Questions tagged [forcing]
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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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, ...
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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 ...
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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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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 ...
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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 ...
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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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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 $f$ ...
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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 ...
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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?
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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?
Thanks....
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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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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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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) ...
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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 $\mathbb{P}\in N$ ...
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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 ...
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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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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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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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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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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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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 order ...
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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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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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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 ...
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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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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 $...
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When forcing with a poset, why do we order the poset in the order that we do?
In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfectly fine, but it ...
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