# Tagged Questions

**2**

votes

**0**answers

422 views

### Boolean-Valued Models vs. the Infinite-valued Logic of Lukasiewicz and set theory

Is anyone familiar with an old paper of C.C. Chang entitled "The Axiom of Comprehension in Infinite-Valued Logic" which shows that the Axiom of Comprehension without parameters is consistent in the ...

**-1**

votes

**1**answer

279 views

### distibution of truth values of all formulas on [0,1]

If one forces with measure algebra, then every formula $\phi(\tau_{1},\tau_{2},...,\tau_{k})$ where $\tau_{i}$'s are names, has a truth value, the measure of which is a real number. It is just a ...

**8**

votes

**2**answers

434 views

### Finite support iterations of $\sigma$-centered forcing notions

I am looking for a proof (or better, a reference) of the following fact:
The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than ...

**6**

votes

**1**answer

345 views

### GCH+ Kurepa Families

I have a couple of questions about known theorems for GCH+Kurepa families.
Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ ...

**5**

votes

**1**answer

193 views

### Permutation models with a class-sized group

I'm working on building a model of ZF where a very weak choice principle fails, in so much as in any permutation model with a set of atoms it is inconsistent that this principle fails.
To get a feel ...

**6**

votes

**0**answers

261 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 ...

**9**

votes

**1**answer

516 views

### Definability of ground model

I have seen mentioned that by a result of Laver, the ground model is definable in any set forcing extension (using parameters).
Does the same hold for class forcing? If it does, in order to establish ...

**3**

votes

**1**answer

415 views

### Forcing and divisibility

A version of this question got a couple of comments but no answer on stackexchange.
I learned the concept of forcing in logic from Boolos & Jeffrey's book Computability and Logic (second edition, ...

**3**

votes

**1**answer

327 views

### Universe-sized groups with only set-sized normal subgroups, their cardinality in a certain range

Let $\kappa$ be an inaccessible cardinal, and let $G$ be a group with $|G| \geq \kappa$. For any cardinal $\lambda \le \kappa$ (regular, say, but not necessary), say $G$ is $\lambda$-simple if for all ...

**6**

votes

**2**answers

459 views

### Natural statements independent from true $\Pi^0_2$ sentences

I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...

**5**

votes

**2**answers

476 views

### Forcing the nonexistence of a certain set

I have a certain set-theoretic axiom (WISC) which follows from Choice (this is a nuking a fly BTW), but which I suspect is independent of ZF. To show this I need to show that a certain set does not ...

**4**

votes

**1**answer

590 views

### Some questions from the paper “Forcing the failure of CH by adding a real” by Shelah and Woodin

1-Let $P=Add(\omega_1, \kappa)$, and let $D$ be a dense open subset of $P$. Then there is a dense subset $S$ of $D$ such that for every $f \in S$ and any $g \in P$, if $domf=domg$ and the set $\{ ...

**3**

votes

**1**answer

327 views

### random real forcing

Let $\kappa$ be an infinite cardinal and let $B$ be the random forcing for adding $\kappa-$many random reals.
what are the elements of $B$.More precisely given a condition $p \in B$, what are the ...

**11**

votes

**4**answers

875 views

### Cantor-Bernstein for notions of forcing

For forcing notions $\mathbb{P}$ and $\mathbb{Q}$ let us write $\mathbb{P}\triangleleft\mathbb{Q}$ if forcing with $\mathbb{Q}$ always adds a $\mathbb{P}$-generic filter over $V$. In other words, ...

**9**

votes

**3**answers

446 views

### A model of CH +$\lnot \diamondsuit$

All of the models of CH which I know of also satisfy $\diamondsuit$. What is the easiest way to produce a model of CH wherein $\diamondsuit$ is false?

**3**

votes

**3**answers

388 views

### “name” for the ground model

I am a beginner of forcing, often I read from some articles something like "$p \Vdash \dot{G}$ is $P$-generic over $\check{M}$" (where $M$ is a countable transitive model, for instance).
Q1. I ...

**1**

vote

**2**answers

247 views

### Substructure Argument for Chain Conditions

Once, someone showed me a nice argument using elementary substructures for proving chain conditions about forcings. It was a basic example, maybe that Cohen forcing is ccc. I've been trying to look up ...

**11**

votes

**2**answers

527 views

### Can we collapse $\omega_1$ to $\omega$ without adding a dominating real?

(Disclaimer: This question was also asked at MSE (http://math.stackexchange.com/questions/71020/can-we-collapse-omega-1-without-adding-a-dominating-real). I'm posting it here because, when I asked it, ...

**3**

votes

**1**answer

227 views

### Is the ordering principle preserved in generic extensions?

The ordering principle says that every set can be linearly ordered.
In a previous question Why are some axioms preserved in generic extensions? Asaf Karagila asked which axioms are preserved in ...

**5**

votes

**2**answers

646 views

### Why are some axioms preserved in generic extensions?

It is a known theorem that for a model of $ZF$, $M$, if $M\models AC$ and $G$ is a $P$-generic filter over $M$, for some $P\in M$, then $M[G]\models AC$.
On the other hand, it is long known that ...

**6**

votes

**0**answers

296 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 ...

**7**

votes

**0**answers

422 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 ...

**10**

votes

**4**answers

861 views

### Forcing over models without the axiom of choice

In the vast majority of papers forcing is always developed over ZFC.
Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain ...

**6**

votes

**0**answers

262 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 ...

