**2**

votes

**1**answer

170 views

### Some random questions about forcing

Are there more general forms of forcing, in any of the following senses?
1) The forcing adds new ordinals to $M[G]$.
2) The forcing is developed on a less or more restrictive form of $\mathbb{P}$ ...

**4**

votes

**1**answer

195 views

### Can a Measureable Cardinal Become the Least Weakly Compact Cardinal in a Forcing Extension?

I am trying to establish whether it is consistent that some property holds at the least weakly compact cardinal. I know that the property holds at measureables.
Hence (hoping everything else goes ...

**10**

votes

**1**answer

344 views

### Does small forcing preserve CH?

Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...

**10**

votes

**2**answers

634 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}, ...

**3**

votes

**1**answer

255 views

### What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech.
Does anyone know when Hechler forcing was first used in a publication?

**11**

votes

**1**answer

304 views

### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...

**8**

votes

**2**answers

276 views

### Forcing with Nontransitive Models

A common approach to forcing is to use countable transitive model $M \in V$ with $\mathbb{P} \in M$ and take a $G \in M$ (which always exists) to form a countable transitive model $M[G]$. Another ...

**5**

votes

**1**answer

260 views

### $\omega$ universally Baire sets, tree representations

I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me.
...

**15**

votes

**0**answers

597 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

347 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

139 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?

**4**

votes

**1**answer

194 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

482 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

110 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

186 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

224 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

155 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

309 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

503 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

322 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

270 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

165 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

343 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

185 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

268 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

325 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

174 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

333 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

709 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

131 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

243 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

283 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

557 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

321 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

143 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

342 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

236 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

163 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

250 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

386 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

238 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

169 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

165 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

267 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

586 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

388 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

369 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:
...

**4**

votes

**2**answers

287 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 ...