**15**

votes

**0**answers

587 views

### Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...

**8**

votes

**2**answers

341 views

### The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...

**7**

votes

**1**answer

138 views

### On the definition of $\alpha$-proper poset

I am reading Uri Abraham's chapter on Proper Forcing in the Handbook of Set Theory and I have a quite trivial question on the definition of $\alpha$-proper forcing. Since there are many equivalent ...

**8**

votes

**2**answers

1k views

### Is there a Hotel California of set-theoretic geology?

Is there a universe which can always be forced to, which never can be forced from?

**3**

votes

**1**answer

193 views

### Adding Generic Reals to Forcing Extensions

I'm following the Jech's Multiple Forcing for a seminar group and I intend to show how to add one or some reals to extensions.
I studied Solovay's model and I can see why learning how to add random ...

**16**

votes

**2**answers

481 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

**2**

votes

**0**answers

108 views

### Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse.
Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...

**6**

votes

**1**answer

181 views

### Failure of Cantor-Bernstein for the Levy Collapse

Related to this question, is it possible to give an example of the failure of Cantor-Bernstein for complete embeddings of forcing notions involving the Levy Collapse $Col(\omega,<\kappa)$? Suppose ...

**2**

votes

**1**answer

220 views

### Forcing Notions with Unknown Real/Cardinal Preserving Situations

Question. Is there any set forcing notion $\mathbb{P}$ in one of the following categories?
(a) $\mathbb{P}$ preserves cardinals but it is still open whether $\mathbb{P}$ adds reals or not.
(b) ...

**5**

votes

**1**answer

151 views

### Function Approximation in c.c.c Forcings without AC in Ground Model

Consider the following basic theorem.
Theorem. If $M$ is a c.t.m of ZFC and $\mathbb{P}$ a c.c.c forcing notion in $M$ and $G$ a $\mathbb{P}$ - generic filter on $M$ then for all $A,B$ in $M$ and for ...

**1**

vote

**3**answers

304 views

### Can the structure of an ultrafilter determine the structure of its ultrapower?

Usually we work with ultrafilters as pure sets without any structure.
Q1. Is there any important notion of structure on an ultrafilter?
Q2. Is there any non-trivial notion of structure on ...

**6**

votes

**2**answers

488 views

### Is the forcing relation defined for mathematical formulas?

Meta-matematical formulas of the language of set-theory (which are not sets, but just sequences of signs) should not be confused with mathematical ones (i.e. formulas coded as sets, e.g. finite ...

**4**

votes

**2**answers

312 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

260 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

159 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

336 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

180 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 ...

**10**

votes

**1**answer

244 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

323 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

172 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 ...

**9**

votes

**3**answers

323 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

688 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

130 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

238 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

279 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

548 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 ...

**8**

votes

**1**answer

314 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

142 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

340 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

234 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

157 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

247 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

374 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

229 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 ...

**5**

votes

**1**answer

167 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

164 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

261 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

579 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 < ...

**4**

votes

**3**answers

369 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

310 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 ...

**12**

votes

**1**answer

358 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

284 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 ...

**4**

votes

**2**answers

170 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

156 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 ...

**8**

votes

**1**answer

232 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

468 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 ...

**2**

votes

**1**answer

220 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.
...

**14**

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

217 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

251 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?