**5**

votes

**1**answer

174 views

### Why relative consistency results by forcing arguments are provable in finitistic metatheory

It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....

**3**

votes

**1**answer

122 views

### Rigid structure which is generically homogeneous

Is it possible to have a structure $T$ in some language which is rigid in $V$, but in a cardinal-preserving extension $T$ is homogeneous (in a suitable sense of the word)?
If this is not possible, is ...

**10**

votes

**1**answer

311 views

### When are generic models not too wild?

This is a question related to ideas raised in http://arxiv.org/abs/1410.1224 and http://arxiv.org/pdf/1405.7456.pdf. Basically, the idea is the following:
Suppose I have a first-order theory $T$. ...

**6**

votes

**1**answer

262 views

### moving up a consequence of PFA

The Proper Forcing Axiom (PFA) implies that every forcing which adds a subset of $\omega_1$ either adds a real or collapses $\omega_2$. Is it consistent that every forcing which adds a subset of $\...

**7**

votes

**1**answer

203 views

### $\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings

Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.)
I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is ...

**3**

votes

**1**answer

242 views

### Representation of meager sets in Cohen extensions

Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $...

**6**

votes

**0**answers

127 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

**1**answer

200 views

### Theorem of Bukovsky characterizing ground models

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$:
(1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in V$...

**9**

votes

**1**answer

298 views

### Two questions about higher Souslin trees

Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$
It is known that in the extension there are ...

**6**

votes

**1**answer

203 views

### Fat stationary sets

Recall a stationary subset $S$ of a regular cardinal $\kappa$ is fat when for every $\alpha < \kappa$, and every club $C$, there is a closed set of order type $\alpha$ contained in $S \cap C$. It ...

**5**

votes

**1**answer

235 views

### Iteration of random reals

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\...

**5**

votes

**1**answer

223 views

### A button for individual reals

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...

**10**

votes

**0**answers

229 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 $j(\omega_1) = \omega_n^V$ ...

**11**

votes

**1**answer

402 views

### Changing cofinalities above supercompact cardinals

Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$
is measurable (or even larger large cardinal if necessary).
Is there a set generic extension of the universe in ...

**11**

votes

**1**answer

338 views

### Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have:
(1) $\kappa$ is singular in $V$ of cofinality $\omega,$
(2) $\kappa$ is regular (and in fact ...

**9**

votes

**1**answer

285 views

### Largeness and arithmetic progression properties of generic reals

Consider the following properties for a subset $A$ of $\mathbb{N}$:
(1) $A$ is large: $\sum_{n \in A}$$ 1\over n$$=\infty,$
(2) $A^\infty=\limsup \frac{|A \cap \{ 1, \dots, n\}|}{n} >0$,
(3) $A_\...

**1**

vote

**1**answer

155 views

### Absoluteness and Tree Representations

Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional ...

**5**

votes

**1**answer

216 views

### A Question on HOD, V and GCH

The theorem 1.1 of the following paper:
Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic.
states that:
Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\...

**5**

votes

**0**answers

166 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

**1**answer

271 views

### Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...

**14**

votes

**1**answer

251 views

### Is the Martin's axiom number $\mathfrak m$ regular

The Martin's axiom number $\mathfrak m$ is the least cardinal $\kappa$ for which $\text{MA}_\kappa(\text{ccc})$ is false, i.e. the least cardinal such that there exists a ccc poset $P$ and a family $\...

**12**

votes

**1**answer

830 views

### Reverse-engineer forcing: am I reinventing the wheel?

In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but
I have a sinking feeling I’m reinventing the wheel; does ...

**7**

votes

**1**answer

187 views

### Can Sacks forcing add a Cohen generic real over $L$?

Motivated by this question Forcing the negation of CH without adding Cohen reals over L and Todd Eisworth's comment, the question is the following:
1) Suppose $V$ has no Cohen generic reals over $L$. ...

**6**

votes

**1**answer

301 views

### Forcing the negation of CH without adding Cohen reals over L

Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?

**4**

votes

**0**answers

145 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 $\...

**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 $\kappa+ 2^\kappa > \kappa^+$)?

**12**

votes

**1**answer

258 views

### When can Power Sets be Limit Cardinals?

My original question (posted in http://math.stackexchange.com/questions/1584430/can-all-power-sets-be-limit-cardinals) was:
Is it possible to create a model of ZFC, so that the cardinality of each ...

**5**

votes

**0**answers

109 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 ...

**6**

votes

**1**answer

266 views

### How many closed measure zero sets are needed to cover the real line, really?

This is a refinement of an earlier question.
This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
For the reader's convenience, I reproduce below the ...

**8**

votes

**1**answer

384 views

### How many closed measure zero sets are needed to cover the real line?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It ...

**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 ...

**8**

votes

**1**answer

180 views

### Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$?

Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values ...

**3**

votes

**1**answer

223 views

### When can you canonically extend an ultrafilter after forcing?

Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes.
Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...

**4**

votes

**0**answers

176 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, \mathcal{P})...

**8**

votes

**1**answer

264 views

### Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$?

Is it consistent that there exists an inaccessible cardinal $\lambda$ and a forcing extension $V[G]$ so that $$V[G]\models\text{There is some non-trivial elementary embedding $j:V_{\lambda}^{V[G]}\...

**8**

votes

**1**answer

270 views

### Transitive models and CH

The following was asked on stackexchange but I think it also belongs here:
http://math.stackexchange.com/questions/1513446/transitive-models-and-ch
Suppose $M, N$ are two countable transitive models ...

**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 $\...

**11**

votes

**1**answer

414 views

### Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits
automatic mutual genericity, if whenever $G,H\subseteq\Q$ are
distinct $V$-generic filters (existing, say, in some forcing
extension ...

**6**

votes

**1**answer

211 views

### Elementary chains in forcing extensions of $M_1$

Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...

**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 ...

**6**

votes

**0**answers

168 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 ...

**26**

votes

**0**answers

575 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 ...

**11**

votes

**1**answer

473 views

### Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:
$\Bbb P_\alpha$ is c.c.c.
$\Bbb P_\alpha$ adds a real ...

**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, ...

**12**

votes

**1**answer

252 views

### Forcings that are not equivalent to Levy collapse

Assume GCH and that $\kappa$ is a regular uncountable cardinal. Let $\mathbb{P}$ be a separative, $<\kappa$-directed closed, nowhere trivial, $\kappa^+$-cc poset of size $\kappa^+$. Must $\mathbb{...

**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 ...

**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 ...

**5**

votes

**1**answer

208 views

### A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable

Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:
given any function $f$ from $\mathbb{R}$ into the families of of subsets ...

**14**

votes

**2**answers

556 views

### Who needs RCS iterations?

According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more ...

**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 $V$.
...