**6**

votes

**1**answer

377 views

### Can the Cohen forcing collapse cardinals?

Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...

**2**

votes

**2**answers

111 views

### proof that “small” sets in an extension by iterated forcing already appear in an earlier stage

In Kunen's book (introduction to independence proofs, ) the following lemma is proved (chapter 8, lemma 5.14):
Assume that in M, $\alpha$ is a limit ordinal,
$( ( \mathbb{P}_\xi : \xi \leq \alpha) , ...

**4**

votes

**0**answers

175 views

### Recursively Pointed Sacks Forcing and Preserving $\omega_1$

Let $\mathbb{P}$ denote recursively pointed Sacks forcing. This is forcing with recursively pointed perfect trees ordered by inclusion. A tree $T \subseteq {}^{<\omega}2$ is recursively pointed if ...

**3**

votes

**1**answer

221 views

### How to change the successor of a singular with a Woodin?

I'm looking for references on how to change the successor of a singular cardinal from "more or less" minimal assumptions. If possible, then without adding bounded subsets to the singular either.
In ...

**6**

votes

**1**answer

209 views

### Consistency of Weak Diamond with a Weak Version of Martin's Axiom

If $S \subset \omega_1$ is stationary, then the weak diamond principle $\Phi(S)$ states that for any $F: 2^{<\omega_1} \to 2$, there is a $g: \omega_1 \to 2$ such that for all $f: \omega_1 \to 2$, ...

**7**

votes

**1**answer

434 views

### Forcing is intuitionistic

The main idea of why it´s necessary a generic filter $G$ to extend a countable transitive $\epsilon$-interpretation (not necessarily a model) $M$ is given by the condition (for which $G$ being a ...

**0**

votes

**1**answer

183 views

### A Question Regarding the Powerset Size Axiom

Consider the the Powerset Size Axiom, that is, the following:
(PSA) ($\forall$x,y) |x|$\lt$|y|$\Rightarrow$$2^{|x|}$$\lt$$2^{|y|}$.
Does there exist a class $\mathscr M$ of models of ZFC such that ...

**3**

votes

**1**answer

184 views

### Does existence of $\omega_1$ subset of reals imply $\omega_1$ choice for subsets of reals?

Suppose there exists a subset of $\Bbb R$ which has cardinality $\omega_1$. Is it then necessarilly true that for every collection of $\omega_1$ subsets of $\Bbb R$ there exists a choice function?
I ...

**3**

votes

**1**answer

188 views

### Is Every New Real in the Silver Extension a Silver Generic Real?

Let $\mathbb{V}$ denote Prikry-Silver forcing. That is, $\mathbb{V}$ is forcing with partial functions $\omega \rightarrow 2$ with coinfinite domain or forcing with uniform trees.
Let $\dot x$ ...

**5**

votes

**2**answers

261 views

### Preservation Results for Iterations of Non-Proper Forcing

Suppose $\mathbb{P}$ is a forcing with the following properties: Let $G \subseteq \mathbb{P}$ be filter generic over $V$, then there exists $A \in V[G]$ such that $V[G]$ thinks $A$ is countable and $A ...

**5**

votes

**2**answers

378 views

### If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
The answer is negative, and in the interests of self-contained ...

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votes

**0**answers

185 views

### A question about $\dot{S^Q}$-semiproperness and revised countable support iterated forcing of length a limit ordinal

For a forcing notion $Q$, let $\dot{S^Q}$ be the $Q$-name for the class of ordinals $\{\kappa : \kappa = \omega_1^{V}$ $or$ $\kappa$ $is$ $a$ $regular$ $uncountable$ $cardinal \}$ in $V^Q$.
We say ...

**6**

votes

**1**answer

380 views

### Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is:
Question 1. Which different approaches to set theoretic forcing are ...

**6**

votes

**0**answers

245 views

### Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.
At most papers ...

**6**

votes

**1**answer

102 views

### $RUCar^{V}$-semiproperness implies properness

This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some ...

**16**

votes

**1**answer

664 views

### Three old questions on the Sacks forcing

I came across the two following Qs in 1970.
Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...

**9**

votes

**0**answers

183 views

### Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...

**5**

votes

**2**answers

216 views

### Iteration of Proper Forcing and Support of Master Conditions

Suppose $\mathbb{P}$ is a definable proper forcing (for instance Sacks forcing). Let $\alpha$ be some ordinal. Let $\mathbb{P}_\alpha$ be the countable support iteration of $\mathbb{P}$ of length ...

**5**

votes

**1**answer

313 views

### Relationship between fragments of the axiom of choice and the dependent choice principles

The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...

**7**

votes

**1**answer

261 views

### Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...

**0**

votes

**0**answers

145 views

### A not defined notion in Friedman's article about Generalized Fubini's Theorem

I intend to study Friedman's article, A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions (http://projecteuclid.org/download/pdf_1/euclid.ijm/1256047607). I think since I had a modern ...

