**26**

votes

**0**answers

562 views

### Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...

**16**

votes

**0**answers

712 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 ...

**12**

votes

**0**answers

285 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 ...

**12**

votes

**0**answers

459 views

### Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...

**11**

votes

**0**answers

201 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 ...

**10**

votes

**0**answers

227 views

### A variant of strong ideals, is it consistent?

Is it consistent relative to large cardinals that there is a precipitous ideal on $\omega_1$ forcing a generic elementary embedding $j : V \to M \subseteq V[G]$, such that $j(\omega_1) = \omega_n^V$ ...

**10**

votes

**0**answers

155 views

### Homogeneity of a variant of Prikry forcing

Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...

**10**

votes

**0**answers

240 views

### When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$
satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.
...

**10**

votes

**0**answers

311 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 ...

**10**

votes

**0**answers

351 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 ...

**10**

votes

**0**answers

425 views

### Existence of a regular subposet which collapses everything except the top cardinal

Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < ...

**9**

votes

**0**answers

157 views

### Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...

**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

**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. ...

**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 ...

**9**

votes

**0**answers

182 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 ...

**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 ...

**8**

votes

**0**answers

269 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 ...

**8**

votes

**0**answers

179 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 ...

**7**

votes

**0**answers

151 views

### $V$ as a $HOD$ of its class generic extension

By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$.
This result is also stated and generalized ...

**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 ...

**7**

votes

**0**answers

349 views

### Sets of reals amenable to each L[x]

If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$?
This can be proved under the Axiom of ...

**6**

votes

**0**answers

125 views

### Forcing in GBC, the ctm approach

There is a nice, detailed survey about forcing in GBC in the appendix of the dissertation of Jonas Reitz. At page 115 the author wrote: " If $ \Gamma $
is a finite collection of sentences forced by $ ...

**6**

votes

**0**answers

190 views

### Adding minimal subsets to $\aleph_\omega$

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...

**6**

votes

**0**answers

166 views

### PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and Short ...

**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, ...

**6**

votes

**0**answers

160 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 ...

**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

**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 ...

**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 ...

**6**

votes

**0**answers

272 views

### What are these sets in Freyd's model?

Recall Freyd's model of $ZF +\neg AC$ (as recounted in MacLane and Moerdijk's book Sheaves in Geometry and Logic): it arises as the Fourman interpretation of the topos of sheaves on a particular ...

**6**

votes

**0**answers

322 views

### Maps between forcing posets

We all know that forcing can be seen (if you like things that way) as a category of sheaves over the poset of forcing conditions equipped with the double negation Grothendieck topology. As such it is ...

**6**

votes

**0**answers

434 views

### A proposed axiom of Laver (updated)

A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom:
(L) Some elementary embedding ...

**6**

votes

**0**answers

317 views

### PFA and the compactness/incompactness of $\omega_2$

There are many consequences of the Proper Forcing Axiom (PFA) which essentially say that $\omega_2$ has a lot of compactness.
For instance, Matteo Viale and Christoph Weiss have a few papers in ...

**5**

votes

**0**answers

165 views

### Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic:
(1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...

**5**

votes

**0**answers

130 views

### Failure of GCH at a strongly compact cardinal

Does Con(ZFC+ there exists a strongly compact cardinal) imply Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)?

**5**

votes

**0**answers

108 views

### Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of
quantum mechanics. On the positive side it demonstrates how the probabilistic
structure of quantum theory follows from its logical ...

**5**

votes

**0**answers

130 views

### Sacks minimality without choice

The usual argument for the minimality result for Sacks forcing uses choice.
Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where ...

**5**

votes

**0**answers

152 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 ...

**4**

votes

**0**answers

143 views

### Two questions about the behavior of the continuum function

The first question asks about the global behavior of the power function in the case of finite gaps.
Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let ...

**4**

votes

**0**answers

175 views

### Theories introduced by a class of forcing notions

The following notion is introduced by Mohammad Golshani. Let $V$ be a model of set theory and let $\mathcal{P}$ be a class consisting of non-trivial forcing notions in $V$. Let
$$Th(V, ...

**4**

votes

**0**answers

245 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

**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 $V[G]$ ...

**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 ...

**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 ...

**4**

votes

**0**answers

150 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 ...

**4**

votes

**0**answers

226 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

**0**answers

99 views

### A question regarding forcing in $NGBC^{-f}$+$BAFA$

Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can ...

**3**

votes

**0**answers

190 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, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, ...

**3**

votes

**0**answers

278 views

### On Successive Regular Cardinals With No Ladders

Definition: Let $\kappa$ be an $\aleph$ cardinal, we say that $\langle f_\alpha\colon\alpha\to\kappa\mid\alpha<\kappa^+\rangle$ is a ladder if every $f_\alpha$ is injective.
Equivalently this is ...