**5**

votes

**1**answer

247 views

### How to find a sub-forcing?

Suppose that $M$ is a model of ZFC, $P\in M$ is a notion of forcing and $G$ is a $P$-generic filter over $M$.
It is a well-known theorem that if $M\subseteq N\subseteq M[G]$, and $N$ is a model of ...

**6**

votes

**2**answers

380 views

### From the product lemma to to a result about powersets

Recall the product lemma from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one factor $\lambda^+$-cc ...

**3**

votes

**1**answer

264 views

### Can we weaken GCH in this class forcing?

I've just stumbled across the following theorem (here):
Theorem Let $P$ be a partial class order in $M$, a transitive model of ZF (resp. ZFC). Suppose for arbitrarily large cardinals $\kappa$ we have ...

**2**

votes

**0**answers

214 views

### Can we prove the completeness of FOL based on forcing?

I asked this question at http://math.stackexchange.com but I didn't get any answer there.
In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a ...

**12**

votes

**1**answer

605 views

### I'll admit it: I don't understand the definition of the Easton product.

I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the ...

**5**

votes

**2**answers

357 views

### Suslin algebras

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...

**6**

votes

**3**answers

363 views

### Why does the Solovay-Tennenbaum theorem work?

I have decided to learn iterations at long last, and I am reading through Jech's Set Theory for now. The standard example after explaining what is an iteration with finite support is the following ...

**5**

votes

**2**answers

287 views

### Forcing with product vs. box product

If we want to add one real number the simplest way to do it is to use Cohen forcing. The poset is $\lbrace p\colon n\to 2\mid n\in\omega\rbrace$ which is a countable set. We can think of this as ...

**7**

votes

**1**answer

249 views

### Which of these relations on partial orders allows us to identify forcing equivalence?

Background
This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing.
In his question, Justin considers a relation $\lhd$ on partial orders (defined ...

**5**

votes

**1**answer

229 views

### Infinite products of forcings

Suppose $M \subseteq N$ are models of ZFC such that $(ORD^\omega)^M = (ORD^\omega)^N$. Let $\langle P_n : n \in \omega \rangle$ be a sequence of countably closed partial orders in $M$, and let ...

**7**

votes

**1**answer

233 views

### Properness of quotient forcing

It is well known that if $P$ is a proper notion of forcing and $\Vdash_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. Is the converse also true, i.e
Suppose that we have ...

**5**

votes

**1**answer

190 views

### On intermediate transitive models for ZFC between M an M[G]

Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). Let $G$ be $B(P)$-generic over $M$, the ...

**5**

votes

**1**answer

342 views

### What goes wrong in Easton forcing if we don't just use regular cardinals?

Recall that Easton forcing was introduced to show that the continuum function at regular cardinals could be anything subject to 'the obvious constraints' (monotonicity etc). However, it is a handy ...

**5**

votes

**2**answers

936 views

### Can Assumptions about forcing produce Mice? [closed]

This is going to take some build up to completely describe what is a very strange question I seem to have walked into by accident:
For every partial order $\mathbb{P}$ and regular cardinal $\lambda ...

**5**

votes

**3**answers

422 views

### Proper class forcing vs forcing with a set of conditions bigger than one's model

This seems like a natural question to ask, but I've not seen it discussed in my reading around (limited to Easton's paper, the third edition of Jech's Set theory and a small handful of articles). What ...

**11**

votes

**1**answer

249 views

### Which forcings preserve (some) determinacy?

The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...

**11**

votes

**1**answer

417 views

### Generic Extensions and $L(V_{\lambda+1})$

Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting
$$L_0(A)=A;$$
$$L_{\alpha+1}(A) = L_\alpha (A)\cup ...

**8**

votes

**2**answers

284 views

### large ccc forcing that preserves CH

Can you name a ccc forcing with the following properties?
1) Atomless and separative
2) The least size of a dense set is large, say at least $\aleph_3$, hopefully as big as you like.
3) Existence ...

**6**

votes

**1**answer

194 views

### Characterization of $\kappa$-closed forcings

For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like. For a partial order $\mathbb{P}$, the ...

**6**

votes

**1**answer

447 views

### Reals added after Cohen forcing II

This is related to my previous question "Reals added after Cohen forcing" Reals added after Cohen forcing
The fact is that what I had in mind by $\lambda-$many Cohen reals was different from what was ...

**6**

votes

**2**answers

346 views

### Solovay's paper from AD+ that all sets are Ramsey

For my research I've been trying to locate Solovay's proof that AD+ implies that all sets are Ramsey (or completely Ramsey, depending on the terminology). I know that it uses Mathias forcing but I ...

**5**

votes

**1**answer

348 views

### Forcing in Ackermann's Set Theory

How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 ...

**4**

votes

**2**answers

221 views

### Equivalence of forcing automorphisms

Suppose that $P$ is a forcing poset in $V$. If $\pi$ is an automorphism of $P$ then $\pi$ extends to automorphisms of the names by induction: $$\pi\dot x = \lbrace(\pi p,\pi\dot y)\mid (p,\dot ...

**8**

votes

**1**answer

370 views

### Can $\omega_1$ be supercompact?

Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"?
In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where ...

**5**

votes

**0**answers

324 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 ...

**5**

votes

**2**answers

272 views

### Measures that are not OD

Is anything known about the consistency strength of the statement:
"There is a normal measure (on a cardinal) that is not ordinal-definable"?
In particular, is it consistent relative to the ...