**6**

votes

**1**answer

194 views

### continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...

**3**

votes

**1**answer

364 views

### Preservation of $\diamondsuit$ by ccc forcings of size $\leq \omega_1$

This is essentially exercise H8 (p.248) of Kunen's Set Theory: An Introduction to Independence Proofs (old edition), or exercise IV.7.58 (p.307) of Kunen's Set Theory (new edition).
Suppose $P$ is ...

**3**

votes

**1**answer

121 views

### Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion.
Fix some dense embedding $ i \colon P ...

**4**

votes

**1**answer

205 views

### forcing square with small conditions

In the paper, Large cardinals and definable counterexamples to the continuum hypothesis, Foreman and Magidor mention a way to force $\square_{\omega_1}$ with countable conditions. (This is used in ...

**5**

votes

**2**answers

206 views

### Borel Sets in Sacks Generic Extension

Let $\mathbb{S}$ denote Sacks forcing. This is forcing with perfect trees or equivalently forcing with uncountable Borel subsets of ${}^\omega 2$ with the relation $\subseteq$.
Let $G \subseteq ...

**5**

votes

**1**answer

330 views

### Embeddings of forcing notions - preserve properness?

Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M $ ...

**5**

votes

**2**answers

233 views

### Forcing $\neg AC$

Sorry if this sounds like a silly reference request, but I wasn't able to track down any. I'm looking for proof, via forcing, that axiom of choice can fail in a model of $ZF$. All of papers I found ...

**4**

votes

**1**answer

305 views

### Question about “Coding the universe”

The following is a result which I know as a weak form of Jensen's coding lemma$^*$ (first published in the book "Coding the universe"; also see http://www.jstor.org/stable/2273986):
For any class ...

**8**

votes

**1**answer

131 views

### Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding.
Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...

**19**

votes

**1**answer

648 views

### If all reals are generic, is the set of reals generic?

Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent:
For every $x\in\Bbb R^V$ there is some $P_x\in W$ such that for some $G\subseteq P_x$ ...

**8**

votes

**2**answers

270 views

### Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...

**8**

votes

**1**answer

343 views

### Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...

**10**

votes

**1**answer

324 views

### Elements of the method of forcing in some papers of N. N. Luzin

In the paper
Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...

**7**

votes

**1**answer

376 views

### On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...

**9**

votes

**2**answers

603 views

### Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...

**4**

votes

**1**answer

118 views

### Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model ...

**2**

votes

**1**answer

222 views

### Partial interpretation of an iteration

Suppose that $\langle\mathbb{P_\alpha,\dot Q_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing.
Let $\dot a$ be a name in $\mathbb P_\delta$, and let $G_\alpha$ be a ...

**11**

votes

**1**answer

366 views

### The independence number

I have been reading about cardinal invariants and I have a question about the independence number $\mathfrak{i}$. In Blass's paper (Combinatorial Characteristics of the Continumm) it is mention that ...

**9**

votes

**1**answer

337 views

### splitting subsets of cardinals

Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add ...

**8**

votes

**1**answer

253 views

### Destroying the Mahloness of a cardinal with $\kappa$.c.c. forcing

Question: Is it possible to have a Mahlo cardinal $\kappa$ such that there is a $\kappa$.c.c. forcing that makes it non-Mahlo?
If this is possible then this forcing must change the cofinality of all ...

**6**

votes

**1**answer

235 views

### Characterization of intermediate submodels of generic extensions

Question 1: Suppose that $V[G]$ is a set generic extension of $V$ by some forcing notion $P\in V,$ and suppose that $W$ is a model of $ZFC, V \subset W \subset V[G].$ Can we find a forcing notion ...

**2**

votes

**1**answer

158 views

### Intermediate Extensions Determined by Reals

Does there exist a forcing $P$ which adds a generic real in the sense that $V[G] = V[x]$ for some $x \in ({}^\omega\omega)^{V[G]}$, and for all reals $y \in ({}^\omega\omega)^{V[G]}$, if $V[y] \neq ...

**5**

votes

**1**answer

311 views

### Ultraproduct of Forcing Extensions & Forcing Extension of Ultraproduct

Notation:
$M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$.
$\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ...

**5**

votes

**2**answers

196 views

### absorption of strategically closed posets

It is a (folklore?) fact that if $\kappa$ is a regular cardinal, and $\mathbb{P}$ is a $\kappa$-closed poset such that $\Vdash_\mathbb{P} |\mathbb{P}| = \kappa$, then $\mathbb{P}$ is equivalent to ...

**7**

votes

**1**answer

199 views

### Preservation of ultrafilters by Sacks forcing

It is well-known that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ...

**10**

votes

**0**answers

233 views

### Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident.
Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...

**4**

votes

**1**answer

151 views

### Absoluteness between $L_\kappa$ and $L$

Working in $L$, suppose $L \models \kappa$ is a cardinal and $(\mathbb{P}, <) \in L_\kappa$. Let $\varphi(x)$ be a $\Sigma_1^1$ formula. Let $\tau \in L_\kappa$ be a $\mathbb{P}$-name for an ...

**2**

votes

**1**answer

173 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

212 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 ...