**6**

votes

**0**answers

241 views

### The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...

**3**

votes

**1**answer

132 views

### Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...

**6**

votes

**0**answers

151 views

### How “small” can an ordinal be made by forcing?

I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ...

**8**

votes

**0**answers

217 views

### On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and ...

**3**

votes

**1**answer

275 views

### On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$.
What I want to know is if then the following ...

**5**

votes

**1**answer

182 views

### Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness

It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...

**2**

votes

**0**answers

129 views

### RCS iteration such that the RCS limit is semi-proper

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that
...

**6**

votes

**0**answers

164 views

### rigidity of $\mathcal P(\omega_1) / NS$ under MA

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...

**6**

votes

**1**answer

197 views

### Reverse of a termspace forcing fact

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable ...

**11**

votes

**1**answer

265 views

### Does stationary reflection imply Mahloness?

Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo?
Remarks:
It is possible for every stationary subset of $\kappa$ to reflect, but ...

**4**

votes

**1**answer

187 views

### Can we always add sets without collapsing cardinals or adding [very] bounded sets?

Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that:
$\Bbb P$ does not add sets of rank ...

**6**

votes

**2**answers

424 views

### Collapsing the cardinals between two singular cardinals

Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$?
If the ...

**0**

votes

**1**answer

262 views

### Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:
"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...

**8**

votes

**1**answer

333 views

### A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists
$\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that
there is no partition of ...

**11**

votes

**1**answer

255 views

### Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.
$\mathbb{P}$ preserves stationary subsets ...

**5**

votes

**1**answer

213 views

### References for Forcing with Side Conditions

I'm looking for some good references about Forcing with Side Conditions, including expository papers that explain the main ideas with some details in order to give me a fairly clear insight of those ...

**8**

votes

**0**answers

268 views

### preserving saturated ideals

A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...

**1**

vote

**1**answer

151 views

### $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$

Let $\kappa$ be an uncountable regular cardibnal. Consider the finite support iteration $(\langle \mathbb{P}_{\alpha}\rangle _{\alpha \leq \kappa},\langle \mathbb{\dot{Q}}_{\alpha}\rangle _{\alpha ...

**9**

votes

**2**answers

531 views

### Preserving $\omega_1$ is Inaccessible to the reals

$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$.
The question is if $\omega_1$ is inaccessible to the reals in $V$ and ...

**11**

votes

**0**answers

199 views

### Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...

**9**

votes

**1**answer

402 views

### Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...

**12**

votes

**0**answers

282 views

### Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...

**10**

votes

**0**answers

264 views

### c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any ...

**6**

votes

**1**answer

333 views

### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...

**7**

votes

**1**answer

200 views

### Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is ...

**11**

votes

**1**answer

313 views

### minimal collapsing without GCH

Suppose $\kappa$ is a regular cardinal. Does there necessarily exist a poset $\mathbb P$ that collapses $\kappa^+$ while preserving all other cardinals?

**24**

votes

**1**answer

1k views

### How hard is it to destroy a diamond? (with a real)

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...

**9**

votes

**1**answer

224 views

### Does $Add(\kappa,1)^L$ ever collapse cardinals?

In general, we know that adding a subset to a regular cardinal $\kappa$ can collapse cardinals. If, for example, there is $\gamma < \kappa$ with $2^\gamma >\kappa$, then $Add(\kappa,1)$ will ...

**6**

votes

**1**answer

164 views

### completions of regular suborders

Suppose $\mathbb{P}$ is a regular suborder of the separative partial order $\mathbb{Q}$ (see below for definitions). Must there always exist some complete boolean algebra $\mathbb{B}$ such that:
...

**6**

votes

**1**answer

250 views

### $\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations:
$\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$;
$\mathfrak{b}$ is the bounding ...

**5**

votes

**1**answer

206 views

### Universally Baire Tree Representation of Projective Sets

In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...

**1**

vote

**0**answers

133 views

### What can be said about a Boolean-valued structure from what the Boolean-valued forcing extension thinks about it?

Suppose that $\phi$ is a formula in the language of set theory such that
there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and ...

**1**

vote

**1**answer

242 views

### The patterns of possibility for nontrivial automorphisms and nontrivial elementary embeddings of the universe

In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the ...