**4**

votes

**3**answers

345 views

### Is every set class generic over a given inner model?

In a paper by B. Mitchell, I stumbled into the following sentence:
"In the summer of 1986 Woodin discovered the second of the forcing orders
associated with a Woodin cardinal, the extender algebra. ...

**3**

votes

**2**answers

343 views

### The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein:
start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...

**13**

votes

**1**answer

646 views

### Forcing over set theory versus forcing over arithmetic

I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...

**42**

votes

**3**answers

2k views

### Forcing as a new chapter of Galois Theory

There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...

**2**

votes

**1**answer

140 views

### $(<\kappa)$-closure for Prikry forcing

A p.o. $\mathcal{P}$ is $(<\kappa)$-closure, if for every decreasing sequence of $<\kappa$ conditions in the forcing $p_0 \geq p_1 \geq \cdots$,
there is a condition that is below all of them. ...

**10**

votes

**1**answer

584 views

### Reals added after Cohen forcing

Let $V_1$ be a generic extension of $V\models GCH$ obtained by adding $\aleph_{\omega}-$many Cohen reals. Then we have the following:
1- In $V_1$ there are $\aleph_{\omega+1}-$many reals,
2- In ...

**5**

votes

**0**answers

524 views

### Mathias forcing with Ramsey ultrafilters, and Cohen reals [closed]

Edit/update: One reason this question never received an answer is because it was founded on a faulty premise! The Blaszczyk-Shelah paper I mentioned below does not prove that Mathias forcing with a ...

**4**

votes

**1**answer

272 views

### a result about Laver property

recently, I am reading Tomek Bartoszynski's book"Set Theory On The Structure Of The Real Line"
There is a lemma I don't understand it's proof.(P244 Lamma 4.6.10),it's original expression as follows.
...

**7**

votes

**2**answers

462 views

### Difference between Laver's and Mathias's forcing

Mathias forcing consists of conditions of the form $(s,A)$ where $s$ is a finite subset of $\omega$, $A$ is an infinite subset of $\omega$ such that $\max(s) < \min(A)$ and $(s,A)\leq (t,B)$ iff ...

**9**

votes

**1**answer

231 views

### Intermediate extension of a Prikry-Silver extension?

Prikry-Silver forcing $\mathbb{V}$ (sometimes just Silver forcing) is the forcing notion consisting of all partial functions $p:\omega\rightarrow 2$ with co-infinite domain. In "Combinatorics on ...

**3**

votes

**1**answer

212 views

### Cohen forcing: why is the cardinality of $\mathcal P \left({\omega}\right)$ in ${\bf L}(\mathcal P \left({\omega}\right))$ independent of G?

This is a part of exercise E4 from Chapter VII of Kunen's Set Theory.
The hint (courtesy of A. Miller) goes like this: let ${\bf P} = Fn(I,2)$, $(I \geq \omega_{1})^M$. Let G be ${\bf P}$-generic ...

**4**

votes

**3**answers

439 views

### Which notions of forcing add a cofinal branch to an $\omega_1$-tree?

I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree.
Question 1:
What kinds of forcings add cofinal branches to $\omega_1$-trees? What kinds of forcings cannot?
...

**13**

votes

**1**answer

596 views

### Can there be an almost-special not-fully-special Aronszajn tree?

Question. Can there be an Aronszajn tree $T$, such that no
c.c.c. forcing extension adds a cofinal branch to $T$, but there is an
$\omega_1$-preserving forcing extension adding a cofinal branch to
...

**16**

votes

**2**answers

2k views

### Two versions of “absolutely ccc”

I have recently been slogging my way through Shelah's "Large continuum, oracles". Essentially from the start there has been a question needling me which I cannot seem to answer.
In the paper, ...

**19**

votes

**3**answers

1k views

### In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?

This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:
"My ...

**3**

votes

**1**answer

228 views

### How large can the power set P(N) be made via forcing?

Given the Constructable Universe L, is there a forcing extension L[G] of L in which P(N) is the 'size' (so to speak) of ORD, the proper class of all ordinals? I am interested in forcing extensions of ...

**5**

votes

**3**answers

529 views

### Background for classic forcing

When I learning forcing theory, I am surprised by various foring definition,I want to know the original purpose, why define partial order in such way and some background material which can help me ...

**8**

votes

**2**answers

318 views

### $\mathfrak{c}$-universal linear order

I've been told once or twice that the following holds:
There is a model of $ZFC+MA+\neg CH$ in which there is a $\mathfrak{c}$-universal linear order embedded in $(\omega^\omega, \le^\ast)$
...

**3**

votes

**0**answers

246 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 ...

**9**

votes

**1**answer

468 views

### Can we change the Lebesgue measure by forcing?

Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...

**6**

votes

**5**answers

829 views

### Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC

Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...

**5**

votes

**2**answers

374 views

### Question about prompt names of ordinals

I asked this question first on math SE and was told that it would better fit here. So:
The following concept is due to Shelah and I have some issues with a claim using this notion:
Suppose that ...

**10**

votes

**2**answers

729 views

### The consistency of Martin's Axiom

In learning about the Consistency of Martin's Axiom through Kunen and Jech with help from other set theorists, I have come to a basic question about marrying these proofs:
What is the connection ...

**2**

votes

**0**answers

455 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 ...