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 ...
Jialiang He's user avatar
8 votes
2 answers
486 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 $(\omega^\omega, \le^\ast)$ ...
Not Mike's user avatar
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5 votes
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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 ...
Asaf Karagila's user avatar
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12 votes
1 answer
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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 ...
Asaf Karagila's user avatar
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6 votes
6 answers
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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 ...
Tobias Neukom's user avatar
5 votes
2 answers
406 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 $\...
Stefan Hoffelner's user avatar
10 votes
2 answers
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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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2 votes
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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 ...
Thomas Benjamin's user avatar
-1 votes
1 answer
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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 ...
sonicyouth's user avatar
13 votes
2 answers
1k views

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 $(...
Goldstern's user avatar
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7 votes
1 answer
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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 $\kappa^+$ ...
Ioannis Souldatos's user avatar
5 votes
1 answer
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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 ...
David Roberts's user avatar
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6 votes
0 answers
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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 ...
David Roberts's user avatar
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12 votes
1 answer
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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 ...
mikeyus's user avatar
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3 votes
1 answer
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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, ...
Michael Hardy's user avatar
3 votes
1 answer
379 views

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 ...
8 votes
3 answers
726 views

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 $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
Kaveh's user avatar
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5 votes
2 answers
601 views

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 ...
David Roberts's user avatar
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5 votes
1 answer
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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 $\{ \...
Mohammad Golshani's user avatar
5 votes
1 answer
637 views

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 ...
Mohammad Golshani's user avatar
13 votes
4 answers
1k 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, $\...
Justin Palumbo's user avatar
9 votes
3 answers
539 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?
Rumpertumskin79's user avatar
5 votes
3 answers
845 views

"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 ...
sonicyouth's user avatar
2 votes
2 answers
287 views

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 ...
anon's user avatar
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13 votes
2 answers
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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,...
Noah Schweber's user avatar
4 votes
1 answer
425 views

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 ...
Stefan Geschke's user avatar
6 votes
2 answers
868 views

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 ...
Asaf Karagila's user avatar
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7 votes
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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 ...
David Roberts's user avatar
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7 votes
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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}...
Everett Piper's user avatar
17 votes
5 answers
2k 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 ...
Asaf Karagila's user avatar
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6 votes
0 answers
400 views

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 ...
Amit Kumar Gupta's user avatar
7 votes
1 answer
851 views

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 ...
Asaf Karagila's user avatar
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20 votes
3 answers
2k 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. ...
Noah Schweber's user avatar
7 votes
1 answer
673 views

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 ...
matteo viale's user avatar
7 votes
2 answers
882 views

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 ...
kvagk's user avatar
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9 votes
2 answers
781 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 ...
3 votes
3 answers
627 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 ...
user avatar
5 votes
3 answers
860 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 ...
Ant emyy Lee's user avatar
5 votes
1 answer
386 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 $\...
Mohammad Golshani's user avatar
10 votes
2 answers
550 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 ...
Peter Krautzberger's user avatar
5 votes
1 answer
343 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 ...
Mohammad Golshani's user avatar
14 votes
3 answers
1k 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 everywhere,...
Mohammad Golshani's user avatar
8 votes
2 answers
919 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 ...
Mohammad Golshani's user avatar
0 votes
1 answer
353 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 ...
beginner's user avatar
22 votes
2 answers
1k 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 votes
1 answer
424 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 ...
Justin Palumbo's user avatar
5 votes
1 answer
416 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 ...
Amit Kumar Gupta's user avatar
3 votes
1 answer
316 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 ...
Stefan Hoffelner's user avatar
5 votes
1 answer
828 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 ...
user8996's user avatar
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7 votes
1 answer
491 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 $f$ ...
Justin Palumbo's user avatar