**3**

votes

**0**answers

188 views

### What algebraic identities does the iteration of forcing operation satisfy?

Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, ...

**4**

votes

**0**answers

244 views

### Is there some absoluteness between the Boolean valued universe $V^{B}$ and $V$?

It is well known that if $\phi$ is a $\Delta_{1}$-formula and $x_{1},..,x_{n}$ in $V$ and $V[G]$ is a forcing extension, then $V\models\phi(x_{1},...,x_{n})$ if and only if ...

**4**

votes

**1**answer

141 views

### $\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$

Asumme tha in $M$, $CH$ holds and $\kappa>\aleph_0$ and $\kappa^{\aleph_0}=\kappa$. Let $K$ be $Fn(\kappa,2)$-generic over $M$.
Question:
Then we can say in $M[K]$ that:
$(i)$ ...

**2**

votes

**1**answer

121 views

### “Namba forcing adds reals” independent of $ZFC + \neg CH$?

I know that, in the presence of $CH$, Namba forcing does not add reals. But when $CH$ fails, is it consistent that it still does not add reals?

**14**

votes

**2**answers

675 views

### Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...

**8**

votes

**2**answers

272 views

### Is there a version of Miller forcing “guided by” an ultrafilter?

It’s well known that Mathias forcing factors as a two-step iteration $P(\omega)/\mathrm{fin}\ast \mathbb{M}_{\dot U}$, where $\mathbb{M}_{\dot U}$ is Mathias forcing guided by the generic ultrafilter ...

**8**

votes

**0**answers

178 views

### History of preservation theorems in forcing theory

For my honours thesis, I am studying a general preservation theorem using a framework provided by Shelah. I am mainly concerned about revised countable support iteration of $\dot{S}$-semiproper ...

**6**

votes

**1**answer

470 views

### Adding sets not containing arithmetic progressions of length three by forcing

Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where:
1) $s\in 2^{<\omega}$,
2) $N\in \mathbb{N}$,
3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...

**6**

votes

**1**answer

287 views

### Tree property and singular strong limit cardinals

I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question:
Theorem. It is consistent, relative to the existence of large cardinals, that ...

**-2**

votes

**1**answer

232 views

### Forcing and $\mathbb{P}$-name [closed]

If $\mathbb{P}$ be a poset, $\dot{Q}$ a $\mathbb{P}$-name and $\mu$ an infinite cardinal such that $\Vdash 0<|\dot{Q}|\leq\mu$.
$(a)$ Exist names $\langle \dot{q}_\alpha\rangle_{\alpha<\mu}$ ...

**1**

vote

**1**answer

253 views

### Confusion with proof about a fact $\mathbb{P}$-name [closed]

Let $\mathbb{P}$ be poset.
Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function ...

**4**

votes

**1**answer

152 views

### presaturated ideals

In this paper, Gitik and Shelah make the following claim (part of Proposition 1.5):
Claim (Gitik-Shelah): Suppose $\kappa < \lambda$ are regular, $2^\lambda = \lambda^+$, and $D$ is a normal ...

**5**

votes

**2**answers

654 views

### A specific Model of ZFC

In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...

**4**

votes

**2**answers

327 views

### Proving results about complete Boolean algebras in ZFC using Boolean valued models

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly ...

**4**

votes

**1**answer

103 views

### The GCH in a reverse Easton support iteration

I am trying to understand the proof that the GCH can first fail at a weakly compact cardinal. We assume the GCH and that there exists a weakly compact cardinal $\kappa$, and we construct a reverse ...

**5**

votes

**1**answer

341 views

### stationary tower forcing

It is known that if $\delta$ is a Woodin cardinal and $\kappa < \delta$, then the stationary tower forcing $\mathbb Q^\kappa_{<\delta}$ preserves cardinals up to $\kappa$ and forces $\delta = ...

**6**

votes

**1**answer

163 views

### Pseudo-Prikry sequences vs Prikry sequences

Definition: Let $V\subseteq W$ be two transitive models of $ZFC$. A pseudo-Prikry sequence, $s$, at a cardinal $\kappa$ for $(V, W)$ is an $\omega$-sequence, cofinal at $\kappa$ such that for every ...