**41**

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 ...

**21**

votes

**1**answer

372 views

### When does the choice of the generic matter?

It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...

**18**

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 ...

**18**

votes

**2**answers

1k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...

**18**

votes

**1**answer

434 views

### Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...

**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 ...

**16**

votes

**2**answers

432 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

**16**

votes

**2**answers

687 views

### Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following:
Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$
In the paper "The ...

**16**

votes

**2**answers

568 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 ...

**15**

votes

**2**answers

2k views

### Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound ...

**15**

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, ...

**14**

votes

**9**answers

2k views

### Examples of ZFC theorems proved via forcing

This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it ...

**14**

votes

**6**answers

1k views

### The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...

**14**

votes

**4**answers

1k views

### A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...

**14**

votes

**2**answers

410 views

### Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...

**14**

votes

**1**answer

435 views

### Is every class that does not add sets necessarily added by forcing?

We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. ...

**14**

votes

**2**answers

665 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 ...

**13**

votes

**8**answers

3k views

### Forcing as a tool to prove theorems

It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to ...

**13**

votes

**1**answer

561 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 ...

**13**

votes

**1**answer

558 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
...

**13**

votes

**1**answer

618 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 ...

**13**

votes

**1**answer

338 views

### Random reals and strongly meager sets

Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see this question).
The notion of strong measure zero sets has its dual concept in the category branch -- ...

**12**

votes

**3**answers

332 views

### Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...

**12**

votes

**3**answers

672 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. ...

**12**

votes

**0**answers

225 views

### capturing small sets in small factors

Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:
...

**11**

votes

**4**answers

856 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, ...

**11**

votes

**3**answers

786 views

### When can we detect forcing?

First, a rather broad question: has there been any work on what, given a model $M$ of set theory, we can say about those models of set theory $N$ and posets $\mathbb{P}$ such that $\mathbb{P}\in N$ ...

**11**

votes

**2**answers

517 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, ...

**11**

votes

**1**answer

560 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 ...

**11**

votes

**1**answer

408 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 ...

**11**

votes

**1**answer

438 views

### Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals?

The question is whether, when you add a Cohen subset to a cardinal
$\kappa$, that cardinal becomes a characteristic of the resulting forcing extension $V[G]$. Or can there be strange instances in ...

**11**

votes

**0**answers

368 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 ...

**10**

votes

**2**answers

865 views

### Forcing over an arbitrary model of ZFC

I’m learning set theory and forcing in the (french) book from Jean-Louis Krivine “Théorie des ensembles”.
Given a countable transitive model $\mathscr{M}$ of ZFC together with a poset $P$, he ...

**10**

votes

**1**answer

1k views

### Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...

**10**

votes

**1**answer

315 views

### V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...

**10**

votes

**2**answers

517 views

### Adding a random real makes the set of ground model reals meager

This is a question about forcing. I have seen the following fact mentioned in multiple places, but have not been able to find a proof: if a random real is added to a transitive model of ZFC, then in ...

**10**

votes

**4**answers

837 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 ...

**10**

votes

**1**answer

379 views

### Forcing mildly over a worldly cardinal.

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a ...

**10**

votes

**2**answers

667 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 ...

**10**

votes

**1**answer

212 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 ...

**10**

votes

**0**answers

311 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 ...

**9**

votes

**2**answers

576 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 ...

**9**

votes

**3**answers

443 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?

**9**

votes

**2**answers

430 views

### Consistency Results Separating Three Cardinal Characteristics Simultaneously

(For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer here; Andreas Blass's handbook article is an excellent reference.)
Problem 2.3 ...

**9**

votes

**1**answer

507 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 ...

**9**

votes

**1**answer

418 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 ...

**9**

votes

**1**answer

185 views

### Class Forcing and Genericity: Predense sets vs Dense classes

In short my question is: why do we use definable classes in the definition of genericity for class forcing, instead of predense sets.
To elaborate, in Sy's book and indeed other sources on the ...

**9**

votes

**1**answer

222 views

### Suslin trees in ccc ideals

A countably complete ideal $I$ on a set $Z$ ideal is c.c.c. when there is no uncountable family of pairwise disjoint $I$-positive subsets of $Z$. If such an ideal exists, then there exists a weakly ...

**9**

votes

**1**answer

204 views

### Can “syntactic forcing” add ordinals?

Kunen, in paragraph VII.9 of his book talks about forcing "via syntactical models", where the we do not use set models of ZFC. Still, the functional $x \mapsto \check x$ can be defined as usual and ...

**9**

votes

**1**answer

461 views

### Resembling the Levy Collapse

Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V_\kappa$ and $\Vdash_{\mathbb{P}} \kappa = \aleph_1$, where $\mathbb{P}$ ...