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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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 ...
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$\mathfrak{c}$-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 $(\omega^\omega, \le^\ast)$
...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 $(...
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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 $\kappa^+$ ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
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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 ...
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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 $\{ \...
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Random real forcing
Let $\kappa$ be an infinite cardinal and let $B$ be the random forcing for adding $\kappa$-many random reals.
Question: What are the elements of $B$. More precisely given a condition $p \in B$, what ...
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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, $\...
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A model of CH +$\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?
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"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 ...
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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 ...
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Can we collapse $\omega_1$ to $\omega$ without adding a dominating real?
(Disclaimer: This question was also asked at MSE (https://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,...
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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 ...
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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 ...
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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 ...
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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 $j:V_{\lambda+1}...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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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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 ...
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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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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 $\...
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Destroying the P-filter-property
It is known that if a forcing notion is proper, then every P-filter will generate a P-filter in the generic extension (see, e.g., Shelah, Proper and Improper Forcing, VI.5)
On the other hand, if we ...
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$< \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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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 everywhere,...
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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 ...
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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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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 ...
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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 ...
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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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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 $f$ ...