**6**

votes

**1**answer

357 views

### Symmetric extensions and class forcing

Suppose $V\models ZFC$ and $P\in V$ is a poset of forcing conditions.
It is a basic theorem in forcing that $V[G]\models ZFC$ for any generic extension by a $V$-generic filter $G$.
It is also known ...

**12**

votes

**3**answers

707 views

### A limit to Shoenfield Absoluteness

Shoenfield's Absoluteness Theorem states that if $\phi$ is any $\Sigma^1_2$ sentence of second-order arithmetic, then $\phi$ is absolute between any two models of $ZF$ which share the same ordinals. ...

**7**

votes

**1**answer

421 views

### complete embeddings of boolean algebras and preservation of stationarity

Define a complete embedding of Boolean algebra as an homomorphism of Boolean algebras which preserves also the sup and inf operations. Notice that if $\mathbb{B}$ and $\mathbb{D}$ are complete boolean ...

**4**

votes

**2**answers

399 views

### Mutually generics

Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of ...

**9**

votes

**2**answers

590 views

### Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem:
Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...

**3**

votes

**3**answers

491 views

### For a given partial order, how many generic extensions?

For a givien partial order, how many generic extensions might exist? In other words, for a boolean valued model class which dreams of a generic extension, how many unique generic objects exist for a ...

**4**

votes

**3**answers

519 views

### On the independence of the Kurepa Hypothesis

Kurepa Hypothesis says there is a Kurepa tree, which is a $\omega_1$-tree has at least $\omega_2$ many branches.
It is known that beginning from a model with an inaccessible cardinal $\kappa$, after ...

**5**

votes

**1**answer

330 views

### Adding large sets by countable conditions preserving the GCH

Let $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties:
1-$P$ preserves GCH and the strong inaccessibility of $\kappa$,
2-$P$ adds a subset of ...

**7**

votes

**2**answers

370 views

### Destroying the P-filter-property

It is known that if a forcing notion is proper, then every P-filter will generate a P-filter in the generic extension (see, e.g., Shelah, Proper and Improper Forcing, VI.5)
On the other hand, if we ...

**3**

votes

**1**answer

242 views

### $< \aleph_1-$support Product of Cohen forcings

Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$
1- Does this forcing preserve ...

**14**

votes

**2**answers

722 views

### Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails ...

**7**

votes

**2**answers

751 views

### Size of stationary sets

What can we say about the size of stationary subsets of $P_{\kappa}(\lambda)$ for infinite cardinals $\kappa, \lambda,$ especially when $\kappa=\aleph_1.$
Please give me some references, if there are ...

**0**

votes

**1**answer

289 views

### some arguments concerning forcing over V

In some papers, the author wants to show 1 forces (A implies B), i.e., for every generic G,
(A implies B) holds in V[G], as follows: Suppose 1 forces A, then let G be a generic filter over V and show ...

**16**

votes

**2**answers

575 views

### How “much” does (Grigorieff) forcing destroy an ultrafilter?

Introduction. I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction ...

**7**

votes

**1**answer

279 views

### Tree Version of Hechler Forcing

In their recent (2009) paper Eventually Different Functions and Inaccessible Cardinals, Brendle and Löwe consider a 'tree version' of the Hechler forcing. This forcing $\mathbb{D}$ consists of ...

**5**

votes

**1**answer

302 views

### Does a generic normal measure extend the club filter?

This question is related to this one. The setup is as follows:
In $V$, $\kappa$ is supercompact and $\mathbb{P} = \mathrm{Coll}(\kappa, \aleph_2)$ is the Levy collapse. $G$ is $(V,P)$-generic, and ...

**2**

votes

**1**answer

243 views

### Gluing functions together in the generic extension

I just read a quite sketchy proof, where some details were skipped and it appears to me that the proof uses some of the following things. However a (probably very easy) question came up:
Assume that ...

**4**

votes

**1**answer

401 views

### Generic filter over $V$

I re-read Jechs chapter about forcing, and got a question. There he characterizes a (what he calls) modern way to make the forcing argument legitimate which (I think) goes like this:
It is pointed ...

**7**

votes

**1**answer

321 views

### Restricted Versions of Hechler Forcing

Given $\mathcal{F}\subseteq\omega^\omega$, let $\mathbb{D}(\mathcal{F})$ denote the
forcing which is much like Hechler forcing, but now in the conditions $\langle
s,f\rangle$ we require that $f$ ...

**8**

votes

**2**answers

502 views

### Statements forced by one condition of a poset, but not the whole thing

In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...

**7**

votes

**3**answers

855 views

### Kunen's use of Countable Transitive Models

Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...

**7**

votes

**1**answer

654 views

### Probabilities independent of ZFC?

Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...

**5**

votes

**1**answer

843 views

### What is random real forcing?

Hi all,
Can someone please explain the idea and the main steps in a random real forcing?
- what makes it (the new real) different from adding a Cohen real?
- is there a good reference for it?
...

**13**

votes

**1**answer

648 views

### Impact of discrepancy between Kunen's and Jech's definition of iterated forcing on full support iterations

One of the first things we usually learn when we study iterated forcing is that we can force over a model of ZFC + GCH to make the continuum function ($\lambda \mapsto 2^{\lambda}$) restricted to some ...

**6**

votes

**1**answer

599 views

### Iterated forcing and CH

I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert ...

**17**

votes

**5**answers

1k views

### What is the generic poset used in forcing, really?

I'm not a set theorist, but I understand the 'pop' version of set-theoretic forcing: in analogy with algebra, we can take a model of a set theory, and an 'indeterminate' (which is some poset), and add ...