**9**

votes

**1**answer

240 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

171 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

167 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

269 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

590 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

393 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

311 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

370 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

288 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

174 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

157 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

261 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

490 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

223 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

225 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?

**6**

votes

**2**answers

258 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 ...

**5**

votes

**1**answer

207 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

150 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

464 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

393 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 ...

**7**

votes

**1**answer

264 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

208 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

279 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

235 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

363 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

225 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:
...

**3**

votes

**3**answers

388 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 ...

**4**

votes

**1**answer

179 views

### what kind of ordinal is the degree of strongness of a partially strong cardinal (Edited and revised)

For an infinite cardinal $\kappa$ and an ordinal $\lambda>\kappa,$ $\kappa$ is called $\lambda-$strong, if there is a non-trivial elementary embedding $j: V \rightarrow M$ with $crit(j)=\kappa$ ...

**4**

votes

**2**answers

310 views

### Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
Definition (1): If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is ...

**7**

votes

**1**answer

193 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 ...

**5**

votes

**1**answer

194 views

### Can we always permute Cohen reals?

Consider the Cohen forcing, and suppose that $\dot x,\dot y$ are names for reals, which are not in the ground model (i.e. $1$ forces that neither is in the ground model).
Can we always find an ...

**8**

votes

**1**answer

161 views

### Why does an $\varepsilon$-incomplete $Q$ remain so in the forcing extension of any $\varepsilon$-complete $P$?

This is Shelah's simpler notion of 'completeness' for not adding reals via forcing. Here $P$ and $Q$ are forcing notions.
Let $S_{\aleph_0}(\kappa)$ be the set of all countable subsets of a cardinal ...

**5**

votes

**1**answer

310 views

### How Random is Cohen?

Suppose that $V$ is a universe of $\sf ZFC$, and $c$ is a Cohen generic real over $V$. Is it possible that $c$ is also generic in other senses? I know that it can't be random or Sacks or whatnot ...

**13**

votes

**6**answers

1k views

### The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...

**6**

votes

**2**answers

159 views

### Separation of almost disjoint families by ground model almost disjoint families

Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model.
Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ ...

**14**

votes

**3**answers

551 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 ...

**7**

votes

**6**answers

527 views

### Applications of forcing in model theory

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

**7**

votes

**1**answer

325 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?

**17**

votes

**1**answer

692 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 ...

**5**

votes

**1**answer

148 views

### Generalization of Levy-Solovay theorem to kappa-c.c. forcings

The Levy-Solovay theorem says that if $\kappa$ is measurable, then it remains measurable in the extension by a small forcing ($|\mathbb{P}|<\kappa$). Is still true if we replace ...

**6**

votes

**1**answer

668 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

293 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

340 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

151 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

267 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

259 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

494 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 ...