $V$, $P$ is a proper forcing notion and
$\Vdash_P `` F$ generates an ultrafilter"
Then the ultrafilter ...

**8**

votes

**1**answer

251 views

### Genericity by names

If $P$ is a notion of forcing in $M$, then $G$ is a $P$-generic filter over $M$ if $G\subseteq P$ is a filter, and for every $D\in M$ which is a dense subset of $P$, $G\cap D\neq\varnothing$.
...

**8**

votes

**1**answer

303 views

### Preservation of some stationary sets by sufficiently closed forcing

The following statement can be proven using elementary submodels and sufficiently generic conditions:
"If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, ...

**23**

votes

**2**answers

600 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 ...

**9**

votes

**1**answer

381 views

### What is known about equiconsistency of PFA and existence of supercompact cardinals?

Question: What is the last status of known partial results and new approaches on the problem of equiconsistency of PFA and existence of supercompact cardinals?

**4**

votes

**0**answers

150 views

### What is the meaning of restricting a Boolean value to a subalgebra?

$\require{AMScd}$
I am studying the proof that the ordering principle does not imply the axiom of choice in Jech's book "The Axiom of Choice" (Section 5.5). Let $P$ be the set of finite partial ...

**10**

votes

**0**answers

351 views

### Intermediate submodels without Boolean algebras

My question is motivated by the following well-known fact regarding intermediate submodels of generic extensions. I would like to know if it can be proven using posets without the need for Boolean ...

**5**

votes

**1**answer

270 views

### A question about the first Cohen model

Consider the first Cohen model, i.e. let $M$ be a countable transitive model of ZFC + $V=L$, let $\mathbb P$ be the poset consisting of finite partial functions from $\omega\times\omega$ to $2$, let $...

**4**

votes

**1**answer

174 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 $\kappa$?

**5**

votes

**1**answer

261 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

**1**answer

463 views

### V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...

**9**

votes

**1**answer

267 views

### Can “syntactic forcing” add ordinals?

Kunen, in paragraph VII.9 of his book talks about forcing "via syntactical models", where the we do not use set models of ZFC. Still, the functional $x \mapsto \check x$ can be defined as usual and is,...

**5**

votes

**1**answer

184 views

### extending elementary embeddings

Suppose $j : M \to N$ is an elementary embeddings between transitive models of ZFC. Everyone knows that if $G$ is $\mathbb{P}$-generic over $M$, $H$ is $j(\mathbb{P})$-generic over $N$, and $j[G] \...

**5**

votes

**1**answer

176 views

### Intermediate submodels and the continuum hypothesis

Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails.
Question 1. Is there a model $W$ such that:
1) $V \subseteq W \subseteq V[G],$
2) $W\models CH,$
...

**1**

vote

**1**answer

289 views

### $(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization:
Definition 1: Let $M$ be a $\...

**2**

votes

**1**answer

632 views

### Subsets of Real Numbers (Edited & Revised Version)

Question 1: Is it consistent with $\text{ZF}$ that only countable subsets of $\mathbb{R}$ are well-orderable?
Question 2: Is it consistent that for some $\lambda$, $\aleph_0 < \lambda < 2^{\...

**4**

votes

**3**answers

456 views

### Minimal Generalized Continuum Hypothesis & Axiom of Choice

It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$.
...

**5**

votes

**2**answers

316 views

### Does ZF have an initial model?

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has an "initial member" $M$ if each member of $\mathcal{K}$ is a forcing extension of $M$ for some partial order $\...

**15**

votes

**2**answers

478 views

### capturing small sets in small factors

Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:
...

**5**

votes

**2**answers

312 views

### Prevalent singular cardinals hypothesis

The following notion is introduced by Assaf Rinot:
Definition. A singular cardinal $\kappa$ is a prevalent singular
cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with $|\mathbb{A}...

**4**

votes

**2**answers

187 views

### On the number of models between a ground model and its forcing extension

Let $\kappa$ be a (finite or infinite) cardinal. Assuming consistency of $\text{ZFC}$ (and probabely some additional assumptions) is the following consistent with $\text{ZFC}$?
There is a countable ...

**3**

votes

**2**answers

161 views

### Joint Forcing Extension Property

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions $\...

**7**

votes

**1**answer

291 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 ...

**8**

votes

**2**answers

584 views

### Is the tree of large cardinals linear?

Kanamori in the introduction of his book "The Higher Infinite" says:
"The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a ...

**1**

vote

**1**answer

231 views

### Forcing Language

I asked the following question (slightly paraphrased) about a week ago in Stack exchange but no one knew the answer to the particular question. I was hoping someone here might be able to help me.
"...

**15**

votes

**4**answers

1k views

### A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...

**5**

votes

**1**answer

239 views

### $\Sigma_1$ Statements and Forcing Extensions

Assuming consistency of $\text{ZFC}$ (with some large cardinal axiom), is the following statement consistent with $\text{ZFC}$?
Any $\Sigma_1$ statement with parameters $\omega_1,\omega_2$ which ...

**4**

votes

**1**answer

260 views

### Consistency Strength of the Failure of Square on Singular Cardinals

Q1. What is the consistency strength of the failure of square on singular cardinals?
Q2. What are known as partial results in this direction?

**7**

votes

**2**answers

283 views

### Possible Choices for Cofinality of $\aleph_n$ without Choice

$\text{ZFC}$ proves that each $\aleph_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such cardinals....

**5**

votes

**1**answer

227 views

### An explicit construction of reals added after some forcing notions

Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals ...

**5**

votes

**0**answers

152 views

### Impact of Supercompacts on Measurables

It is consistent that the least measurable cardinal can carry exactly one normal measure but in almost all models for this theory there is no supercompact cardinal. It seems existence of a ...

**7**

votes

**4**answers

468 views

### A Special Pair of Models for ZFC (New Version)

Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
(1) $M\subseteq N$
(2) $\aleph_{1}^{N}=\aleph_{1}^{M}$
(3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$
Update: According to Peter's useful ...

**6**

votes

**2**answers

456 views

### The First Failure of GCH in Large Cardinals Smaller than Measurables

A well known theorem by Scott says:
If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\...

**7**

votes

**1**answer

277 views

### Countable Product of Class Forcing Notions

Is the following consistent?
There are definable class forcing notions $\lbrace \mathbb{P}_{n}\rbrace_{n\in \omega}$ such that:
The product of any finitly many of them preserves $\text{ZFC}$ and ...

**4**

votes

**0**answers

227 views

### random real forcing, independent real

Assume all independent reals that are added by random real forcing. Take enumeration of each independent real. Is the family of all enumerations dominating?

**3**

votes

**4**answers

313 views

### What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$.
It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...

**4**

votes

**1**answer

243 views

### Adding large sets not containing countable ground model sets

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension).
Before I state the question, let me add ...

**6**

votes

**2**answers

416 views

### Question about Woodin's stationary tower

Suppose that $\delta$ is a Woodin cardinal and that $\kappa$ is the critical point of the generic embedding $j:V\rightarrow M$ after forcing with the stationary tower ($\kappa$ can be $\omega_1$ or $\...

**2**

votes

**1**answer

255 views

### Are measures of a measurable cardinal measurable? (Edited and Updated Version)

Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
$\mathbb{M}_{\...

**3**

votes

**3**answers

401 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 ...