**9**

votes

**0**answers

233 views

### Proving regularity properties from forcing axioms

It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...

**9**

votes

**1**answer

343 views

### singularize the least inaccessible?

Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while ...

**8**

votes

**1**answer

403 views

### Adding a real with infinite conditions

Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.
Does $\Bbb P$ ...

**7**

votes

**1**answer

201 views

### Generic filters of inverse limits

Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper Fragile measurability, Journal of Symbolic Logic 59 (1994) 262-282.
There, $\mathbb ...

**4**

votes

**0**answers

126 views

### a game with generic filters

The present question is a follow-up to this one. Assume GCH holds in $V$ and suppose $G \subseteq Add(\omega_1,\omega_2)$ is generic over $V$. For any $g \subseteq Add(\omega_1,\alpha)$ in $V[G]$ ...

**6**

votes

**0**answers

199 views

### Core model for supercompact cardinals and iteration trees

I have a few somehow related questions:
Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...

**13**

votes

**3**answers

556 views

### Products of Cohen forcings

Let $\lambda$ be an infinite cardinal. Does the full support product of $\lambda$ copies of $Add(\omega, 1)$ collapse $2^\lambda$ to $\aleph_0$?
For $\lambda = \omega$, it is known to be true (it is ...

**14**

votes

**2**answers

520 views

### Preservation of properness

Suppose $\mathbb P$ is a countably closed forcing, and $\mathbb Q$ is a c.c.c. forcing that adds reals. Is $\mathbb P$ still proper in $V^{\mathbb Q}$?

**3**

votes

**1**answer

281 views

### Some questions on (non)-measurable sets without AC

In his answer to a Math Stack Exchange question of Katlus, Asaf Karagila wrote the following:
"It is a theorem that from $ZF+DC+$"$\aleph_1$$\le$$|$$\mathbb R$$|$" we can prove that there is an ...

**11**

votes

**1**answer

239 views

### counterexample regarding quotient algebras in forcing

Suppose $A$ and $B$ are complete subalgebras of a complete boolean algebra $C$. Let $G \subseteq A$ be generic. In the extension $V[G]$, we can define the quotient algebras $B/G$ and $C/G$ in the ...

**26**

votes

**5**answers

1k views

### Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end ...

**6**

votes

**0**answers

263 views

### A new cardinality living in every forcing extension?

This question is motivated by the papers http://arxiv.org/abs/1405.7456 and http://arxiv.org/abs/1410.1224.
Say that a set $X$ is "generically presentable" over $V\models ZF$ if there is some ...

**1**

vote

**1**answer

216 views

### Sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$

Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish ...

**4**

votes

**4**answers

1k views

### Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...

**9**

votes

**0**answers

186 views

### Preserving Jonsson cardinals

I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ...

**11**

votes

**1**answer

592 views

### Is forcing computable?

By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...

**34**

votes

**1**answer

938 views

### Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...

**11**

votes

**2**answers

356 views

### Semiproper but not proper

Assume V=L. Is there a semi-proper notion of forcing that is not proper?
Namba forcing isn't semi-proper in L, and Prikry forcing isn't even available there.

**3**

votes

**1**answer

178 views

### CCC Forcing and $\omega_1$ conditions

I have a question about the proof of the Lemma 7.2 in the paper
I. Juhász, P. Koszmider and L. Soukup,
A first countable, initially $\omega_{1}$-compact but non-compact space,
Topology and its ...

**2**

votes

**1**answer

144 views

### A question regarding forcing extensions

Can one, for an infinite set A in ZFC, use forcing to add so many generic subsets of A as to make the collection of all subsets of A a proper class? Consider now a model $M$ of ZFC and use $Add( , ...

**1**

vote

**2**answers

148 views

### A Question Regarding Defining Generic Extensions of ZF and ZFC in Morse-Kelly Set Theory

It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example:
MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) ...

**7**

votes

**2**answers

381 views

### When can we reach a real by forcing?

I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such ...

**2**

votes

**1**answer

243 views

### At what level of the analytic hierarchy do Cohen reals lie?

In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:
i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only ...

**5**

votes

**2**answers

397 views

### Gorelic's Forcing for large Lindelöf spaces with points $G_\delta$

I am trying to understand a step for proving that there exists large Hausdorff Lindelöf Spaces with points $G_\delta$ using forcing. I am following Isaac Gorelic's "The Baire Category And Forcing ...

**8**

votes

**2**answers

382 views

### centeredness in forcing iterations

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we ...

**8**

votes

**1**answer

336 views

### A question about three forcings

Let $P_1$ be the finite support iteration of random forcings of length $\omega$. Let $P_2 = \text{Random} \times \text{Random}$ and $P_3 = \text{Random} \times \text{Cohen}$. Are $P_i, P_j$, for $1 ...

**8**

votes

**1**answer

307 views

### On $V$-decisive and weakly homogeneous forcings

Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...

**9**

votes

**1**answer

318 views

### Canonical functions in set theory and their applications

Given regular cardinal $\kappa>\omega,$ we can define the canonical functions $f_\alpha: \kappa\to \kappa,$ for $\alpha<\kappa^+.$
Some of their properties are presented in Chapter 22 of the ...

**6**

votes

**1**answer

422 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

116 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

180 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

227 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

219 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

472 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

187 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

197 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

193 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

274 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

390 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 ...

**4**

votes

**0**answers

191 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

389 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 ...

**7**

votes

**0**answers

252 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

105 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

680 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

196 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

225 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

329 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

264 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 ...

**6**

votes

**1**answer

200 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

371 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 ...