Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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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 ...
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3answers
489 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
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3answers
507 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 ...
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1answer
323 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
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2answers
367 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 ...
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1answer
239 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 ...
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2answers
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 ...
7
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2answers
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
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1answer
286 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
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2answers
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 ...
7
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1answer
275 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
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1answer
300 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 ...
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1answer
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
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1answer
395 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
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1answer
317 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
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2answers
489 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
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3answers
821 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
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1answer
602 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
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1answer
803 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
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1answer
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 ...
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5answers
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 ...
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2answers
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 ...
7
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2answers
661 views

failure of $\square(\kappa)$ at an inaccessible $\kappa$

How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where $\square(\kappa)$ is defined as follows: There is a sequence $(C_i:i< \kappa)$ such that: (1) $C_{i+1} = ...
4
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1answer
134 views

Is genericity essential to “things which are forced are true in the extension” or only to its converse?

Genericity is still a little bit mysterious to me, although not as much as it used to be. Here is a rough paraphrase of Theorem 3.5 of Kunen's Set Theory: an Introduction to Independence Proofs ...
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4answers
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The history of Proper Forcing

What were the initial motivations of the use of the proper forcing.?
5
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1answer
350 views

Does “antichain” mean something different in set-forcing than in lattice theory?

On page 3 of Introduction to Lattices and Order, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise incomparable elements: The ordered set P is an ...
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3answers
544 views

Characterizing forcings that don't add any dominating reals

Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ eventually dominates $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) ...
3
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2answers
487 views

A problem about posets similar to Suslin's problem

Suslin's problem is: Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$? The answer is that it's independent of ZFC. The related ...
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2answers
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If Q is a subset of the plane of size less than continuum, then does every closed F in Q extend to a closed connected G in the plane with the same trace on Q? (Or is this independent of ZFC?)

This question arises in connection with this MO question and especially with Sergei Ivanov's wonderful answer, which showed that for any countable set $Q\subset\mathbb{R}^2$ and every closed set ...
11
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3answers
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 $\mathbb{P}\in N$ ...
3
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3answers
396 views

Complexity of the statement 'P is proper'

Assume that $(P,\le)$ is a notion of forcing. There are several ways to define what it means for $P$ being proper and I would like to know: What is the complexity (in terms of the Levy-Hierarchy) of ...
3
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1answer
294 views

Equivalent definitions of $(M,P)$-genericity

Hi! I started to read the chapter 31 in Jechs book about proper forcing. Unfortunately it is written in a rather sketchy way and I do have some issues in proving a lemma about two equivalent ...
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2answers
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The consistency of ZFC + CH gives the ability to travel to a universe which models ZFC + \neg CH? [closed]

Why does forcing seem to be so vacuously true? It seems like you are just reversing the subset containment of the model of ZFC + CH to be the other way in the poset. So, why is this valid? Why are ...
5
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1answer
621 views

Strange question about Hechler

Recently during a lecture, my professor mentioned that forcing over any poset which is countable, separative, and atomless, is essentially the same as forcing over the Cohen poset, that is to say ...
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9answers
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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 ...
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1answer
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An eventually different function adding no Solovay real nor dominating function?

Definitions I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one). A ...
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2answers
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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 ...
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356 views

A sequence of generic filters that does not come from an iteration

Fix a countable transitive model $M$ of ZFC. In my answer to this question I indicated that there are forcing iterations $((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and ...
4
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2answers
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A question about iterated forcing

I'm trying to get a better grasp of iterated forcing, and I ran across the following problem: 0) Let $P_\alpha$ be posets in a c.t.m. $M$, $\alpha<\beta$, and for each $\alpha$ let $G_\alpha$ be ...
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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 ...
4
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1answer
284 views

Preservation of properties under countable-support iterations

In the following question a property (of a forcing notion) is preserved by a CS-iteration if the following implication holds: If Pa has this property (for every ordinal a< d, for d being the ...
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2answers
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confusion about forcing

I learned from Kunen's book, besides forcing over countable transitive model (c.t.m.), there is an another way to do forcing, called the "syntactic method", i.e. forcing over V. Fixed a partial ...
4
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1answer
135 views

Equivalent definitions of M-genericity.

I'm trying to learn about forcing, and have heard that there are several equivalent ways to define genericity. For instance, let M be a transitive model of ZFC containing a poset (P, ≤). Suppose G ...
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3answers
674 views

Tractability of forcing-invariant statements under large cardinals

It is usual to mention theorems of the kind: Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi ...
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2answers
448 views

Collapsing cardinals before the first inaccessible

This is again a question about forcing. Start in $L$, the constructible universe. CH holds. Let $\lambda$ be an inaccessible cardinal, also let $\lambda$ > $\aleph_0$. For each $\alpha < \lambda$, ...
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How can I force the continuum to be weakly compact?

Since the continuum has to be a regular/singular uncountable cardinal, and since by Easton Theorem it can be anything we want it to be, what is the forcing that makes the continuum a weakly compact ...
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8answers
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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 ...
5
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1answer
283 views

The closure of a generic ultrapower

Let $I$ be a normal ideal on $P_{\kappa} (\lambda)$. Let $V$ denote our ground model. Now we force with the $I$-positive sets, and if $G$ is the resulting generic filter, it can be shown that $G$ is a ...
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2answers
390 views

Question about an example of forcing where we adjoin a new set of natural numbers

Forcing is quite new to me and there is a basic example in Jech that I don't understand. Let $P$ be the following notion of forcing: the forcing conditions are 0-1 sequences and $p$ is stronger than ...
5
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2answers
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What are the Martin's Maximum consequences of Namba forcing?

It is known that Namba forcing is stationary-preserving and hence can be used in the setting of Martin's Maximum. Does this result in any striking consequences?